numerical control over complex analytic singularities...be diﬃcult to exaggerate the importance of...

Numerical Control

over

Complex Analytic Singularities

by David B. Massey

For my mother, Mary Alice Massey,

and in memory of my grandparents:

Leslie Ellsworth Porter William Walter MasseyMary Frances Porter Bessie Ann Massey

PREFACE

In 1983, I began work on my dissertation, “Families of Hypersurfaces with One-dimensionalSingular Sets”, at Duke University. In that paper, I attempted to describe two numbers whichone could effectively calculate from the defining equation of a hypersurface with a one-dimensionalsingular set – two numbers which control the topology and geometry in a similar fashion to howthe Milnor number controls the topology and geometry for isolated hypersurface singularities.

Since that time, my work has centered around finding numerical data which “control” varioustopological and geometric properties of complex analytic singularities.

In 1987, while at The University of Notre Dame, I defined the Le cycles and the Le numbersfor non-isolated hypersurface singularities. The Le numbers are a generalization of the Milnornumber, and they control the singularities; the constancy of the Le numbers in a family impliesthe constancy of the Milnor fibres in the family, and also implies Thom’s af condition holds. Mywork on Le numbers from 1987 to the present is contained in my recent monograph, Le Cyclesand Hypersurface Singularities.

In 1988, I came to Northeastern University, and was immediately asked by Terry Gaffney howto generalize the Le numbers of a hypersurface to the case of complete intersections. My answer tothis was that I thought the generalization would have two distinct pieces: the first piece should bea method for associating numbers to an arbitrary constructible complex of sheaves on a complexanalytic space – one should recover the Le numbers of a hypersurface by applying this new methodto the complex of vanishing cycles of the defining equation of the hypersurface. The second pieceshould be to decide what complex of sheaves should play the role of the sheaf of vanishing cyclesin the case of a complete intersection.

This first piece – finding a method for associating numbers to a constructible complex of sheaves– is described in my 1994 paper, “Numerical Invariants of Perverse Sheaves”. As the title indicates,a number of results for Le numbers only generalize nicely in the case where the underlying complexis actually a perverse sheaf.

After this first piece was completed, it became apparent that the second piece of the generaliza-tion was not to replace the sheaf of vanishing cycles by some other complex. Rather, one shouldcontinue to use the vanishing cycles a function, but the function could now have an arbitrarilysingular domain. This changed the problem to one of finding a sufficiently algebraic characteriza-tion of the vanishing cycles – one that actually allows one to effectively produce the numbers thatshould control the singularities.

The first part of such an algebraic description of the vanishing cycles appears in my paper“Hypercohomology of Milnor Fibres”, and the final piece appears in the paper “Critical Points ofFunctions on Singular Spaces”.

Having completed all of the above pieces, I thought it would be a relatively simple matter tomerge them into one coherent whole; thus, I began writing a second book during the 1995-1996academic year. Little did I suspect that the details, refinements, and corrections would take solong. Even now, I have only just realized that the correct topological treatment should use the

v

vi

micro-local theory of Kashiwara and Schapira. Hence, I could be delayed longer by writing anappendix on micro-local theory, and by rewriting all of my Morse theory proofs in terms of themicro-local theory. As the micro-local theory would add another level of complexity to an alreadycomplicated work, I have decided to leave the topological part of this work for a later volume.

Thus, what appears in this book is the general algebraic machinery (Vogel cycles and gapcycles), a mildly rewritten version of Le Cycles and Hypersurface Singularities in terms ofthis general set-up, the generalization of the Milnor number to functions on arbitrarily singularspaces, the generalization of the Le numbers and cycles to functions on arbitrarily singular spaces,and new generalized Le-Iomdine formulas and results on Thom’s af condition. Moreover, thecorrect treatment of these topics requires the inclusion of appendices on intersection theory andthe derived category.

One reason that I feel obligated to include the appendix on intersection theory is to correct thestupidest sentence that I have ever had published. In Le Cycles and Hypersurface Singular-ities, I attempted to give a quick summary of the needed intersection theory; I wrote “If we havetwo irreducible subschemes V and W in an open subset U of some affine space, V and W are saidto intersect properly in U provided that codim V ∩W = codim V +codim W ; when this is the case,the intersection product of [V ] and [W ] is defined by [V ] · [W ] = [V ∩ W ].” This statement is, ingeneral, quite false. Moreover, I knew that it was false, and certainly never used it anywhere inthe book; I have no idea how I wrote such a ridiculous thing. Hopefully, the included appendixwill eliminate any confusion that I may have caused.

Many people have contributed to the results which appear here. But, since I have thanked themin the individual works listed above, I will not give this extensive list here. However, it wouldbe difficult to exaggerate the importance of Terry Gaffney’s and Le Dung Trang’s contributionsto this book. Not only have they helped me with or given me many results, but their continuingenthusiasm for my work is an incredible motivating force.

I should also thank two former graduate students at Northeastern University, Mike Green andRobert Gassler; conversations with them contributed greatly to my own understanding. Finally, Iwant to thank some friends who have helped me greatly in ways not directly related to mathematics:General and Mrs. Hannon, Jenn Hannon, John Hannon, and Jed Hannon (the entire Hannonfamily), Mike Roberts, Tim Roberts, and especially Chad Brazee.

David B. MasseyBoston, MAApril 26, 2000

TABLE OF CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I. Algebraic Preliminaries: Gap Sheaves and Vogel Cycles

Chapter 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 1. Gap Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Chapter 2. Gap Cycles and Vogel Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 3. The Le-Iomdine-Vogel Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Chapter 4. Summary of Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Part II. Le Cycles and Hypersurface Singularities

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Chapter 1. Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 2. Elementary Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Chapter 3. A Handle Decomposition of the Milnor Fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 4. Generalized Le-Iomdine Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Chapter 5. Le Numbers and Hyperplane Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Chapter 6. Thom’s af Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 7. Aligned Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Chapter 8. Suspending Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 9. Constancy of the Milnor Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

vii

Chapter 10. Another Characterization of the Le Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Part III. Isolated Critical Points of Functions on Singular Spaces

Chapter 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 1. Critical Avatars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 2. The Relative Polar Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Chapter 3. The Link between the Algebraic and Topological Points of View . . . . . . . . . . . 128

Chapter 4. The Special Case of Perverse Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Chapter 5. Thom’s af Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Chapter 6. Continuous Families of Constructible Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Part IV. Non-Isolated Critical Points of Functions on Singular Spaces

Chapter 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Chapter 1. Le-Vogel Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Chapter 2. Le-Iomdine Formulas and Thom’s Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Chapter 3. Le-Vogel Cycles and the Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Appendix A. Analytic Cycles and Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Appendix B. The Derived Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Appendix C. Privileged Neighborhoods and Lifting Milnor Fibrations . . . . . . 221

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

viii

ABSTRACT

The Milnor number is a powerful invariant of an isolated, complex, affine hypersurface singu-larity. It provides data about the local, ambient, topological-type of the hypersurface, and theconstancy of the Milnor number throughout a family implies that Thom’s af condition holds andthat the local, ambient, topological-type is constant in the family. Much of the usefulness of theMilnor number is due to the fact that it can be effectively calculated in an algebraic manner.

The Le cycles and numbers are a generalization of the Milnor number to the setting of complex,affine hypersurface singularities, where the singular set is allowed to be of arbitrary dimension. Aswith the Milnor number, the Le numbers provide data about the local, ambient, topological-typeof the hypersurface, and the constancy of the Le numbers throughout a family implies that Thom’saf condition holds and that the Milnor fibrations are constant throughout the family. Again, muchof the usefulness of the Le numbers is due to the fact that they can be effectively calculated in analgebraic manner.

In this work, we generalize the Le cycles and numbers to the case of hypersurfaces inside arbitraryanalytic spaces. We define the Le-Vogel cycles and numbers, and prove that the Le-Vogel numberscontrol Thom’s af condition. We also prove a relationship between the Euler characteristic of theMilnor fibre and the Le-Vogel numbers. Moreover, we give examples which show that the Le-Vogelnumbers are effectively calculable.

In order to define the Le-Vogel cycles and numbers, we require, and include, a great deal ofbackground material on Vogel cycles, analytic intersection theory, and the derived category. Also,to serve as a model case for the Le-Vogel cycles, we recall our earlier work on the Le cycles of anaffine hypersurface singularity.

1991 Mathematics Subject Classification. 32B15, 32C35, 32C18, 32B10

Key words and phrases. Gap sheaf, Vogel cycle, Milnor fibre and number, Le cycles and numbers,vanishing cycles, perverse sheaves, Thom’s af condition, Le-Vogel cycles and numbers

ix

OVERVIEW

The Milnor number of an affine complex analytic hypersurface with an isolated singularityhas been a ridiculously successful invariant: it can be effectively calculated, it determines thehomotopy-type of the Milnor fibre, and its constancy in a family controls much of the geometryand topology of the family.

It is no wonder that there have been myriad attempts to generalize the Milnor number tothe cases where the singularity is non-isolated or where the underlying space is arbitrary. Fromthe topological side, one might suspect that the Betti numbers or the Euler characteristic of theMilnor fibre might be reasonable substitutes for the Milnor number. From a differential geometrypoint-of-view, one can consider various notions of indices of vector fields. From the algebraic side,there are sheaf-theoretic generalizations of the Milnor number. A nice, but by no means complete,expository discussion of the Milnor number and its generalizations is contained in [Te1].

Our own work on generalizing Milnor numbers began with the Le varieties, Le cycles, and Lenumbers of a non-isolated affine hypersurface singularity; this work appeared in [Mas6], [Mas8],[Mas9], and [Mas14]. If U is an open subset of Cn+1, f : U → C is an analytic function, andz := (z0, . . . , zn) is a linear choice of coordinates for Cn+1, then the Le numbers, λ0

f,z, λ1f,z, . . . , λ

nf,z,

have a number of very desirable properties.Let s denote the dimension of the critical locus of f at a point p ∈ f−1(0). Then, the Le numbers,

λif,z, are zero for i > s, and if s = 0, then λ0

f,z is precisely the Milnor number. More generally, allof the Le numbers are effectively calculable, and the Milnor fibre has a handle decomposition inwhich the number of handles attached of a given index is given by the corresponding Le number.The constancy of the Le numbers in a family implies that Thom’s af holds for the total space (theunion of the members of the family) and that the Milnor fibrations are constant in the family. Allof these properties, and more, are proved in Part II of this book.

There is only one question which is addressed by the results of this book: how does onegeneralize the Le numbers of an analytic function to the setting where the underlyingspace is no longer affine, but, rather, is an arbitrarily singular analytic space?

Obviously, to answer this, we need to consider how the Le numbers are defined. The Le numbersare intersection numbers of the Le cycles with affine linear subspaces defined by the coordinatechoice z. Hence, if we could generalize the Le cycles, Λi

f,z, then we would know how to generalizethe Le numbers.

The Le cycles are defined by looking at the relative polar varieties ([L-T2], [Te3], and [Te4])of f , with the correct cycle structure, and using them to give a “decomposition” of the Jacobianideal of f into a collection of cycles. In a more general setting, this decomposition has been studiedby Vogel [Vo] and van Gastel [Gas1], [Gas2]. The first problem, when the underlying space isarbitrary, is that there are three competing definitions for the polar cycles – all of which involvegap sheaves, a notion first studied in [Si-Tr].

Part I of this book discusses these three competing definitions in the general context of decom-posing any ideal, not necessarily one related to the Jacobian of a function. In Part I, we provethat the Le-Iomdine formulas, which are such an important result on Le numbers, actually hold ina much more general setting.

In Part II, we give all of our previously known results on Le numbers, but give the proofs in1

2 DAVID B. MASSEY

terms of our work in Part I.In Part III, we deal with the second major problem when the ambient space, X, is singular: how

does one define an analog of the Milnor number for an isolated critical point? For that matter, whatdoes an “isolated critical point” even mean in this setting? As we shall see, the derived categoryand the vanishing cycles of the constant sheaf along f will be unavoidable tools for answering thesequestions.

It turns out that if the constant sheaf C•X is perverse (up to a shift), then the whole theory

becomes much easier; for instance, this would be the case if X was a local complete intersection.However, for arbitrary spaces, the fact that the complex links of strata can have non-trivial coho-mology in more then one degree leads us to take perverse cohomology of the constant sheaf withall possible shifts. In a sense, perverse cohomology lets us decompose the relevant topological dataabout X into a collection of “positive” and “negative” pieces, and then when we work with thesepieces, data from the various strata cannot cancel each other, because they all have the same sign.

A result that appears at the end of Part III is a generalization of the result of Le and Saito thatthe constancy of the Milnor number in a family implies Thom’s af condition [Le-Sa]; of course,in our theorem, the underlying space is arbitrary.

With Parts I and III out of the way, and using Part II as a guide, it is relatively trivial to defineour generalization of the Le cycles and numbers in Part IV. We refer to these new gadgets as theLe-Vogel cycles, or LeVo cycles for short. There are collections of LeVo cycles for various shifts– these shifts correspond to degrees in which the complex links of strata of X have non-trivialcohomology. The indexing on the shifts is set-up in such a way that local complete intersectionshave non-zero LeVo cycles only when the shift is zero.

In Part IV, we prove an incredibly general Le-Saito type result and prove a result relating theLeVo numbers to the Euler characteristic of the Milnor fibre.

Appendices A and B contain background information (without proofs) on intersection theoryand the derived category. The intersection theory that we need is very simple – we need onlyproper intersections of analytic cycles in affine space; this is what is described in Appendix A.Appendix B contains more information than we really need; it is sort of a working mathematiciansguide to the derived category, perverse sheaves, and vanishing cycles.

Appendix C contains some extremely technical arguments which are needed in Part II, where weprove that constancy of the Le numbers in a family implies the constancy of the Milnor fibrations.We believe that these arguments would only serve to obstruct the exposition in Part II; hence, wehave relegated them to an appendix.

Finally, a word on what is not contained in this book. One will not find most of the ex-tremely topological results on Le numbers extended to the general setting of the LeVo numbers.This includes results on handle-decompositions, Morse inequalities, and the constancy of the LeVonumbers implying constancy of the Milnor fibrations. While we certainly believe that we can gen-eralize most of these results, the correct treatment appears to require the micro-local of Kashiwaraand Schapira [K-S1], [K-S2]. As this would add significant length and complexity to an alreadylong and difficult work, we have elected to place such results in a future book.

Part I. ALGEBRAIC PRELIMINARIES:

GAP SHEAVES AND VOGEL CYCLES

Chapter 0. INTRODUCTION

Throughout this book, our primary algebraic tool consists of a method for taking a coherent sheafof ideals and decomposing it into pure-dimensional “pieces”. Actually, we begin with an orderedset of generators for the ideal, and produce a collection of pure-dimensional analytic cycles, theVogel cycles, which seem to contain a great deal of “geometric” data related to the original ideal.Part I of this book contains the construction of the Vogel cycles; it is, regrettably, very technical innature. The Vogel cycles are defined using gap sheaves, together with the associated analytic cycleswhich they define, the gap cycles. A gap sheaf is a formal device which gives a scheme-theoreticmeaning to the analytic closure of the difference of an initial scheme and an analytic set.

If the underlying space is not Cohen-Macaulay, the main technical problem is that there are,at least, three different reasonable definitions of the gap sheaves and cycles; we select as “the”definition the one that works most nicely in inductive proofs. We show, however, that if one re-chooses the functions defining the ideal in a suitably “generic” way, then all competing definitionsfor the gap cycles and Vogel cycles agree.

In Chapter 3 of this part, we prove some extremely general Le-Iomdine-Vogel formulas; as weshall see in later chapters, these formulas are an amazingly effective tool for transforming problemsabout a given singularity into problems involving a singularity of smaller dimension.

The reader who wishes to bypass this technical portion of the book can jump to the Summaryof Part I, which begins on page 31.

3

4 DAVID B. MASSEY

Chapter 1. GAP SHEAVES

Let W be analytic subset of an analytic space X and let α be a coherent sheaf of ideals in OX

.At each point x of V (α), we wish to consider scheme-theoretically those components of V (α) whichare not contained in |W |. This leads one to the notion of a gap sheaf. Our primary references forgap sheaves are [Si-Tr] and [Fi].

Let β be a second coherent sheaf of ideals in OX

. We write αx for the stalk of α in OX,x .

Definition 1.1. Let S be the multiplicatively closed set OX,x −

⋃p where the union is over all

p ∈ Ass(OX,x/αx) with |V (p)| * |W |. Then, we define αx¬W to equal S−1αx ∩ OX,x . Thus,

αx¬W is the ideal in OX,x consisting of the intersection of those (possibly embedded) primary

ideals, q, associated to αx such that |V (q)| * |W |.Now, we have defined αx¬W in each stalk. By [Si-Tr], if we perform this operation simultane-

ously at all points of V (α), then we obtain a coherent sheaf of ideals called a gap sheaf ; we writethis sheaf as α¬W . If β is any coherent sheaf of ideals such that W = supp(OX/β), then

α¬W =∞⋃

k=0

(α : βk).

If V = V (α), we let V ¬W denote the scheme V (α¬W ). It is important to note that the schemeV ¬W does not depend on the structure of W as a scheme, but only as an analytic set. The schemeV ¬W is sometimes referred to as the analytic closure of V − W [Fi, p.41]; this is certainly thecorrect, intuitive way to think of V ¬W .

We find it convenient to extend this gap sheaf notation to the case of analytic sets (reducedschemes) and analytic cycles.

Hence, if Z and W are analytic sets, then we let Z¬W denote the union of the components ofZ which are not contained in W ; if C =

∑mi[Vi] is an analytic cycle in a complex manifold M

and W is an analytic subset of M , then we define C¬W by

C¬W =∑

Vi 6⊆W

mi[Vi].

If α is a coherent sheaf of ideals in OM , C is a cycle in M , and W is an analytic subset of M , thenclearly [V (α)¬W ] = [V (α)]¬W and |C¬W | = |C|¬W .

The following properties of gap sheaves are immediate from the definition.

Proposition 1.2.

i) αx = βx for all x ∈ X −W if and only if α¬W = β¬W ;

ii) if αi is a finite collection of coherent ideals in OX , then ∩(αi¬W ) = (∩αi)¬W ;

iii) if V (α) ∩ (X −W ) is reduced, then the sheaf of ideals of functions vanishing on the analyticset V (α) ∩ (X −W ) is α¬W .

PART I. ALGEBRAIC PRELIMINARIES 5

Later, the reader may wonder why we do not define something analogous to a gap sheaf, butwhere we keep those components which are contained in a given analytic set, W , instead ofthrowing them away.

On the level of schemes, we can not make this approach work; the primary ideals in a primarydecomposition (of a given ideal) which define varieties contained in W would not be independent ofthe decomposition. We could just take the isolated primary ideals which define varieties containedin W , but this disposes of too much algebraic structure. Similarly, we could consider V ¬(V ¬W ),which would not dispose of all embedded components, but would eliminate embedded componentscontained in both W and V ¬W .

However, even this device would not aid us much later; as we shall see – beginning with Definition2.14 – we need to deal more with the intersection product on analytic cycles, and not so much withprimary decompositions.

The following lemma is very useful for calculating V ¬W .

Lemma 1.3. Let (X,OX

) be an analytic space, let α, β, and γ be coherent sheaves of ideals inO

X, let f, g ∈ O

X, and let W , Y , and Z be analytic subsets of X such that Z ⊆ W . Then,

i) α¬W = (α¬Z)¬W , and thus, as schemes, V (α)¬W = (V (α)¬Z)¬W ;

ii) (α + β)¬W = (α¬Z + β)¬W , and thus, as schemes,(V (α) ∩ V (β)

)¬W =

(V (α¬Z) ∩ V (β)

)¬W ;

iii) if V (α + γ) ⊆ W , then((α ∩ β) + γ

)¬W = (β + γ)¬W , and thus, as schemes,((

V (α) ∪ V (β))∩ V (γ)

)¬W =

(V (β) ∩ V (γ)

)¬W ;

iv) if V (α+ < g >) ⊆ W , then (α+ < fg >)¬W = (α+ < f >)¬W , and thus, as schemes,(V (α) ∩ V (fg)

)¬W =

(V (α) ∩ V (f)

)¬W.

v) α¬(W ∪ Y ) = (α¬W )¬Y , and thus, as schemes,

V (α)¬(W ∪ Y ) = (V (α)¬W )¬Y.

The analog of ii) for sets and cycles is also trivial to verify; that is,

(|V (α)| ∩ |V (β)|

)¬W =

((|V (α)|¬Z

)∩ |V (β)|

)¬W,

and, if all intersections are proper,

([V (α)] · [V (β)]

)¬W =

(([V (α)]¬Z

)· [V (β)]

)¬W.

6 DAVID B. MASSEY

Proof. Statements i), ii), iii), and iv) are merely exercises in localization (see [Mas3]). Statementv) is trivial.

Remark 1.4. While it is a trivial observation, it is frequently important and useful to note that, forany coherent sheaf of ideals, α, in O

Xand for any f ∈ O

X, V (α)¬V (f) and V (f) intersect properly

and V (f) contains no embedded subvarieties of V (α)¬V (f); thus, the intersection product cycle[V (α)¬V (f)] · [V (f)] in X is well-defined (without having to mention an ambient manifold) and isequal to [V (< α ¬ V (f) > + < f >)].

If V (α) and V (f) intersect properly, then [V (α)] = [V (α)¬V (f)] and, hence,

[V (α)] · [V (f)] = [V (α)¬V (f)] · [V (f)] = [V (< α ¬ V (f) > + < f >)].

Lemma 1.5. Let X be purely d-dimensional and Cohen-Macaulay. Let f1, . . . , fk ∈ OX and letW be an analytic subset of X. If V (f1, . . . , fk) ¬ W is purely (d−k)-dimensional, then it containsno embedded subvarieties.

Proof. By definition, V (f1, . . . , fk) ¬ W can not have any embedded subvarieties contained in W .At points, p, outside of W , f1, . . . , fk determines a regular sequence in the Cohen-Macaulay ringOX,p ; hence, there are no embedded subvarieties outside of W .

Example 1.6. For the remainder of this chapter, we wish to describe the blow-up of a space alongan ideal; the description via gap sheaves is very nice.

Let (X,OX

) be an analytic space, and let f := (f0, . . . , fk) be an ordered (k+1)-tuple of elementsof O

X. Then, the blow-up of X along f consists of an analytic subspace Blf X ⊆ X × Pk, together

with the projection morphism π : Blf X → X, which is the restriction of the standard projectionfrom X × Pk to X. If we use [w0 : · · · : wk] for homogeneous coordinates on Pk, then the blow-upis given as a scheme by

Blf X := V(wifj − wjfi 06i,j6k

)¬

(V (f0, . . . , fk)× Pk

).

In order to describe the exceptional divisor as a cycle, we need to work on affine coordinatepatches in Pk. We shall describe both the blow-up and the exceptional divisor on each affine patchwj 6= 0.

On the patch wj 6= 0, we use coordinates wi := wi/wj for all i 6= j. Then,

(1.7) wj 6= 0 ∩ Blf X = V (fi − wifji6=j) ¬

(V (fj)× Pk

),

and the exceptional divisor, E, is the cycle defined on each affine patch in the following manner

(1.8) wj 6= 0 ∩ E :=[V

(fi − wifji6=j

)¬

(V (fj)× Pk

) ]·[V (fj)× Pk

].

We have made these definitions with respect to a chosen (k + 1)-tuple f . In fact, the analyticisomorphism-type of the morphism π : Blf X → X only depends on the ideal, I, generated bythe components f0, . . . , fk; this isomorphism-type is referred to as the blow-up of X along I. Ofcourse, the isomorphism-type of the exceptional divisor also depends only on the ideal I, and thisisomorphism-type is simply called the the exceptional divisor of the blow-up of X along I.


Chapter 2. GAP CYCLES AND VOGEL CYCLES

Let X be a d-dimensional analytic space and let f := (f0, . . . , fk) be an ordered (k + 1)-tuple ofelements of O

X. We will define a sequence of cycles, the Vogel cycles ([Vo], [Gas1], [Gas2]) of f ;

these cycles provide effectively calculable data about the coherent sheaf of ideals < f0, . . . , fk >.Before we can define the Vogel cycles, we must first define the gap varieties and gap cycles of f .

It will prove useful (in Part IV) to define gap and Vogel objects with respect to a given cycle.Hence, throughout Part I, we let M denote the cycle

∑l ml[Vl] in X; we assume that this is a

minimal presentation of M – that is, we assume that the Vl are distinct, irreducible analytic subsetsof X and that none of the ml equal zero. In addition, to avoid cancellation of contributions fromvarious Vl, we assume that all of the ml have the same sign, i.e., that ±M > 0.

If X is a union of irreducible components Xi, we will define the gap and Vogel cycles in Xas sums of the gap and Vogel cycles from each Xi; similarly, we will define gap and Vogel cycleswith respect to M simply by taking weighted sums of the gap and Vogel cycles of the irreduciblecomponents. The case of an irreducible space X can be recovered from the cycle case by simplytaking M = [X]. Thus, we find that we need to first define the gap varieties, gap cycles, and Vogelcycles in the case where X is irreducible.

However, even if we assume that the underlying space is irreducible, there is a further compli-cation in the general setting: OX may not be Cohen-Macaulay. This causes numerous problems,for we must worry about embedded subvarieties. To deal with this problem, we introduce threeavatars of gap varieties and examine the relations between them.

We will define the (ordinary) gap varieties,Πi

f

i, the modified gap varieties,

Πi

f

i, and the

inductive gap varieties,Πi

f

i. We shall use the inductive gap varieties to define the Vogel cycles,

but need to make assumptions about the (ordinary) gap varieties in order for the definition tomake sense; the modified gap varieties are merely a convenient tool for proving results about Πi

f

and Πif .

Definition 2.1. Assume that X is irreducible (though, not necessarily reduced). For all i, wedefine the gap varieties, the modified gap varieties, and the inductive gap varieties of f , which wedenote by Πi

f , Πif , and Πi

f , respectively.

First, if i < d− (k + 1) or i > d, we set Πif = Πi

f = Πif = ∅.

We define Πdf := X¬V (f), and Πd

f = Πdf := X¬V (fk).

For d− (k + 1) 6 i < d, the i-th gap variety of f , Πif , is defined as

Πif := V (fi+k+1−d, . . . , fk) ¬ V (f).

Note that Πd−(k+1)f = ∅.

For d− (k + 1) < i < d, the i-th modified gap variety of f , Πif , is defined as

Πif := V (fi+k+1−d, . . . , fk) ¬ V (fi+k−d);

we define Πd−(k+1)f := ∅.

8 DAVID B. MASSEY

For d− (k+1) < i < d, the i-th inductive gap variety of f , Πif , is defined by downward induction

(recall that Πdf is defined above)

Πif :=

(Πi+1

f ∩ V (fi+k+1−d))¬ V (fi+k−d);

we define Πd−(k+1)f := ∅.

Naturally, we define the i-th gap cycle, modified i-th gap cycle, and inductive i-th gap cycle of fto be the cycles defined by these schemes, i.e.,

[Πi

f

],[Πi

f

], and

[Πi

f

], respectively.

If X is a union of irreducible components Xj and f ∈ (OX

)k+1, then we define the i-th gapcycle of f by

[Πi

f

]:=

∑j

[Πi

f|Xj

], the i-th modified gap cycle of f by

[Πi

f

]:=

∑j

[Πi

f|Xj

], and the

i-th inductive gap cycle of f by[Πi

f

]:=

∑j

[Πi

f|Xj

].

We define the i-th gap set of f , the i-th modified gap set of f , and the i-th inductive gap set of f tobe

∣∣∣[Πif

]∣∣∣, ∣∣∣[Πif

]∣∣∣, and∣∣∣[Πi

f

]∣∣∣, respectively. We will write simply∣∣Πi

f

∣∣, ∣∣Πif

∣∣, and∣∣Πi

f

∣∣, respectively.

Finally, we need to define gap cycles and sets with respect to the cycle M . We define the i-thgap cycle of f with respect to M by Πi

f (M) :=∑

l ml

[Πi

f|Vl

], the i-th modified gap cycle of f with

respect to M by Πif (M) :=

∑l ml

[Πi

f|Vl

], and the i-th inductive gap cycle of f with respect to M

by Πif (M) :=

∑l ml

[Πi

f|Vl

].

Of course, we define the associated gap sets with respect to M to be the sets underlying thevarious gap cycles.

Note that we have not defined gap varieties for f unless X is irreducible.

The following proposition gives a number of basic results and interrelationships between thevarious gap varieties.

Proposition 2.2. Let X be irreducible. Then,

i) there is an inclusion of sets∣∣Πi

f

∣∣ ⊆ ∣∣Πif

∣∣ ⊆ ∣∣Πif

∣∣, ∣∣Πif

∣∣ is purely i-dimensional, and all componentsof

∣∣Πif

∣∣ and∣∣Πi

f

∣∣ have dimension at least i;

ii) there is an equality of schemes Πif =

(Πi+1

f ∩ V (fi+k+1−d))¬V (f);

iii) if i 6 d, then∣∣Πi−1

f

∣∣ ⊆ ∣∣Πif

∣∣ and∣∣Πi−1

f

∣∣ ⊆ ∣∣Πif

∣∣;iv) the sets V (fi+k+1−d) and

∣∣Πi+1f

∣∣ intersect properly, and there is an equality of cycles[Πi

f

]=

([Πi+1

f

]·[V (fi+k+1−d)

])¬ V (fi+k−d);

v) if there is an equality of sets∣∣Πi

f

∣∣ =∣∣Πi

f

∣∣, then the schemes Πif and Πi

f are equal up to embeddedsubvariety, and so there is an equality of cycles

[Πi

f

]=

[Πi

f

];

vi) if there is an equality of sets∣∣Πj

f

∣∣ =∣∣Πj

f

∣∣ for all j > i + 1, then[Πi

f

]6

[Πi

f

];


vii) if there is an equality of schemes Πi+1f = Πi+1

f , then there is an equality of schemes Πif = Πi

f .

Proof. i) is obvious from the definitions. ii) follows immediately from Lemma 1.3.ii (using V (f) forboth Z and W ). v) is immediate.

Proof of iii): ii) implies∣∣Πi−1

f

∣∣ ⊆ ∣∣Πif

∣∣. That∣∣Πi−1

f

∣∣ ⊆ ∣∣Πif

∣∣ follows from the inductive definition.

Proof of iv): By definition, Πi+1f has no components or embedded subvarieties contained in

V (fi+k+1−d). Thus,[Πi+1

f ∩ V (fi+k+1−d)]

=[Πi+1

f

]·[V (fi+k+1−d)

]. The desired conclusion

follows.

Proof of vi): By downward induction on i. Note first that[Πd

f

]6

[Πd

f

], since they are, in fact,

equal. Suppose now that i < d and that there is an equality of sets∣∣Πj

f

∣∣ =∣∣Πj

f

∣∣ for all j > i + 1.From induction, we know that

[Πi+1

f

]6

[Πi+1

f

]. Thus,[

Πif

]=

[Πi+1

f ∩ V (fi+k+1−d)]¬ V (f) >

[Πi+1

f ∩ V (fi+k+1−d)]¬ V (fi+k−d).

Since iv) tells us that∣∣Πi+1

f

∣∣ intersects V (fi+k+1−d) properly, we may apply [Fu, 8.2.a] toconclude that

[Πi+1

f ∩V (fi+k+1−d)]

>[Πi+1

f

]·[V (fi+k+1−d)

](the presence of embedded varieties

in Πi+1f can cause a strict inequality). Therefore,[

Πif

]>

([Πi+1

f

]·[V (fi+k+1−d)

])¬ V (fi+k−d).

Now, applying our inductive hypothesis and iv), we conclude that[Πi

f

]6

[Πi

f

].

Proof of vii): We have

Πif =

(Πi+1

f ∩ V (fi+k+1−d))¬ V (fi+k−d) =

(Πi+1

f ∩ V (fi+k+1−d))¬ V (fi+k−d).

By 1.3.i, this equals((

Πi+1f ∩ V (fi+k+1−d)

)¬ V (f)

)¬ V (fi+k−d). By ii) of this proposition, this

last expression equals Πif ¬ V (fi+k−d) =

(V (fi+k+1−d, . . . , fk) ¬ V (f)

)¬ V (fi+k−d). Applying

1.3.i again, we find that Πif = V (fi+k+1−d, . . . , fk) ¬ V (fi+k−d) = Πi

f .

Note that 2.2.i implies that∣∣Πi

f

∣∣ and∣∣Πi

f

∣∣ are purely i-dimensional if and only if they arei-dimensional.

We wish to define the Vogel cycles now. However, before we can do this, we need to decidewhich of the different gap cycles to use to define the Vogel cycles. As a preliminary step, we firstdefine the sets which will underlie the Vogel cycles.

Definition 2.3. Assume that X is irreducible. If i 6= d, then we define the i-th Vogel set of f ,Di

f , to be the union of the irreducible components of∣∣Πi+1

f ∩ V (fi+k+1−d)∣∣ which are contained in

V (f); by 2.2.ii, this is equivalent to

Dif =

∣∣Πi+1f ∩ V (fi+k+1−d)

∣∣ −∣∣Πi

f

∣∣.

10 DAVID B. MASSEY

We set Ddf =

∅, if f 6≡ 0X, if f ≡ 0.

Note that, if i < d− (k + 1) or i > d, then Dif = ∅.

If X is a union of irreducible components Xj, we define Dif :=

⋃j Di

f|Xj

.

We define the i-th Vogel set of f with respect to M to be Dif (M) :=

⋃j Di

f|Vl

.

Proposition 2.4. Every component of Dif (M) has dimension at least i and |M | ∩ V (f) =⋃

i Dif (M). If Πi

f (M) is i-dimensional and C is an i-dimensional irreducible component of |M | ∩|V (f)|, then C ⊆ Di

f (M).If X is irreducible of dimension d, then, for all i 6 d− 1,∣∣Πi+1

f ∩ V (f)∣∣ =

⋃k6i

Dkf .

Proof. We may work on each irreducible set, Vl, separately; therefore, we assume that we are inthe case where X is irreducible and M = [X].

That every component of Dif has dimension at least i follows immediately from the fact that

each component of Πi+1f has dimension at least i + 1 (by 2.2.i).

By definition X =∣∣Πd

f

∣∣∪Ddf . Hence, V (f) =

∣∣Πdf ∩V (f)

∣∣∪Ddf and so, the equation V (f) =

⋃i

Dif

follows once we show the final claim of the proposition.

Suppose that i 6 d− 1. Then,∣∣Πi+1f ∩ V (f)

∣∣ =∣∣Πi+1

f ∩ V (fi+k+1−d) ∩ V (f)∣∣ =

∣∣(Πif ∪Di

f

)∩ V (f)

∣∣ =∣∣Πi

f ∩ V (f)∣∣ ∪Di

f .

As Πif is eventually empty, the desired conclusion follows.

Finally, suppose that C is an i-dimensional irreducible component of |V (f)| and Πif is i-dimen-

sional. Then, C is contained in a component C ′ of |V (fi+k+2−d, . . . , fk)|; such a C ′ necessarily hasdimension at least i + 1. Thus, C ′ cannot be contained in V (f). It follows that C ′ is contained inΠi+1

f . Therefore,

C ⊆ C ′ ∩ V (fi+k+1−d) ⊆∣∣Πi+1

f ∩ V (fi+k+1−d)∣∣ =

∣∣Πif

∣∣ ∪Dif .

If Πif is i-dimensional, then – since C ⊆ V (f) and is i-dimensional – it follows that C 6⊆

∣∣Πif

∣∣, andso C ⊆ Di

f .

Below, we prove the Dimensionality Lemma in which we state as hypotheses/conclusions that“∣∣Πi

f (M)∣∣ is i-dimensional” and “Di

f (M) is i-dimensional”. Since sets cannot be negative-dimen-sional, for i < 0, we mean that the respective set is empty. Note that 2.4 implies that Di

f (M) ispurely i-dimensional if and only if it is i-dimensional.


Lemma 2.5 (Dimensionality Lemma). The following are equivalent:

i) for all i,∣∣Πi

f (M)∣∣ is i-dimensional;

ii) for all i,∣∣Πi

f (M)∣∣ =

∣∣Πif (M)

∣∣;iii) for all i,

∣∣Πif (M)

∣∣ =∣∣Πi

f (M)∣∣.

In addition, these equivalent conditions imply

iv) for all i, Dif (M) is i-dimensional;

and, for all p ∈ |M | ∩ V (f), there exists a neighborhood of p in which iv) implies i), ii), and iii).

Proof. Again we may consider each component appearing M separately; hence, we may assumethat X is irreducible and M = [X].

As all the statements are set-theoretic, to cut down on notation, we shall omit the vertical linesaround the various gap sheaves.

We will show that i) and iii) are each equivalent to ii), that i) implies iv), and that, near pointsof V (f), iv) implies i).

i)⇒ ii): Assume i). From the definition of Πif , what we need to show is: if C is a component of

V (fi+k+1−d, . . . , fk), then C is contained in V (f) if and only if C is contained in V (fi+k−d). AsV (f) ⊆ V (fi+k−d), one implication is trivial, and so what we must show is that if C is a componentof V (fi+k+1−d, . . . , fk) and C ⊆ V (fi+k−d), then C ⊆ V (f).

Suppose not. As C is a component of V (fi+k+1−d, . . . , fk), the dimension of C is at least i. IfC 6⊆ V (f), then – by definition – C is a component of Πi

f . But C is also contained in V (fi+k−d),and so C is a component of Πi

f ∩ V (fi+k−d) = Πi−1f ∪ Di−1

f . As C is not contained in V (f), weconclude that C is a component of Πi−1

f of dimension at least i. This contradicts i).

ii)⇒ i): Assume ii). From Definition 2.1, Πif is purely i-dimensional for i > d. Suppose that i0 is

the largest integer i (less than d) such that Πif is not purely i-dimensional. Then, Πi0+1

f is purely(i0 + 1)-dimensional and, by ii), the set Πi0+1

f is equal to V (fi0+k+2−d, . . . , fk) ¬ V (fi0+k+1−d).Hence, the intersection Πi0+1

f ∩ V (fi0+k+1−d) is proper, and so Πi0+1f ∩ V (fi0+k+1−d) is purely

i0-dimensional. As there is an equality of sets Πi0+1f ∩ V (fi0+k+1−d) = Πi0

f ∪Di0f , this contradicts

the fact that Πi0f is not purely i0-dimensional.

iii)⇒ ii): Assume iii). Then ii) follows immediately from the fact that Πif ⊆ Πi

f ⊆ Πif (see 2.2.i).

ii)⇒ iii): Assume ii). The proof is by induction. iii) is certainly true by definition for i > d.Now, suppose that Πi

f = Πif for i > m, where m 6 d. We need to show that Πm−1

f = Πm−1f . We

have

Πm−1f =

(Πm

f ∩ V (fm+k−d))¬ V (fm+k−1−d) =

(Πm

f ∩ V (fm+k−d))¬ V (fm+k−1−d).

12 DAVID B. MASSEY

By combining the definition of Πmf as V (fm+k+1−d, . . . , fk) ¬ V (f) with Lemma 1.3.ii, we conclude

that (Πm

f ∩ V (fm+k−d))¬ V (fm+k−1−d) = V (fm+k−d, . . . , fk) ¬ V (fm+k−1−d)

and so, Πm−1f = Πm−1

f . By ii), this implies that Πm−1f = Πm−1

f and we are finished.

i)⇒ iv): Assume i), and suppose that i0 is such that Di0f is not purely i0-dimensional. Then,

Πi0+1f ∩ V (fi0+k+1−d) is not purely i0-dimensional. As Πi0+1

f is purely (i0 + 1)-dimensional byassumption, it follows that V (fi0+k+1−d) contains a component, C, of Πi0+1

f . As C is a componentof Πi0+1

f , C is not contained in V (f).Thus, C is a component of Πi0+1

f ∩ V (fi0+k+1−d) = Πi0f ∪Di0

f which is not contained in V (f),and so C is an (i0 + 1)-dimensional component of Πi0

f – this contradicts our assumption.

iv)⇒ i): Assume iv), and that we are interested in the germ of the situation at a point p ∈ V (f).Let i0 be the smallest i such that Πi

f is not purely i-dimensional. By Proposition 2.2.i, Πi0f must

have dimension at least i0 + 1. Thus, since p ∈ V (f), Πi0f ∩ V (fi0+k−d) has dimension at least

i0. But, as sets, Πi0f ∩ V (fi0+k−d) = Πi0−1

f ∪Di0−1f , and by assumption Di0−1

f is purely (i0 − 1)-dimensional. Therefore, we conclude that Πi0−1

f has dimension at least i0 – a contradiction of thechoice of i0.

Remark 2.6. Our phrasing of Lemma 2.5 is the most elegant, and is in the form that we will usuallyneed. However, it is occasionally helpful to note that our proof does not require that one knowsi), ii), or iii) for all i. Specifically, what our proof actually shows is that:

• if∣∣Πi−1

f (M)∣∣ is (i− 1)-dimensional, then

∣∣Πif (M)

∣∣ =∣∣Πi

f (M)∣∣;

• if∣∣Πi

f (M)∣∣ =

∣∣Πif (M)

∣∣ for all i > k, then∣∣Πk

f (M)∣∣ is k-dimensional;

• if∣∣Πi

f (M)∣∣ =

∣∣Πif (M)

∣∣, then∣∣Πi

f (M)∣∣ =

∣∣Πif (M)

∣∣;• if

∣∣Πif (M)

∣∣ =∣∣Πi

f (M)∣∣ for all i > m, and

∣∣Πm−1f (M)

∣∣ =∣∣Πm−1

f (M)∣∣, then

∣∣Πm−1f (M)

∣∣ =∣∣Πm−1f (M)

∣∣; in particular, if∣∣Πi

f (M)∣∣ =

∣∣Πif (M)

∣∣ for all i > m− 1, then∣∣Πi

f (M)∣∣ =

∣∣Πif (M)

∣∣for all i > m− 1;

• if∣∣Πi

f (M)∣∣ is i-dimensional and

∣∣Πi+1f (M)

∣∣ is (i + 1)-dimensional, then Dif (M) is

i-dimensional; and

• if p ∈ |M | ∩ V (f),∣∣Πi−1

f (M)∣∣ is (i − 1)-dimensional at p, and Di−1

f (M) is (i − 1)-dimensional at p, then

∣∣Πif (M)

∣∣ is i-dimensional at p.

Definition 2.7. If the equivalent conditions i), ii), and iii) of Lemma 2.5 hold, we say that thegap sets of f with respect to M have the correct dimension.

If the equivalent conditions i), ii), iii), and iv) of Lemma 2.5 hold at a point p ∈ |M | ∩V (f), wesay that the Vogel sets of f with respect to M have the correct dimension at p. We say simply that


the Vogel sets of f with respect to M have the correct dimension provided that they have correctdimension at all points of |M | ∩ V (f).

Remark 2.8. Note that, since every component of Dif (M) has dimension at least i (see 2.4), if the

Vogel sets all have correct dimension at p, then all the Vogel sets have correct dimension at pointsnear p.

Note also that if the gap sets have correct dimension, then the Vogel sets have correct dimension.Moreover, since we are interested only in what happens near V (f), the natural assumption for usto make seems like it should be that the Vogel cycles have correct dimension. However, our usualassumption will be that gap sets have correct dimension; for 2.5 tells us that, in a neighborhoodof V (f), these assumptions are equivalent, and requiring the gap sets to have the correct dimensionsaves us from having to state over and over again that we take a small neighborhood of a point ofV (f).

It is important to remember that one implication of the Vogel and gap sets having correctdimension is that Di

f (M), Πif (M), Πi

f (M), and Πif (M) are all empty if i < 0, and Π0

f (M) =Π0

f (M) = Π0f (M) = ∅ at points of |M | ∩ V (f).

Finally, consider the special case where p is an isolated point of |M | ∩ V (f). Then, 2.4 impliesthat, near p, Di

f (M) = ∅ if i > 1, and D0f (M) = p. Thus, 2.5 implies that the gap sets and the

Vogel sets have correct dimension at p.

Proposition 2.9. If X is irreducible and Cohen-Macaulay, and all of the gap sets of f have correctdimension, then, for all i, the schemes Πi

f , Πif , and Πi

f are equal.

Proof. By 2.2.v, if the gap sets have correct dimension, then the schemes Πif and Πi

f are equalup to embedded subvariety. By Lemma 1.5, Πi

f and Πif have no embedded subvarieties; therefore,

they are equal as schemes.To prove that the scheme structure of Πi

f agrees with the other two, we must, of course, useinduction. Let d denote the dimension of X. For i > d, we know that Πi

f = Πif = Πi

f .Suppose, inductively, that Πi+1

f = Πi+1f = Πi+1

f . Then, 2.2.vii tells us that Πif = Πi

f and, by thefirst paragraph above, we know that this equals Πi

f .

While we have been selecting (k+1)-tuples, f , our primary object of interest is, in fact, the ideal< f > generated by the f0, . . . , fk. As far as the ideal < f > is concerned, the functions comprising fmay not be suitably generic. However, as we shall see, to obtain a well-behaved ordered collectionof generators, one only needs to replace (f0, . . . , fk) by generic linear combinations of the fi’sthemselves. However, the term “generic” here is used in a non-standard way; what we need is toreplace f0 by a generic linear combination, then – fixing this new f0 – replace f1 by a generic linearcombination, and so on. Since “generic” should always mean open and dense in some topology,we will define a new, convenient one.

Definition 2.10. The pseudo-Zariski topology (pZ-topology) on a topological space (X, T ) is anew topological space (X, TpZ) given by U ∈ TpZ if and only if U is empty or is an open, densesubset in (X, T ). (One verifies easily that this, in fact, yields a topology on X.)

Given two topological spaces X and Y , let πX and πY denote the projections from X × Y onto

14 DAVID B. MASSEY

X and Y , respectively. The inductive pseudo-Zariski topology (IPZ-topology) on X × Y is givenby: W ⊆ X×Y is open in the IPZ-topology if and only if πX(W) is open in the pZ-topology on Xand, for all x ∈ πX(W), πY

(W ∩ π−1

X (x))

is open in the pZ-topology on Y . (It is trivial to verifythat this is a topology on X × Y , and that a non-empty open set in the IPZ-topology on X × Yis a dense set in the cross-product topology on X × Y .)

Finally, given a finite number of topological spaces X1, X2, . . . , Xm, the IPZ-topology onX1×X2×· · ·×Xm is given inductively by using the IPZ-topology on each product in the expression((

(X1 ×X2)×X3

)× · · · ×Xm−1

)×Xm.

A generic linear reorganization of a (k + 1)-tuple f is a matrix product fA, where the matrix Ais invertible and is an element of some given generic subset in the IPZ-topology on the (k +1)-foldproduct Ck+1 × · · · × Ck+1 (where we consider each column of A to be contained in one copy ofCk+1).

Note that, if X1 = X2 = · · · = Xm = CN (or PN ), then the IPZ-topology on the product ismore fine than the Zariski topology, but sets which are open in the IPZ-topology need not be openin the classical topology on the product.

Proposition/Definition 2.11. If X is irreducible, then, for all p ∈ X, for a generic linearreorganization, f , of f , the gap sets of f all have correct dimension at p and, for all i, there is anequality of schemes Πi

f= Πi

f= Πi

fin a neighborhood of p.

Therefore, for p ∈ |M |, for a generic linear reorganization, f , of f , the gap sets of f with respectto |M | all have correct dimension at p and, for all i, there is an equality of cycles Πi

f(M) =

Πif(M) = Πi

f(M) at p.

If we are working in the algebraic category, then we may produce such generic linear reorgani-zations globally.

We refer to a reorganization f such that the above equality of cycles holds as an agreeablereorganization of f (with respect to M at p) (for it makes the various cycle structures agree).

Proof. Assume that X is irreducible. We fix a point p ∈ X. Our sole reason for stating the results“at p” is that, at several places in the proof, we will need to know that certain analytic sets havea finite number of analytic components. This is, of course, guaranteed near a given point or in thealgebraic category. Hence, throughout the proof, we will make no further reference to working ina neighborhood of p, but will assume that all of the analytic sets that arise have a finite numberof components.

We first show:

(†) for a generic linear reorganization, f , of f , for all i, V (fi+k−d) contains no component orembedded subvariety of Πi

f.

We produce the (k+1)-tuple f one element at a time, by downward induction. If f is identicallyzero on X, then (†) is trivial. So, suppose that one of the fi does not vanish on X. Then, for ageneric linear combination fk := a0f0 + · · ·+akfk, fk does not vanish on X. Thus, V (fk) containsno component or embedded subvariety of Πd

f.

Now, suppose that we have made generic linear reorganizations of f to produce f , and that


V (fi+k−d) contains no component or embedded subvariety of Πif

for all i > m. Then, for every

component or embedded subvariety, W , of Πm−1

f, W is contained in V (fm+k−d, . . . , fk), but there

exists some fj with j < m + k − d such that W 6⊆ V (fj). Thus, a generic linear combination ofthe f ’s will not vanish on any component of embedded subvariety of Πm−1

f. This proves (†).

As Πi−1

f=

(Πi

f∩V (fi+k−d)

)¬V (f) by 2.2.ii, (†) implies that the Vogel sets of f all have correct

dimension. We show that Πif

= Πif

= Πif

by downward induction on i.

When i = d, the statement is clear. Assume now that Πi+1

f= Πi+1

f= Πi+1

f. Then,

Πif

=(Πi+1

f∩ V (fi+k+1−d)

)¬V (fi+k−d) =

(Πi+1

f∩ V (fi+k+1−d)

)¬V (fi+k−d),

which, by 1.3.i, equals((

Πi+1

f∩ V (fi+k+1−d)

)¬V (f)

)¬V (fi+k−d).

Therefore, applying 2.2.ii, followed by (†), we conclude that Πif

= Πif¬ V (fi+k−d) = Πi

f.

As Πif

= Πif

for all i, by applying 2.2.vii, we conclude that Πif

= Πif

= Πif.

We now wish to endow the Vogel sets a cycle structure. First, we need the following easyproposition.

Proposition 2.12. If X is irreducible, and∣∣Πj−1

f

∣∣ is (j − 1)-dimensional for all j > i, then[Πj

f

]=

[Πj

f

]for all j > i, and there is an equality of cycles given by

[Πi

f

]=

([Πi+1

f

]·[V (fi+k+1−d)

])¬ V (f).

Therefore, on X−V (f), all of V (fk), V (fk−1), . . . , V (fi+k+1−d) intersect properly and, on X−V (f),[Πi

f

]= V (fk) · V (fk−1) · . . . · V (fi+k+1−d).

Proof. Using Remark 2.6, we see that∣∣Πj−1

f

∣∣ being (j − 1)-dimensional for all j > i implies that[Πj

f

]=

[Πj

f

]for all j > i.

Now, by 2.2.iv, the statement[Πi

f

]=

([Πi+1

f

]·[V (fi+k+1−d)

])¬ V (f) is equivalent to the set-

theoretic statement∣∣Πi

f

∣∣ =(∣∣Πi+1

f

∣∣ ∩ V (fi+k+1−d))¬ V (f). This set-theoretic statement follows

easily from 2.5.iii; for it tells us that(∣∣Πi+1f

∣∣ ∩ V (fi+k+1−d))¬ V (f) =

(∣∣Πi+1f

∣∣ ∩ V (fi+k+1−d))¬ V (f)

and 2.2.ii tells us that this equals∣∣Πi

f

∣∣. Applying 2.5.iii again yields the desired equality of cy-cles.

16 DAVID B. MASSEY

Remark 2.13. It is tempting to write that[Πi

f

]=

(V (fk) · V (fk−1) · . . . · V (fi+k+1−d)

)¬V (f). We

could do this if we were willing to use intersection theory with non-proper intersections; this seemsespecially innocuous when the non-proper part of the intersection lies in a portion that we aregoing to throw away, as it does here. Nonetheless, we do not want wish to write any formulasinvolving intersection theory which are not discussed in Appendix A.

However, 2.12 does tell us that, on X,[Πi

f

]is the closure in X of this cycle on X − V (f). This

cycle structure turns out to be the correct one to use in order to endow the Vogel sets with a cyclestructure.

However, in order to guarantee that the cycles we define actually have as their underlying setsthe Vogel sets of f , we only define the Vogel cycles when the gap sets (or Vogel sets) have thecorrect dimensions and, even then, we must restrict ourselves to what happens in a neighborhoodof V (f).

Definition 2.14. If X is irreducible, and∣∣Πj−1

f

∣∣ is (j − 1)-dimensional at each point in V (f) forall j > i, then we define the i-th Vogel cycle of f , ∆i

f , to be the sum of the components of([Πi+1

f

]·[V (fi+k+1−d)

])−

[Πi

f

]which intersect V (f). In other words, if([

Πi+1f

]·[V (fi+k+1−d)

])−

[Πi

f

]=

∑j

pj

[Wj

],

then ∆if =

∑Wj∩V (f) 6=∅

pj

[Wj

].

If∣∣Πj−1

f|Vl

∣∣ is (j − 1)-dimensional at each point in V (f|Vl) for all j > i and for all components Vl

of |M |, then we say that the i-th Vogel cycle of f with respect to M is defined and its definition is∆i

f (M) :=∑

l

ml ∆if|Vl

.

Note that the Dimensionality Lemma implies that there is no difference between saying thatall the Vogel sets have correct dimension and that all the Vogel cycles are defined; we prefer tosay that the Vogel cycles are defined, as the Vogel cycles are the objects in which we are mostinterested.

We have defined the Vogel cycles to consist of pieces which intersect V (f); however, 2.12 yieldsimmediately:

Proposition 2.15. If ∆if (M) is defined, then each ∆i

f|Vl

is non-negative and purely i-dimensional.

Moreover,∣∣∆i

f (M)∣∣ = Di

f (M) ⊆ |M | ∩ V (f).

Remark 2.16. If X is irreducible, Proposition 2.12 and Proposition 2.15, together with the Di-mensionality Lemma, tell us how the Vogel cycles should be calculated; we will describe this now,


omitting the square brackets for the cycles.

One begins with Πdf = X¬V (fk); thus, Πd

f is either 0 or X. Next, one calculates the intersectionΠd

f · V (fk). This intersection cycle has components contained in V (f) and components which arenot contained in V (f). By 2.12, the sum of the components which are not contained in V (f)is precisely Πd−1

f and the sum of the components which are contained in V (f) is ∆d−1f . Having

calculated Πdf ·V (fk) = Πd−1

f +∆d−1f , we use our newly found Πd−1

f in the next step: the calculationof Πd−1

f · V (fk−1). One proceeds downward inductively.The subtle point in the above description is that, if one is working in a neighborhood of a point

of V (f), one may check while performing the calculation that the Vogel sets,∣∣∆i

f

∣∣, have correctdimension. For, by splitting the intersections into pieces which are contained in V (f), and pieceswhich are not, we are actually obtaining a cycle ∆i

f whose underlying set is precisely Dif (this

follows from 2.2.ii). Thus, one proceeds with the inductive calculation described above, and thenchecks that the calculated ∆i

f have correct dimension, which then tells one that the calculation isactually correct.

Consider the special case where p is an isolated point of |M | ∩ V (f). As we saw in Remark 2.8,it is automatic that the Vogel cycles are defined at p, and only ∆0

f can be non-zero.

Example 2.17. We continue to suppress the square brackets around cycles. Let X = C5 and let

f = (f0, f1, f2, f3, f4) = (−2ux2, −2vx2, −2wx2, −3x2 − 2x(u2 + v2 + w2), 2y).

(The reason for the strange, seemingly pointless, coefficients is that we will use this example laterin a different context. See Example II.2.4.) Then, V (f) = V (x, y) and Π5

f = C5.

Π5f · V (f4) = Π5

f · V (−2y) = V (y).

As V (y) is not contained in V (f), Π4f = V (y), and we continue.

Π4f ·V (f3) = V (y) ·V (−3x2−2x(u2 +v2 +w2)) = V (−3x−2(u2 +v2 +w2), y)+V (x, y) = Π3

f +∆3f .

Π3f · V (f2) = V (−3x− 2(u2 + v2 + w2), y) · V (−2wx2) =

V (−3x− 2(u2 + v2), w, y) + 2V (u2 + v2 + w2, x, y) = Π2f + ∆2

f .

Π2f · V (f1) = V (−3x− 2(u2 + v2), w, y) · V (−2vx2) =

V (−3x− 2u2, v, w, y) + 2V (u2 + v2, w, x, y) = Π1f + ∆1

f .

Π1f · V (f0) = V (−3x− 2u2, v, w, y) · V (−2ux2) = V (u, v, w, x, y) + 2V (u2, v, w, x, y) = 5[0] = ∆0

f .

Hence, we find the Vogel sets all have correct dimension, and so the Vogel cycles are definedand ∆3

f = V (x, y), ∆2f = 2V (u2 + v2 + w2, x, y), ∆1

f = 2V (u2 + v2, w, x, y), and ∆0f = 5[0].

18 DAVID B. MASSEY

Remark 2.18. Suppose that all the Vogel cycles of f are defined and k + 1 > d. Consider thetruncated d-tuple ftr := (fk+1−d, . . . , fk); we claim that, in a neighborhood of V (f), |V (f)| = |V (ftr)|and both f and ftr will produce the same Di, ∆i, Πi, Πi, and Πi for all i (all of them will be emptyfor i < 0).

It is immediate from the definitions that Πif = Πi

ftrand Πi

f = Πiftr

. We would know that, nearV (f), Πi

f = Πiftr

and, hence, that Dif = Di

ftrand ∆i

f = ∆iftr

, if we could show that there is anequality of sets |V (f)| = |V (ftr)|.

This is easy; by definition of Πif , |V (ftr)| = |V (f)| ∪ |Π0

f |. As we are assuming that Π0f is 0-

dimensional (and, of course, has no components contained in V (f)), there is a neighborhood ofV (f) in which |V (f)| = |V (ftr)|.

Suppose that all the Vogel cycles of f are defined and k + 1 < d. Consider the extended d-tuplefex := (f0, . . . , f0, . . . , fk) (where there are d− k occurrences of f0); clearly, |V (f)| = |V (fex)|, andf and fex will produce the same Di, ∆i, Πi, Πi, and Πi for all i (all of them will be empty fori < d− (k + 1)).

Looking at the two cases above, we see that, if all the Vogel cycles are defined, the whole theoryremains unchanged if we assume that d = k + 1, i.e., if we assume that the dimension of theunderlying space X is exactly equal to the number of functions in our tuple f .

Proposition 2.19. Suppose that X is irreducible of dimension k + 1. Let s := dimp V (f) > 0.

If∣∣Πj−1

f

∣∣ is (j−1)-dimensional for all j > s, then, in a neighborhood of p, all of V (fk), V (fk−1),. . . , V (fs+1) intersect properly,[

Πs+1f

]= V (fk) · V (fk−1) · . . . · V (fs+1),

and ∆sf equals the sum of those components of V (fk) · V (fk−1) · . . . · V (fs+1) · V (fs) which are

contained in V (f).

In particular, if p is an isolated point in V (f), then

(∆0

f

)p

=(V (fk) · V (fk−1) · . . . · V (f1) · V (f0)

)p.

Proof. If∣∣Πj−1

f

∣∣ is (j − 1)-dimensional for all j > s, then 2.12 tells us that, on X − V (f),V (fk), V (fk−1), . . . , V (fs+1) intersect properly and[

Πs+1f

]= V (fk) · V (fk−1) · . . . · V (fs+1).

However, Πs+1f is purely (s+1)-dimensional, and every component of V (fk, . . . , fs+1) has dimension

at least s + 1. As s = dimp V (f), it follows that there is a neighborhood of p in which the closureof |V (fk, . . . , fs+1) − V (f)| = |V (fk, . . . , fs+1)|, and thus in which V (fk), V (fk−1), . . . , V (fs+1)intersect properly and [

Πs+1f

]= V (fk) · V (fk−1) · . . . · V (fs+1).


Therefore, [Πs+1

f

]· V (fs) = V (fk) · V (fk−1) · . . . · V (fs+1) · V (fs)

and 2.12 tells us that the components of this that are contained in V (f) are precisely ∆sf .

Recalling Remark 2.8, if p is an isolated point in V (f), then all of the gaps sets have correctdimension at p, which implies that Π0

f is empty. Thus,

(∆0

f

)p

=(V (fk) · V (fk−1) · . . . · V (f1) · V (f0)

)p

follows at once from the above.

We now prove a theorem which gives the basic relation between Vogel cycles and the blow-up.In fact, we show that the Vogel cycles are representatives of the Segre classes, as defined in [Fu,§4.2]. In the generic case, this is Theorem 3.3 of [Gas1], and is also proved in Lemma 2.2 of [G-G].However, we are interested in cases which may not be quite so generic.

Theorem 2.20. Let X be an irreducible analytic subset of an analytic manifold U , let π : Blf X →X denote the blow-up of X along f (see Example 1.6), and let Ef denote the corresponding excep-tional divisor.

If Ef properly intersects U × Pm × 0 in U × Pk for all m, then

i) the Vogel cycles of f are defined;

ii) there exists a neighborhood Ω of V (f) such that, for all m, Blf X intersects Ω× Pm × 0properly in Ω× Pk; and

iii) inside Ω, for all i,

Πi+1f = π∗(Blf X · (U × Pi+k+1−d × 0))

and∆i

f = π∗(Ef · (U × Pi+k+1−d × 0)),

where the intersection takes place in U × Pk and π∗ denotes the proper push-forward.

Moreover, for all p ∈ X, there exists an open neighborhood W of p in U such that, for a genericlinear reorganization, f , of f , Ef properly intersects W × Pm × 0 inside W × Pk for all m. Inthe algebraic category, we may produce such generic linear reorganizations globally, i.e., such thatEf properly intersects U × Pm × 0 inside U × Pk for all m.

Proof. We show the last two statements first. As in 2.11, the reason that we can only make localstatements in the analytic case is because we must worry about analytic sets having an infinitenumber of irreducible components. For all p ∈ X, π−1(p) is compact, and so, any analytic set canhave only a finite number of irreducible components which meet π−1(p). In the algebraic setting,we know that we have a finite number of irreducible components globally. For notational ease, weassume in the following paragraph, in the analytic case, that U is rechosen as small as necessary at

20 DAVID B. MASSEY

each stage so that U×Pk contains a finite number of analytic components (of any specified analyticset) which intersect π−1(p); this will mean that we will write U in place of the open neighborhoodW which appears in the statement of the theorem.

Now, as each point in each component of Ef cannot have all of its homogeneous coordinatesequal to zero, for each component ν of Ef , there exists a homogeneous coordinate wk(ν) suchthat V (wk(ν)) properly intersects ν. Therefore, for generic (a0,0, . . . , a0,k) ∈ Ck+1, the linear formwk := a0,0w0 + · · · + a0,kwk is such that V (wk) contains no component of Ef . We continue inthis manner; for generic (a1,0, . . . , a1,k) ∈ Ck+1, the linear form wk−1 := a1,0w0 + · · · + a1,kwk issuch that V (wk−1) contains no component of Ef ∩V (wk). Continuing, we produce a generic linearreorganization, w, of w such that, for all m, Ef properly intersects V (wm+1, . . . , wk) inside U×Pk.This proves the last two claims of the theorem.

We now prove i), ii), and iii) of the theorem.

We use [w0 : · · · : wk] as homogeneous coordinates on Pk. Let η : Blf X → Pk denote therestriction of the projection. Until the end of the proof, we shall simply write fj in place of fj π;no confusion will arise, since it is clear that we must mean fj π when the domain is contained inBlf X.

Certainly, π−1 induces an isomorphism from Πi+1f − V (f) to

η−1(Pi+k+1−d × 0)− Ef = Blf X ∩ (U × Pi+k+1−d × 0)− Ef .

Hence, Πi+1f is purely (i + 1)-dimensional if and only if

Blf X ∩ (U × Pi+k+1−d × 0)− Ef

is purely (i+1)-dimensional. But, every component of Blf X∩(U ×Pi+k+1−d×0) has dimensionat least i + 1, while – by hypothesis – Ef ∩ (U × Pi+k+1−d × 0) is purely i-dimensional. Thus,

Blf X ∩ (U × Pi+k+1−d × 0)− Ef = Blf X ∩ (U × Pi+k+1−d × 0),

and every component has dimension at least i + 1. As Ef is locally defined in Blf X by a singleequation and Ef ∩ (U × Pi+k+1−d × 0) is purely i-dimensional, it follows that Blf X ∩ (U ×Pi+k+1−d × 0) is purely (i + 1)-dimensional, for all i, at all points which lie in Ef . This provesii) from the statement of the theorem, and proves that Πi+1

f is purely (i + 1)-dimensional, for alli, at all points of V (f), and so the Vogel cycles are defined. This proves i).

Note that the Dimensionality Lemma and the above paragraphs imply that, in a neighborhoodof any point p ∈ V (f),

(*) Blf X ∩ V (wi+k+2−d, . . . , wk) = Blf X ∩ V (wi+k+2−d, . . . , wk)− V (fi+k+1−d).

Let p be a point in V (f). As the Vogel cycles are defined, there exists a neighborhood of psuch that X − V (f), V (fk) − V (f), . . . , V (fi+k+2−d) − V (f) all intersect properly and π inducesan isomorphism [

Blf X − E]·[V (fk)− E

]· . . . ·

[V (fi+k+2−d)− E

] ∼=[X − V (f)

]·[V (fk)− V (f)

]· . . . ·

[V (fi+k+2−d)− V (f)

].


By the Dimensionality Lemma, no component of this intersection is contained in V (fi+k+1−d), andso we conclude that Πi+1

f is equal to

π∗

([Blf X − V (fi+k+1−d)

]·[V (fk)− V (fi+k+1−d)

]· . . . ·

[V (fi+k+2−d)− V (fi+k+1−d)

]).

We claim that this implies the first equality of the theorem:

(†) Πi+1f = π∗

(Blf X · V (wi+k+2−d, . . . , wk)

),

in a neighborhood of any point in V (f).

To see this, note that Blf X − V (fi+k+1−d) ⊆ wi+k+1−d 6= 0. On the open set, W ⊆ U × Pk,where fi+k+1−d 6= 0 and wi+k+1−d 6= 0, there is an equality of schemes

Blf X = V

(fj

fi+k+1−d− wj

wi+k+1−d

)j 6=i+k+1−d

.

At points of W,

fj

fi+k+1−d− wj

wi+k+1−d

j 6=i+k+1−d

is easily seen to be a regular sequence. There-

fore, on W, the cycle [Blf X] is equal to the intersection product of the cycles[V

(fj

fi+k+1−d− wj

wi+k+1−d

)]j 6=i+k+1−d

.

Moreover, on W, for j > i + k + 2− d,[V

(fj

fi+k+1−d− wj

wi+k+1−d

)]·[V (fj)

]=

[V

(fj

fi+k+1−d− wj

wi+k+1−d, fj

)]=

[V (fj , wj)

]=

[V (fj)

]·[V (wj)

]=

[V

(fj

fi+k+1−d− wj

wi+k+1−d, wj

)]=

[V

(fj

fi+k+1−d− wj

wi+k+1−d

)]·[V (wj)

].

Hence, on W,

[Blf X] ·[V (fk)

]· . . . ·

[V (fi+k+2−d)

]= [Blf X] ·

[V (wk)

]· . . . ·

[V (wi+k+2−d)

]=

[Blf X] ·[V (wi+k+2−d, . . . , wk)

],

and so (†) follows from our previous paragraphs and (∗).Now, by definition, ∆i

f + Πif = Πi+1

f · V (fi+k+1−d). Applying (†) and the push-forward for-mula (see Appendix A.14) – which we may use since V (fi+k+1−d π) properly intersects Blf X ∩V (wi+k+2−d, . . . , wk) by (*) – we conclude that

∆if + Πi

f = π∗(V (fi+k+1−d π) · Blf X · V (wi+k+2−d, . . . , wk)

).

By the Dimensionality Lemma, ∆if consists of those components of the proper push-forward which

are contained in V (f). Hence, we will have proved the second equality of the theorem if we can

22 DAVID B. MASSEY

show that the components of V (fi+k+1−d π) · Blf X · V (wi+k+2−d, . . . , wk) which are containedin Ef are equal to Ef · V (wi+k+2−d, . . . , wk).

On the open set where wi+k+1−d 6= 0, Ef is defined to be V (fi+k+1−d π) · Blf X. Thus, it isenough to show that V (fi+k+1−d π) ·Blf X ·V (wi+k+2−d, . . . , wk) has no components contained inEf which are also contained in V (wi+k+1−d). However, by hypothesis, V (wi+k+1−d, . . . , wk) prop-erly intersects Ef , and so every component of Ef ∩ V (wi+k+1−d, wi+k+2−d, . . . , wk) has dimensioni− 1. As every component of V (fi+k+1−d π) ·Blf X ·V (wi+k+2−d, . . . , wk) has dimension at leasti, we are finished.

Remark 2.21. Note that the proof of 2.20 shows that, for each i, if Ef properly intersects U ×Pi+k+1−d × 0 in U × Pk, then Πi+1

f is (i + 1)-dimensional near V (f) – the point being that wedo not need to assume that we have proper intersections for all i.

The following corollary follows immediately from Theorem 2.20.

Corollary 2.22 (The Segre-Vogel Relation). Let X be an analytic subset of an analyticmanifold U , and let π : U×Pk → U denote the projection. Assume that M is purely d-dimensional.For each Vl appearing in M , consider BlfVl ⊆ Vl×Pk ⊆ U×Pk, and let El

f denote the correspondingexceptional divisor. Let BlfM :=

∑l ml[BlfVl] and Ef (M) :=

∑l ml[El

f ].

If |Ef (M)| properly intersects U × Pm × 0 in U × Pk for all m, then

i) the Vogel cycles of f with respect to M are defined;

ii) there exists a neighborhood Ω of |M |∩V (f) such that, for all m, |Blf M | intersects Ω×Pm×0properly in Ω× Pk; and

iii) inside Ω, for all i,

Πi+1f (M) = π∗(Blf M · (U × Pi+k+1−d × 0))

and∆i

f (M) = π∗(Ef (M) · (U × Pi+k+1−d × 0)),


Moreover, for all p ∈ |M | ∩ X, there exists an open neighborhood W of p in U such that, fora generic linear reorganization, f , of f , |Ef (M)| properly intersects W × Pm × 0 inside W × Pk

for all m. In the algebraic category, we may produce such generic linear reorganizations globally,i.e., such that |Ef (M)| properly intersects U × Pm × 0 inside U × Pk for all m.

Corollary 2.23. Let X be an analytic subset of an analytic manifold U . Assume that M is purely(k + 1)-dimensional.

Then, continuing with the notation from the previous corollary, the multiplicity of p × Pk inEf (M) is

(∆0

f(M)

)p, where f is a generic linear reorganization of f .


Moreover, if p is an isolated point in |M | ∩ V (f), then p × Pk is the unique component ofEf (M) over p and the multiplicity of p × Pk in Ef (M) is

(∆0

f (M))p; if, in addition, there is a

regular sequence f := (f0, . . . , fk) in OU such that f := f|X , then(∆0

f (M))p

=(M · V (f)

)p.

Proof. We use the notation from 2.22.

That the multiplicity of p × Pk in Ef (M) is(∆0

f(M)

)p

follows immediately from 2.22.iii, forwe will be able to pick some copy of P0 ⊆ Pk so that, over p, |Ef (M)| ∩ (U × P0) = p × P0.

If p is an isolated point in |M | ∩ V (f), then since Ef (M) ⊆ U × Pk is purely k-dimensional,the only component of Ef (M) over p has to be p×Pk, and so the proper intersection conditionof 2.22 is automatically satisfied over a neighborhood of p. Thus, the multiplicity of p × Pk inEf (M) is

(∆0

f (M))p.

By 2.19, if we restrict to each Vl, then(∆0

f

)p

=(V (fk) · V (fk−1) · . . . · V (f1) · V (f0)

)p

=

(Vl · V (fk) · V (fk−1) · . . . · V (f1) · V (f0)

)p

= (Vl · V (f))p,

where we used that f is a regular sequence for the last equality (see Appendix A, section 4 for thisequality and the one before it).

Definition 2.24. We call a generic linear reorganization of f , such as appears in Corollary 2.22,a Vogel reorganization of f with respect to M .

A generic linear reorganization of f which is both agreeable and Vogel is called unifying.

Remark 2.25. Theorem 3.3 of [Gas1] actually shows that, by replacing f by a generic lineartransformation applied to f , one obtains a unifying f ; the point being that the linear transformationis actually generic, not just generic in the IPZ topology. However, as one can see in the proof of2.20, proving that one can use an IPZ-generic transformation to obtain a suitable f is quite trivial,and is actually what one uses in examples.

24 DAVID B. MASSEY

Chapter 3. THE LE-IOMDINE-VOGEL FORMULAS

As in the previous chapter, X will denote an analytic space of dimension d contained in ananalytic manifold U , f := (f0, . . . , fk) will be an ordered (k + 1)-tuple of elements of O

X, and

M =∑

l ml[Vl] will be an analytic cycle in X such that ±M > 0.We wish to examine the effect on the Vogel cycles of adding scalar multiples of a large power

of a new function g : X → C to f0. The formulas that we derive are a powerful tool for inductiveproofs.

Throughout most of this chapter, we will be making the assumption that the Vogel cycles of fhave correct dimension; as discussed in Remark 2.18, this means that we may as well assume thatthe number of elements of f is exactly d. Therefore, we will find it convenient to let n := d − 1,and then write that the dimension of X is n + 1 and that f = (f0, . . . , fn). Moreover, as all of ourresults will concern gap and Vogel cycles, the contributions from various irreducible componentsof M will simply add, and so – for simplicity – we will make the assumption that X is irreducible(though not necessarily reduced) and prove most results in the case where M = [X].

Since we will be assuming that the gap sets have correct dimension, ∆0f will be purely 0-

dimensional, and for any p ∈ X, we write(∆0

f

)p

for the coefficient (possibly zero) of p appearingin the cycle ∆0

f .

The following lemma relates the Vogel cycles of (f0, . . . , fn) to the Vogel cycles of (f1, . . . , fn, g),where g is a new function. We think of this as relating the Vogel cycles of f to the Vogel cycles of frestricted to V (g) – the elimination of f0 corresponds to the drop in dimension of the ambient space.As we shall see later, this “restriction” lemma is an essential step in proving the Le-Iomdine-Vogel(LIV) formulas.

Lemma 3.1 (The Restriction Lemma). Let X be an irreducible analytic space of dimensionn+1, and let f := (f0, . . . , fn) ∈

(OX

)n+1. Let g ∈ OX , let h := (f1, . . . , fn, g), and let p ∈ V (f , g).

i) Suppose that Π1f is 1-dimensional at p. Then, Π1

f properly intersects V (g) at p if and only ifV (h) = V (f , g) as germs of sets at p.

ii) Suppose that the Vogel sets of f have correct dimension at p, that V (h) = V (f , g) as germsof sets at p, and that V (g) properly intersects Di

f at p for all i > 1.Then, dimpV (h) =

(dimpV (f)

)− 1 provided that dimpV (f) > 1, V (g) properly intersects Πi

f

at p for all i, the Vogel sets of h have correct dimension at p, and, for all i such that 1 6 i 6 n,there are equalities of germs of cycles at p given by

Πih = Πi+1

f · V (g) and ∆ih = ∆i+1

f · V (g).

In addition, when i = 0, we have the following equality of germs of cycles

∆0h =

(Π1

f · V (g))

+(∆1

f · V (g)).

Proof.

Proof of i): As germs of sets at p,

V (f1, . . . , fn, g) =(Π1

f ∪ V (f))∩ V (g) =

(Π1

f ∩ V (g))∪ V (f , g).


Since Π1f is purely 1-dimensional at p, that Π1

f properly intersects V (g) at p is equivalent toΠ1

f ∩ V (g) being empty or equal to p. As p ∈ V (f , g), we have proved i).

Proof of ii): By 2.4, V (f) =⋃

Dif . As the Vogel cycles have correct dimension and those of

dimension at least one are properly intersected by V (g) at p, we conclude that dimpV (h) equals(dimpV (f)

)− 1 provided that dimpV (f) > 1.

To see that V (g) properly intersects Πif at p, we work solely with germs of sets at p. For i 6 0,

Πif = ∅, and so there is nothing to prove. We now proceed with a proof by contradiction. Let m

be the smallest i such that Πif does not properly intersect V (g) at p. Note that i) implies that

m > 2, and we have that dimpΠmf ∩ V (g) = m. Thus, dimpΠm

f ∩ V (fm−1) ∩ V (g) > m. However,Πm

f ∩V (fm−1) = Πm−1f ∪Dm−1

f , and so we would have to have that either dimpΠm−1f ∩V (g) > m−1

or dimpDm−1f ∩ V (g) > m− 1; the first possibility is excluded by definition of m, and the second

possibility is excluded by hypothesis. Thus, we have shown that V (g) properly intersects Πif at p

for all i.To show that the Vogel sets of h have correct dimension at p, we once again work on the

level of germs of sets. By definition, Πih = V (fi+1, . . . , fn, g) ¬ V (h). One of our assumptions

is that V (h) = V (f , g); hence, Πih = V (fi+1, . . . , fn, g) ¬ V (f , g). We apply 1.3.iii to obtain

Πih =

(Πi+1

f ∩ V (g))¬ V (f , g). However, V (f , g) contains no components of Πi+1

f ∩ V (g), forΠi+1

f ∩ V (g) is purely i-dimensional, while – as sets – Πi+1f ∩ V (f) ∩ V (g) =

( ⋃m6i Dm

f

)∩ V (g),

which has dimension less than i. Therefore, as germs of sets at p, Πih = Πi+1

f ∩ V (g) and is purelyi-dimensional, and so the Vogel sets of h have correct dimension at p.

We wish to see that, for 1 6 i 6 n, Πih = Πi+1

f ·V (g) at p. As we saw above, this equality holdsfor the underlying sets and neither set has a component contained in V (f). Therefore, it is enoughto show that the cycles Πi

h and Πi+1f ·V (g) are equal on X−V (f). Applying Remark 2.13, we find

that both of these cycles on X − V (f) are given by V (fi+1) · . . . · V (fn) · V (g).Finally, for 0 6 i 6 n− 1, ∆i

h = Πi+1h · V (fi+1)− Πi

h. Thus, for 1 6 i 6 n− 1,

∆ih = Πi+2

f · V (g) · V (fi+1)− Πi+1f · V (g) =

(Πi+1

f + ∆i+1f

)· V (g)− Πi+1

f · V (g) = ∆i+1f · V (g).

When i = 0, we have

∆0h = Π1

h · V (f1) = Π2f · V (g) · V (f1) =

(Π1

f + ∆1f

)· V (g).

When i = n, ∆nh = Πn+1

h · V (g) − Πnh = Πn+1

h · V (g) − Πn+1f · V (g). We need to show that

Πn+1h − Πn+1

f = ∆n+1f .

If f ≡ 0, then Πn+1f = 0, ∆n+1

f = [X], and – as V (g) properly intersects ∆n+1f – we conclude

that h 6≡ 0 and so Πn+1h = [X]. Thus, if f ≡ 0, Πn+1

h − Πn+1f = ∆n+1

f . If f 6≡ 0, then Πn+1f = [X],

∆n+1f = 0, and – as V (g) properly intersects Πn+1

f – we conclude that h 6≡ 0 and so Πn+1h = [X].

Thus, if f 6≡ 0, Πn+1h − Πn+1

f = ∆n+1f .

Definition 3.2. Suppose that p ∈ Π1f ∩ V (g) and that dimpΠ1

f = 1. Let η be an irreduciblecomponent (with its reduced structure) of Π1

f which passes through p.If η ∩ V (g) is zero-dimensional at p, then we define the gap ratio of η at p (for f with respect

to g) to be the ratio of intersection numbers

(η · V (fk+1−d)

)p(

η · V (g))p

.

26 DAVID B. MASSEY

If η ∩ V (g) is not zero-dimensional at p (i.e., if η ⊆ V (g)), then we define the gap ratio of η atp (for f with respect to g) to be 0.

A gap ratio (at p for f with respect to g) is any one of the gap ratios of any component ofΠ1

f ∩ V (g) through p.

If p ∈ V (g), but p 6∈ Π1f , then we say that all the gap ratios are zero.

Finally, a gap ratio at p for f with respect to g and the cycle M is a gap ratio (at p for f withrespect to g) of f|Vl

for some Vl appearing in M .

Lemma 3.3. Let X be an irreducible analytic space of dimension n + 1, let f := (f0, . . . , fn) ∈(OX

)n+1, let g ∈ OX , and let p ∈ V (g). Let a be a non-zero complex number, and let j > 1 be aninteger.

If j is greater than or equal to the maximum gap ratio at p for f with respect to g, then, for allbut (possibly) a finite number of complex a,

i) Π1f properly intersects V (f0 + agj) at p, and

(∆0

f

)p

=(Π1

f · V (f0 + agj))p.

Moreover, if we have the strict inequality that j is greater than the maximum gap ratio at p forf with respect to g, then i) holds for all non-zero a; in particular, this is the case if j > 1+

(∆0

f

)p.

ii) Suppose that Π1f is 1-dimensional at p, and that p ∈ V (f , g). Then, Π1

f properly intersectsV (f0 +agj) at p if and only if there is an equality of germs of sets at p given by V (f1, . . . , fn, f0 +agj) = V (f , g).

iii) Suppose that p ∈ V (f , g) and that, at p, there is an equality of germs of sets given byV (f1, . . . , fn, f0 + agj) = V (f , g), the Vogel sets of f all have correct dimension, and that, for alli > 1, V (g) properly intersects each Di

f .If 1 6 i 6 n, then, at p, Πi+1

f properly intersects V (f0 +agj), the Vogel sets of the (n+1)-tuple(f1, . . . , fn, f0 + agj) have correct dimension, and there is an equality of germs of cycles given by

Πi(f1,...,fn,f0+agj) = Πi+1

f · V (f0 + agj).

Proof.

i) Recall that Π1f is purely one-dimensional at p. Thus, we may write the cycle

[Π1

f

]as

∑mν [ν],

where each ν is a reduced, irreducible curve at p. Let αν(t) denote a local parameterization of ν

such that αν(0) = p. Then, to show that Π1f properly intersects V (f0 +agj) at p, we need to show

that, for all ν, (f0+agj)|αν (t)6≡ 0. To show that

(∆0

f

)p

=(Π1

f ·V (f0+agj))p, we need to show that(

Π1f · V (f0)

)p

=∑

mν

([ν] · V (f0 + agj)

)p; calculating intersection numbers as in A.9 of Appendix

A, we find that what we need to show is that, for all ν, multtf0(αν(t)) = multt((f0 + agj) αν)(t).Thus, we may prove both the proper intersection statement and the intersection formula at thesame time by proving this multiplicity statement.


Clearly, multt((f0 +agj) αν)(t) = minmultt(f0 αν)(t), multt(gj αν)(t)

, unless the lowest

degree terms of f0(αν(t)) and −a(gjαν)(t) are precisely equal. As multt(f0αν)(t) =([ν]·V (f0)

)p

and multt(gjαν)(t) = j([ν]·V (g)

)p, we conclude that multt((f0+agj)αν)(t) = multt(f0αν)(t) if

j is greater than the maximum gap ratio, and that this equality holds when j equals the maximumgap ratio except for the finite number of values of a which would cause cancellation of the lowestdegree terms. This proves i).

ii) This follows immediately by applying Lemma 3.1.i with the g of the lemma replaced byf0 + agj .

iii) This follows immediately by applying Lemma 3.1.ii with the g of the lemma replaced byf0 + agj .

Theorem 3.4 (The Le-Iomdine-Vogel formulas). Suppose that each Vl appearing in M hasdimension n + 1. Let f := (f0, . . . , fn) ∈

(OX

)n+1, let g ∈ OX , and let p ∈ |M | ∩ V (f , g). Let a

be a non-zero complex number, let j > 1 be an integer, and let h := (f1, . . . , fn, f0 + agj).Suppose that the Vogel cycles of f with respect to M are defined at p, and that V (g) properly

intersects each of the Vogel cycles, ∆if (M), at p for all i > 1.

If j is greater than or equal to the maximum gap ratio at p for f with respect to g and M , thenfor all but (possibly) a finite number of complex a, in a neighborhood of p:

i) there is an equality of sets given by |M | ∩ V (h) = |M | ∩ V (f , g),

ii) dimp(|M | ∩ V (h)) =(dimp

(|M | ∩ V (f)

))− 1 provided that dimp(|M | ∩ V (f)) > 1,

iii) the Vogel cycles of h with respect to M exist at p, and

iv) ∆0h(M) = ∆0

f (M) + j(∆1

f (M) · V (g))

and, for 1 6 i 6 n− 1, ∆ih(M) = j

(∆i+1

f (M) · V (g)).

Moreover, if we have the strict inequality that j is greater than the maximum gap ratio at p forf with respect to g and M , then these equalities hold for all non-zero a; in particular, this is thecase if j > 1 + maxl

(∆0

f|Vl

)p.

Proof. The assumption that all ml have the same sign prevents cancellation of contributions fromvarious Vl; thus, the assumption that V (g) properly intersects each ∆i

f (M) implies that V (g)properly intersects each ∆i

f|Vl

for all l. Therefore, we are reduced to considering the case of Lemma

3.3, where X is irreducible and M equals [X].

Now, the equality of sets in i) is precisely 3.3.ii; the statement concerning dimpV (h) followsfrom this equality of sets, combined with the facts that V (f) =

⋃Di

f (see 2.4) and that V (g)properly intersects the non-zero-dimensional Vogel cycles of V (f).

Now, suppose that 0 6 i 6 n−1. By definition, Πih+∆i

h = Πi+1h ·V (fi+1). By 3.3.iii, this equals

Πi+2f ·V (f0+agj)·V (fi+1). By definition of the Vogel cycles, this equals

(Πi+1

f +∆i+1f

)·V (f0+agj).

As∣∣∆i+1

f

∣∣ ⊆ V (f0), ∆i+1f · V (f0 + agj) = ∆i+1

f · V (agj) = j(∆i+1

f · V (g)). Therefore, we have

28 DAVID B. MASSEY

shown that

(†) Πih + ∆i

h =(Πi+1

f · V (f0 + agj))

+ j(∆i+1

f · V (g)).

If i = 0, then Πih = 0, and the first equality of iv) of the theorem follows from (†) and 3.3.i.

If 1 6 i 6 n − 1, then 3.3.iii tells us that(Πi+1

f · V (f0 + agj))

= Πih; cancelling Πi

h from eachside of (†) yields the second equality of the theorem.

Remark 3.5. A principal use of the LIV formulas is in families; one requires something about theconstancy of the Vogel cycles of f in the family, and the LIV formulas imply the constancy of theVogel cycles of a tuple of function with a smaller zero locus.

However, it is possible to use these formulas “in reverse” – to calculate the Vogel cycles ofh(a,j) := (f1, . . . , fn, f0 + agj) and have them tell us about the Vogel cycles of (f0, . . . , fn). Thedifficulty of applying the LIV formulas in this manner is that it is not so easy to know when jis greater than or equal to the maximum gap ratio. We discuss this problem below, using thenotation from the theorem.

Suppose that the Vogel cycles of f are defined at p, and that V (g) properly intersects each ofthe Vogel cycles, ∆i

f , at p for all i > 1. Assume that, in a neighborhood of p, there is an equalityof sets given by V (h(a,j)) = V (f , g) (we are still assuming that a 6= 0).

By assuming that V (g) properly intersects ∆if for i > 1, we are assuming that we can calculate

the Vogel sets of f in dimensions one and higher. While it would be nice to be able to proceedwithout this assumption, there seems to be no way to avoid it. Notice that, if we could calculate(∆0

f )p, then we would know that the LIV formulas hold for j > (∆0f )p. However, (∆0

f )p is typicallymore difficult to calculate than (∆1

f · V (g))p. So, we will assume that we can also calculate theintersection number (∆1

f · V (g))p, and then consider the problem of how can one tell when j islarge enough for the LIV formulas to hold using data gathered from ∆0

h(a,j)and (∆1

f · V (g))p.

Our best answer is that:

if j >(∆0

h(a,j)

)p− j

(∆1

f

)p, then the LIV formulas hold, and so

(∆0

f

)p

=(∆0

h(a,j)

)p− j

(∆1

f

)p.

To see this, note that the proof of 3.4 shows that(∆0

h(a,j)

)p− j

(∆1

f

)p

=(Π1

f · V (f0 + agj))p.

We claim that, if j >(Π1

f ·V (f0 +agj))p, then j > (∆0

f )p =(Π1

f ·V (f0))p

and so the LIV formulashold. This is easy; calculating intersection numbers as in the proof of 3.3,(

Π1f · V (f0 + agj)

)p

> min(

Π1f · V (f0)

)p, j

(Π1

f · V (g))p

.

The desired conclusion follows.

One might hope that if(∆0

h(a,j+1)

)p−

(∆0

h(a,j)

)p

=(∆1

f

)p

(which would be true if the LIVformulas held), then one could, in fact, conclude that the LIV formulas do hold. Unfortunately,the situation is slightly more complicated than this.

Let us call (a, j) an exceptional pair if there exists a component ν of Π1f at p such that(

ν · V (f0 + agj))p6= min

(ν · V (f0)

)p, j

(ν · V (g)

)p

.


Looking at the proofs of 3.3 and 3.4, it is easy to see that, if (a, j) is not an exceptional pair, then(∆0

h(a,j+1)

)p−

(∆0

h(a,j)

)p

>(∆1

f

)p

with equality if and only if j is greater than or equal to themaximum gap ratio. Hence, if it were not for the existence of exceptional pairs, one could simplymake a table of values of

(∆0

h(a,j)

)p

for fixed a and increasing j, and when a difference betweensuccessive entries is exactly

(∆1

f

)p, one would have identified the maximum gap ratio and would

know that the LIV formulas hold beyond that value for j.On the other hand, if (a, j) is an exceptional pair, it is quite possible that

(∆0

h(a,j+1)

)p−(

∆0h(a,j)

)p

=(∆1

f

)p

and still j is smaller than the maximum gap ratio. Of course, this can onlyhappen once for each possible exceptional pair, and the number of exceptional pairs is certainly nomore than the number of components of Π1

f through p. Thus, if we know the number of componentsof Π1

f through p, call this number c, and we make a table of values of(∆0

h(a,j)

)p, once we see a

difference between successive values equalling(∆1

f

)p

more than c times, we know that j is highenough for the LIV formulas to hold.

Alternatively, and only pseudo-rigorously, if one selects the constant a “randomly”, then a willnot be part of an exceptional pair and so,

(∆0

h(a,j+1)

)p−

(∆0

h(a,j)

)p

>(∆1

f

)p

with equality ifand only if j is greater than or equal to the maximum gap ratio. This approach is particularlywell-suited for computer calculation.

The following lemma is related to 3.3 and 3.4 and will be of use to us later.

Lemma 3.6 . Suppose that each Vl appearing in M has dimension n + 1, let f := (f0, . . . , fn) ∈(OX

)n+1, let p ∈ |M | ∩ V (f), and suppose that the Vogel cycles of f with respect to M at p exist.Let a be a non-zero complex number, and let j > 1 be an integer. Let π denote the projection

from C×X to X, and let w denote the projection from C×X to C.Then, the Vogel sets of h := (wj , f1 π, . . . , fn π, f0 π + awj) with respect to C × M have

correct dimension at (0,p), for all i 6 n + 1, C× Πif (M) properly intersects V (f0 π + awj), and

there is an equality of germs of cycles at (0,p) given by

Πih(C×M) =

(C× Πi

f (M))· V (f0 π + awj).

Proof. As usual, we instantly reduce ourselves to the case where X is irreducible and M = [X].

That C× Πif properly intersects V (f0 π + awj) is obvious.

First, note that V (h) = 0 × V (f). Suppose that 1 6 i 6 n + 1. Then,

Πih = V (fi π, . . . , fn π, f0 π + awj) ¬

(0 × V (f)

).

We have V (fi π, . . . , fn π) =(C×Πi

f

)∪ V (f π), and V (f π)∩ V (f0 π + awj) ⊆ 0× V (f).

Applying 1.3.iii, we find that

Πih =

(C×Πi

f

)∩ V (f0 π + awj) ¬

(0 × V (f)

).

Now, near p,(C× Πi

f

)∩ V (f0 π + awj) is purely i-dimensional, and – not only does it have no

components contained in 0 × V (f) – in fact, it has no components contained in C × V (f); for,

30 DAVID B. MASSEY

by 2.4, the set Πif ∩ V (f) equals the union of all of the Vogel sets of dimension less than or equal

to i − 1. Therefore, the set Πih equals the set

(C × Πi

f

)∩ V (f0 π + awj) and, hence, the Vogel

sets of h have correct dimension at (0,p).As we saw above,

∣∣Πih

∣∣ =∣∣Πi

h

∣∣ has no components contained in C× V (f). Thus, to prove thatthe cycles Πi

h and(C × Πi

f

)· V (f0 π + awj) are equal, it is enough to prove the equality on(

C×X)−

(C× V (f)

). Once again, we apply Remark 2.13 and find that both cycles are equal to

V (fi π) · . . . · V (fn π) · V (f0 π + awj)

on(C×X

)−

(C× V (f)

).


Chapter 4. SUMMARY OF PART I

Let W be analytic subset of an analytic space X and let α be a coherent sheaf of ideals in OX

.Let V denote the scheme V (α). Then, the gap sheaf V ¬W is the analytic closure of V −W ; thatis, V ¬W is the scheme obtained from V by removing any components or embedded subvarietiescontained in W .

Let X be a d-dimensional irreducible (though not necessarily reduced) analytic space and letf := (f0, . . . , fk) ∈ (O

X)k+1. The i-th gap variety of f , Πi

f , is defined as

Πif := V (fi+k+1−d, . . . , fk) ¬ V (f),

if d− (k + 1) < i < d. Similarly, the i-th modified gap variety of f , Πif , is defined as

Πif := V (fi+k+1−d, . . . , fk) ¬ V (fi+k−d),

if d− (k + 1) < i < d. The i-th inductive gap variety of f , Πif , is defined by downward induction

Πdf =

X, if f 6≡ 0∅, if f ≡ 0

andΠi

f :=(Πi+1

f ∩ V (fi+k+1−d))¬ V (fi+k−d),

if d− (k + 1) < i < d.

If X is irreducible and Cohen-Macaulay, and each Πif is i-dimensional,then all three types of

gap varieties are equal. If X is an arbitrary irreducible space, then, locally, we may replace eachmember of the tuple f by a “generic” linear combination of the elements of f to obtain a new tuple,a generic linear reorganization of f , for which the gap sheaves, modified gap sheaves, and inductivegap sheaves are all equal.

If X is irreducible of dimension d and each Πif is i-dimensional, then, on X − V (f),

[Πi

f

]=

V (fk) ·V (fk−1) · . . . ·V (fi+k+1−d); hence, on X,[Πi

f

]is the closure in X of this cycle on X−V (f).

If X is a union of irreducible components, X =⋃

j Xj , then we do not define gap sheaves, butonly gap cycles. Writing

[V

]for the cycle defined by a scheme V , we define the i-th gap cycle

of f by[Πi

f

]:=

∑j

[Πi

f|Xj

], the i-th modified gap cycle of f by

[Πi

f

]:=

∑j

[Πi

f|Xj

], and the i-th

inductive gap cycle of f by[Πi

f

]:=

∑j

[Πi

f|Xj

].

More generally, if we have an analytic cycle M :=∑

l ml[Vl] in X, where all of the ml have thesame sign, then we define the various gap cycles relative to M by taking the sum of the appropriategap cycles restricted to each of the Vl, weighted by the ml. The requirement that all of the ml

have the same sign prevents the cancellation of contributions from the various Vl.

The modified gap varieties and cycles are merely an intermediate tool. The inductive gapvarieties are what we actually use to define (below) our primary objects of study: the Vogel cycles.However, the hypotheses that must be satisfied before we can define the Vogel cycles include,crucially, the hypothesis that each gap set

∣∣Πif

∣∣ has dimension i. Thus, while one can safely forgetthe definition of the modified gap varieties, both the gap varieties and inductive gap varieties areimportant for our future results.

32 DAVID B. MASSEY

If X is irreducible of dimension d and each Πif is i-dimensional, then the i-th Vogel cycle of f ,

∆if is given by

∆if =

([Πi+1

f

]·[V (fi+k+1−d)

])−

[Πi

f

].

If X is a union of irreducible components, then the i-th Vogel cycle is obtained by summing the i-thVogel cycles of all of the irreducible components (as in the definition of the gap cycles). Similarly,one obtains Vogel cycles with respect to a given cycle M by taking the weighted sum of the Vogelcycles of f restricted to each subvariety appearing in M .

If each ∆if is i-dimensional (which one can obtain locally by replacing f by a generic linear

reorganization), then each Vogel cycle, ∆if , is purely i-dimensional, non-negative, and is contained

in V (f). Moreover, V (f) =⋃

i

∣∣∆if

∣∣. Thus, we think of the Vogel cycles as decomposing V (f) onthe level of cycles.

We proved the important Segre-Vogel Relation: Let X be an irreducible, d-dimensional, analyticsubset of an analytic manifold U , let f = (f0, . . . , fk) ∈ (O

X)k+1, let π : Blf X → X denote the

blow-up of X along f , and let Ef denote the corresponding exceptional divisor.

If Ef properly intersects U × Pm × 0 in U × Pk for all m, then Vogel cycles of f are definedand, in a neighborhood of V (f), for all i,

Πi+1f = π∗(Blf X · (U × Pi+k+1−d × 0))

and∆i

f = π∗(Ef · (U × Pi+k+1−d × 0)),


Moreover, for all p ∈ X, there exists an open neighborhood W of p in U such that, for a genericlinear reorganization, f , of f , Ef properly intersects W×Pm ×0 inside W×Pk for all m. In thealgebraic category, we may produce such generic linear reorganizations globally, i.e., such that Ef

properly intersects U × Pm × 0 inside U × Pk for all m.

What we have just stated is the Segre-Vogel Relation for an irreducible space X, as it appearsin Theorem 2.20. We give a more general version with respect to a pure-dimensional cycle inCorollary 2.22.

Finally, we derived the Le-Iomdine-Vogel (LIV) formulas: Let X be an irreducible analytic spaceof dimension n + 1, let f := (f0, . . . , fn) ∈

(OX

)n+1, let g ∈ OX , and let p ∈ V (f , g). Let a be anon-zero complex number, let j > 1 be an integer, and let h := (f1, . . . , fn, f0 + agj).

Suppose that the Vogel cycles of f are defined at p, and that V (g) properly intersects each ofthe Vogel cycles, ∆i

f , at p for all i > 1.

If j is sufficiently large, then there is an equality of sets given by V (h) = V (f , g), dimpV (h) =(dimpV (f)

)− 1 provided that dimpV (f) > 1, the Vogel cycles of h exist at p, and

∆0h = ∆0

f + j(∆1

f · V (g))

and, for 1 6 i 6 n− 1,∆i

h = j(∆i+1

f · V (g)).


In particular, if j > 1+(∆0

f

)p, then these conclusions hold. Once again, there is a more general

version of this result with respect to the cycle M .

Fundamental Concepts from Part I:

Gap sheaf, V ¬W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1

Gap variety (resp. modified, inductive), Πif (resp. Πi

f , Πif ) . . . . . . . . . . . 2.1

Vogel set, Dif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3

Correct dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.7

Pseudo-Zariski topology, (resp. inductive) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10

Generic linear reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10

Agreeable reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11

Vogel cycle, ∆if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.14

Segre-Vogel relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22

Vogel reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.24

Unifying reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24

Gap ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2

Le-Iomdine-Vogel formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4

34 DAVID B. MASSEY

f

Figure 0.1. The Milnor Fibration inside a ball

Part II. LE CYCLES AND HYPERSURFACE SINGULARITIES


The Le numbers and Le cycles generalize the data given by the Milnor number of an isolatedhypersurface singularity. In this introduction, we wish to quickly review why the Milnor numberof an isolated hypersurface singularity is important. We will then give some previously-knowngeneral results on non-isolated hypersurface singularities, and indicate the types of results thatcan be obtained by the machinery contained in the remainder of Part II. We shall also describehow the results from Part I enter into the development of this Le cycle machinery.

Part II deals with the case of hypersurfaces in open subsets of affine space – this is the casedescribed in [Mas6]. The case where the ambient space is itself allowed to be singular is much moredifficult, and is the problem addressed in Parts III and IV. While we could, of course, concludethe affine results as a corollary of the more general case, we prefer to describe the affine situationfirst, and use it as a guide in developing the general case.

Let U be an open neighborhood of the origin in Cn+1 and let f : (U ,0) → (C, 0) be an analytic

function. Then, the Milnor fibration [Mi3], [Le7], [Ra] of f at the origin is an object of primaryimportance in the study of the local, ambient topology of the hypersurface, V (f) := f−1(0), definedby f at the origin. Milnor defined his fibration on a sphere of radius ε; however, his Theorem 5.11of [Mi3] leads one to consider a more convenient, equivalent, fibration which lives inside the openball of radius ε. Hence, throughout Part II, we will use the Milnor fibration as defined below.

For all ε > 0, letBε denote the open ball of radius ε centered at the origin in C

n+1. For allη > 0, let Dη denote the closed disc centered at the origin in C, and let ∂Dη denote its boundary,which is a circle of radius η. Then, having fixed an analytic function, f , there exists ε0 > 0 suchthat, for all ε such that 0 < ε ε0, there exists ηε > 0 such that, for all η such that 0 < η ηε,

35

Figure 0.3. The coordinate hyperplanes

36 DAVID B. MASSEY

the restriction of f to a mapBε ∩ f−1(∂Dη) → ∂Dη is a smooth, locally trivial fibration whose

diffeomorphism-type is independent of the choice of ε and η.This fibration is called the Milnor fibration of f at the origin and the fibre is the Milnor fibre of

f at the origin, which we denote by Ff,0. Hence, the Milnor fibre is a smooth complex n-manifold(of real dimension 2n). The homotopy-type of the Milnor fibre is an invariant of the local, ambienttopological-type of the hypersurface at the origin.

The Results of Milnor

We keep the notation from above; in particular, U is an open neighborhood of the origin inC

n+1 and f : (U ,0) → (C, 0) is an analytic function (actually, for Milnor, f was required to be apolynomial). We will use Σf to denote the critical locus of the map f .

In [Mi3], Milnor proved the existence of the object that is now called the Milnor fibration.He also proved that the Milnor fibre, Ff,0, has the homotopy-type of a finite n-dimensional CW-complex ([Mi3], Theorem 5.1). This implies that all of the homology groups are finitely-generated,are zero above degree n, and that Hn(Ff,0) is free Abelian.

In addition, Milnor proved that if f has an isolated critical point at the origin, i.e., dim0Σf = 0,then Ff,0 is (n − 1)-connected ([Mi3], Lemma 6.4). Combining this with the previous result, itfollows ([Mi3], Theorem 6.5) that, in the case of an isolated singularity, the Milnor fibre has thehomotopy-type of a finite bouquet (one-point union) of n-spheres; the number of spheres in thisbouquet is the Milnor number and is denoted by µ (or µ0(f), or some other such variant). Inparticular, the reduced homology is trivial except in degree n, and there the homology group is Z

µ

.The Milnor number can be calculated algebraically by taking the dimension as a complex vectorspace of the algebra On+1

0 /J(f), where On+10 denotes the ring of analytic germs at the origin and

J(f) denotes the Jacobian ideal 〈 ∂f∂z0

, . . . , ∂f∂zn

〉.So that we can do an example, there is one final result of Milnor’s that we wish to mention here.

Suppose that f is a weighted homogeneous polynomial (i.e., there exist positive integers r0, . . . , rn

such that f (z0r0 , . . . , zn

rn) is a homogeneous polynomial). Then, ([Mi3], Lemma 9.4) the Milnorfibre, Ff,0, is diffeomorphic to f−1(1).

Example 0.2. As an example, consider f = xyz, which defines a hypersurface in C3 consisting

of the three coordinate planes. Thus, V (f) is a hypersurface with a one-dimensional singular setconsisting of the three coordinate axes.

By the above result on weighted homogeneous polynomials, the Milnor fibre is diffeomorphic to

Figure 0.5. The Whitney umbrella

PART II. LE CYCLES AND HYPERSURFACE SINGULARITIES 37

the set of points where xyz = 1; but, this is where x = 0, y = 0, and x = 1yz .

Thus, Ff,0 is homeomorphic to C∗ × C

∗, where C∗ = C − 0. In particular, Ff,0 is homotopy-

equivalent to the product of two circles, and so has non-zero homology in degrees 0, 1, and 2.

Further Results

We wish to consider another classic example: the Whitney umbrella.

Example 0.4. The Whitney umbrella is the hypersurface in C3 defined by the vanishing of f =

y2 − zx2.

Here, we have drawn the picture over the real numbers – this is the rarely-seen picture thatexplains the word “umbrella” in the name of this example. The “handle” of this umbrella is notusually drawn when one is in the complex setting, for the inclusion of this line gives the impressionthat the local dimension of the hypersurface is not constant; something which is not possible overthe complex numbers. A second reason why one rarely sees the above picture is that one frequentlyencounters the Whitney umbrella as a family of nodes degenerating to a cusp; this representationis achieved by making the analytic change of coordinates z = x + t to obtain f = y2 − x3 − tx2

(see Example 1.12).

To determine the homotopy-type of the Milnor fibre of the Whitney umbrella at the origin, weneed a new result.

The result we need is that if we have an analytic function g(z0, . . . , zn) and a variable y, disjointfrom the z’s, then the Milnor fibre of y2 + g(z0, . . . , zn) is homotopy-equivalent to the suspensionof the Milnor fibre of g. By an abuse of language, one frequently says that the singularity ofy2 + g(z0, . . . , zn) is the suspension of the singularity of g.

So, in our example, Fy2−zx2,0 is homotopy-equivalent to the suspension of Fzx2,0. But, as zx2

is homogeneous,

Fzx2,0∼=

(z, x) | zx2 = 1

=

(1x2

, x

) ∣∣∣∣ x = 0

∼= C

∗.

Thus, Fy2−zx2,0 is homotopy-equivalent to the suspension of a circle, i.e., the Milnor fibre of f atthe origin is homotopy-equivalent to a 2-sphere.

38 DAVID B. MASSEY

The suspension result used above is a special case of a much more general result proved invarious forms: first by Sebastiani and Thom in [Se-Th], then by Oka in [Ok] and Sakamoto in[Sak]. This result states that:

Theorem 0.6 (Sebastiani-Thom Result). If f : (U ,0) → (C, 0) and g : (U ′,0) → (C, 0) areanalytic functions, then the Milnor fibre of the function h : (U × U ′,0) → (C, 0) defined byh(w, z) := f(w) + g(z) is homotopy-equivalent to the join, Ff,0 ∗ Fg,0, of the Milnor fibres of fand g.

This determines the homology of Fh,0 in a simple way, since the reduced homology of the joinof two spaces X and Y is given by

Hj+1(X ∗ Y ) =∑

k+l=j

Hk(X) ⊗ Hl(Y ) ⊕∑

k+l=j−1

Tor(Hk(X), Hl(Y )

).

Returning now to Example 0.2 where f = xyz, we see that Ff,0 need not have the homotopy-type of a bouquet of spheres when the singularity is non-isolated. However, there is the generalresult of Kato and Matsumoto [K-M]:

Theorem 0.7. If s := dim0Σf , then Ff,0 is (n − s − 1)-connected; in particular, when s = 0, werecover the result of Milnor.

Moreover, this is the best possible general bound on the connectivity of the Milnor fibre, as isshown by:

Example 0.8. Considerg := z0z1 . . . zs+1 + z2

s+2 + · · · + z2n;

we leave it as an exercise for the reader to verify, using our earlier methods, that this g has ans-dimensional critical locus at the origin and Fg,0 has non-trivial homology in dimension n − s.

Le’s Attaching Result

The result of Kato and Matsumoto can be obtained from a more general result of Le; a resultwhich is one of few which allows calculations concerning the homology of the Milnor fibre for anarbitrary hypersurface singularity.

Let U be an open neighborhood of the origin in Cn+1 and let f : (U ,0) → (C, 0) be an analytic

function. Let L : Cn+1 → C be a generic linear form. Then, it is easy to see that if dim0Σf 1,

then dim0Σ(f|V (L)) = (dim0Σf) − 1.

Now, the main result of [Le1] is:

Theorem 0.9. The Milnor fibre Ff,0 is obtained from the Milnor fibre Ff|V (L),0 by attaching a

certain number of n-handles (n-cells on the homotopy level); this number of attached n-handles is


given by the intersection number(Γ1

f,L· V (f)

)

0, where Γ1

f,Ldenotes the relative polar curve of f

with respect to L.

We will define the polar curve and discuss how to calculate intersection numbers in Chapter 1,but we can already see that Kato and Matsumoto’s result follows inductively from Theorem 0.9since we already know Milnor’s result for isolated singularities and because attaching handles ofindex k does not affect the connectivity in dimensions less than k − 1.

Not only does Le’s result imply Kato and Matsumoto’s, but – assuming that(Γ1

f,L· V (f)

)

0is

effectively calculable – Le’s result enables the calculation of the Euler characteristic of the Milnorfibre, together with some Morse-type inequalities on the Betti numbers of the Milnor fibre; forinstance, the n-th Betti number, bn (Ff,0), must be less than or equal to

(Γ1

f,L· V (f)

)

0.

Unfortunately, the Morse inequalities above are usually far from being equalities. Of course, thereal value of Le’s result is that it allows one to calculate some important information even in thecases where the homotopy-type of the Milnor fibre cannot be determined by other means.

The Result of Le and Ramanujam

As the homotopy-type of the Milnor fibre is an invariant of the local, ambient topological-typeof the hypersurface at the origin, if one has a family of hypersurfaces with isolated singularitiesin which the local, ambient, topological-type is constant, then the Milnor number must remainconstant in the family. In 1976, Le and Ramanujam [L-R] proved the converse of this; we describetheir result now.

LetD be an open disc about the origin in C, let U be an open neighborhood of the origin in

Cn+1, and let f : (

D × U ,

D × 0) → (C, 0) be an analytic function; we write ft for the function

defined by ft(z) := f(t, z). Le and Ramanujam proved:

Theorem 0.10. Suppose that, for all small t, dim0Σft = 0 and that the Milnor number of ft atthe origin is independent of t. Then, for all small t,

i) the fibre-homotopy type of the Milnor fibrations of ft at the origin is independent of t;

and, if n = 2,

ii) the diffeomorphism-type of the Milnor fibrations of ft and the local, ambient, topological-typeof V (ft) at the origin are independent of t.

The Result of Le and Saito

The result of Le and Saito again deals with families of singularities, so we continue with f :

(D × U ,

D × 0) → (C, 0) as above. The result of [Le-Sa] tells one how limiting tangent spaces to

nearby level hypersurfaces of f approach the singularity.

Theorem 0.11. Suppose that, for all small t, dim0Σft = 0 and that the Milnor number of ft at

40 DAVID B. MASSEY

the origin is independent of t. Then,D×0 satisfies Thom’s af condition at the origin with respect

to the ambient stratum, i.e., if pi is a sequence of points inD×U −Σf such that pi → 0 and such

that TpiV (f − f(pi)) converges to some T , then C × 0 = T0(D × 0) ⊆ T .

Generalizing the Milnor Number

So, suppose we have a single analytic function, f : (U ,0) → (C, 0) with a critical locus ofarbitrary dimension s := dim0Σf . What properties would we want generalized Milnor numbers off at 0 to have?

First, associated to f , we want there to be s + 1 numbers which are effectively calculable; callthe numbers λ0

f , . . . , λsf . In the case of an isolated singularity, we want λ0

f to be the Milnor numberof f and all other λi

f to be zero.

For arbitrary s, we would like to generalize Milnor’s result for isolated singularities and showthat the Milnor fibre of f at the origin has a handle decomposition in which the number of attachedhandles of each index are given by the appropriate λi

f .

Finally, we would like to have generalizations of the results of Le and Ramanujam and Le andSaito to families of hypersurface singularities of arbitrary dimension.

The λ0f , . . . , λs

f that we define to achieve these goals are called the Le numbers of f . In order todefine the Le numbers, we shall apply the machinery of Part I to the Jacobian ideal of f ; if we letz := (z0, . . . , zn) be coordinates on U , we obtain an ordered (n+1)-tuple Jz(f) :=

(∂f∂z0

, . . . , ∂f∂zn

),

and the corresponding Vogel cycles,∆i

Jz(f)

i, from Part I (I.2.14) are the Le cycles,

Λi

f,z

i, of f

with respect to z. We then obtain Le numbers, λ0f,z, . . . , λ

sf,z, by intersecting these Le cycles with

affine linear subspaces defined by the coordinate functions (z0, . . . , zn).

Thus, our generalization of the Milnor number depends on a choice of coordinates. Nonetheless,as we shall see, these Le numbers have all of the properties that we expect to have in a generalizationof the Milnor number to functions with non-isolated critical loci on affine spaces.


Chapter 1. DEFINITIONS AND BASIC PROPERTIES

In this chapter, we define and prove some elementary results about the fundamental objectsof study in Part II – the Le cycles and Le numbers. The Le cycles are analytic cycles which, ina sense, decompose the critical locus of an analytic function. The Le numbers are intersectionnumbers of the Le cycles with certain affine linear subspaces.

To define the Le cycles, we first need to define the relative polar cycles, which are the cyclesassociated to the relative polar varieties. The relative polar varieties were studied by Le andTeissier in a number of places (see, for instance, [Te4], [Te5], and [Te7]). Le and Teissier definethe relative polar varieties of a function with respect to generic linear flags, and they usually assumethat the flags have been chosen generically enough so that the relative polar varieties have manyspecial properties.

However, the whole theory seems to behave more nicely if one does not require the flags to bequite so generic, and then works with possibly non-reduced schemes and cycles. This means thatwe will define the relative polar varieties and cycles in terms of gap varieties and cycles, and thenthe Le cycles will be obtained from the corresponding Vogel cycles.

The key features of our definition of the relative polar varieties in terms of gap varieties arethat the polar varieties are not necessarily reduced and that the dimension of the critical locusof the function is allowed to be arbitrary. The reader who is familiar with the works of Le andTeissier ([Te4], [Te5], [Te7]) should note that we index by the generic dimension instead of thecodimension.

There is one further difference between our presentation of the relative polar varieties and thatof Le and Teissier; instead of fixing a complete flag inside the ambient affine space, we fix a linearchoice of coordinates z := (z0, . . . , zn) for C

n+1. We do this because we frequently find it useful tohave the linear functions z0, . . . , zn at our disposal.

Let U be an open subset of Cn+1, let z := (z0, . . . , zn) be a linear choice of coordinates for C

n+1,and let h : (U ,0) → (C,0) be an analytic function. We write Σh for the critical locus of h, i.e.,Σh := V

(∂h∂z0

, . . . , ∂h∂zn

)

Definition 1.1. For 0 k n, the k-th (relative) polar variety, Γkh,z, of h with respect to z is the

scheme V(

∂h∂zk

, . . . , ∂h∂zn

)¬ Σh (see [Mas7], [Mas8], [Mas11]). If the choice of the coordinate

system is clear, we will often simply write Γkh.

If f equals the Jacobian (n + 1)-tuple Jz(h) :=(

∂h∂z0

, . . . , ∂h∂zn

), then Γk

h,z agrees with the gap

variety, Πkf , of f (see Part I, Definition 2.1).

Thus, on the level of ideals, Γkh,z consists of those components of the scheme V

(∂h∂zk

, . . . , ∂h∂zn

)

which are not contained in |Σh|. Note, in particular, that Γ0h,z is empty.

Naturally, we define the k-th polar cycle of h with respect to z to be the analytic cycle[Γk

h,z

].

This agrees with our previous definition of the gap cycles (of the Jacobian tuple) from Part I,Definition 2.1.

42 DAVID B. MASSEY

Clearly, as sets, ∅ = Γ0h,z ⊆ Γ1

h,z ⊆ . . . ⊆ Γn+1h,z = U . In fact, by I.2.2.ii, we have that :

Proposition 1.2.(Γk+1

h,z ∩ V(

∂h∂zk

))¬ Σh = Γk

h,z as schemes, and thus all the components of

the cycle[Γk+1

h,z ∩ V(

∂h∂zk

)]−

[Γk

h,z

]are contained in the critical set of the map h.

As the ideal 〈 ∂h∂zk

, . . . , ∂h∂zn

〉 is invariant under any linear change of coordinates which leavesV (z0, . . . , zk−1) invariant, we see that the scheme Γk

h,z depends only on h and the choice of thefirst k coordinates. At times, it will be convenient to subscript the k-th polar variety with onlythe first k coordinates instead of the whole coordinate system; for instance, we write Γ1

h,z0for the

polar curve.

While it is immediate from the number of defining equations that every component of theanalytic set

∣∣∣Γk

h,z

∣∣∣ has dimension at least k, one usually requires that the coordinate system be

suitably generic so that the dimension of Γkh,z equals k. In this case, we have the following:

Proposition 1.3. If dimpΓkh,z = k, then Γk

h,z has no embedded subvarieties through the point p.

Proof. This is immediate from I.1.5.

Proposition 1.4. If dimpΣh < k, then Γkh,z and V

(∂h∂zk

, . . . , ∂h∂zn

)are equal up to embedded

subvariety and, hence, are equal as cycles at p. If dimpΣh < k and dimpΓkh,z = k, then Γk

h,z and

V(

∂h∂zk

, . . . , ∂h∂zn

)are equal as schemes at p.

If f equals the Jacobian (n + 1)-tuple, Jz(h), and |Γkh,z| is purely k-dimensional for all k, then,

for all k, Γkh,z and the various gap schemes, Πk

f , Πkf , and Πk

f are all equal as schemes.

Proof. As schemes, V := V(

∂h∂zk

, . . . , ∂h∂zn

)consists of the components not contained in Σh –

these comprise Γkh,z – together with those contained in Σh. By the number of defining equations,

every isolated component of V must have dimension at least k. Thus, if dimpΣh < k, the only

components of V which are contained in Σh must be embedded. Therefore, V(

∂h∂zk

, . . . , ∂h∂zn

)

equals Γkh,z up to embedded subvariety and, hence, they are equal as cycles.

But this certainly implies that Γkh,z and V are equal as germs of sets at p. Thus, if dimpΓk

h,z =k, then dimpV = k, i.e., V is a local complete intersection at p. Hence, V has no embeddedsubvarieties at p. The second statement follows.

Since OU is Cohen-Macaulay, the final statement follows immediately from I.2.9 (since, bydefinition, Γk

h,z = Πkf ).

Definition 1.5. If the intersection of Γkh,z and V (z0−p0, . . . , zk−1−pk−1) is purely zero-dimensional


at a point p = (p0, . . . , pn) (i.e., either p is an isolated point of the intersection or p is not in theintersection), then we say that the k-th polar number, γk

h,z(p), is defined and we set γkh,z(p) equal

to the intersection number(Γk

h,z · V (z0 − p0, . . . , zk−1 − pk−1))p

.

(We use the term polar numbers, instead of polar multiplicities, since we are not assuming thatthe coordinates are so generic that this intersection number gives the multiplicities.)

Note that, if γkh,z is defined at p, then it must be defined at all points near p. Note also that,

if γkh,z(p) is defined, then Γk

h,z must be purely k-dimensional at p and so – by 1.3 – Γkh,z has no

embedded subvarieties at p.

Remark 1.6. As sets,

Σ(h|V (z0−p0,...,zk−1−pk−1)) = V

(z0 − p0, . . . , zk−1 − pk−1,

∂h

∂zk, . . . ,

∂h

∂zn

)

= V (z0 − p0, . . . , zk−1 − pk−1) ∩(Σh ∪ Γk

h,z

).

Hence, if γkh,z(p) is defined and p ∈ Σh, then

Σ(h|V (z0−p0,...,zk−1−pk−1)) = V (z0 − p0, . . . , zk−1 − pk−1) ∩ Σh

at p.

We now wish to define the Le cycles. Unlike the polar varieties and cycles, the Le cycles aresupported on the critical set of h itself. These cycles demonstrate a number of properties whichgeneralize the data given by the Milnor number for an isolated singularity.

Definition 1.7. For 0 k n, we define the k-th Le cycle of h with respect to z,[Λk

h,z

], to be

[Γk+1

h,z ∩ V

(∂h

∂zk

)]−

[Γk

h,z

].

If the choice of coordinate system is clear, we will sometimes simply write[Λk

h

]. Also, as we

have given the Le cycles no structure as schemes, we will usually omit the brackets and write Λkh,z

to denote the Le cycle – unless we explicitly state that we are considering it as a set only.Note that, as every component of Γk+1

h,z has dimension at least k + 1, every component of Λkh,z

has dimension at least k. We say that the cycle[Λk

h,z

]or the set

∣∣∣Λk

h,z

∣∣∣ has correct dimension at

a point p provided that∣∣∣Λk

h,z

∣∣∣ is purely k-dimensional at p.

We define the k-th Le number of h at p with respect to z, λkh,z(p), to equal the intersection num-

ber(Λk

h,z · V (z0 − p0, . . . , zk−1 − pk−1))

p, provided this intersection is purely zero-dimensional at

44 DAVID B. MASSEY

p. If this intersection is not purely zero-dimensional at p, then we say that the k-th Le number(of h at p with respect to z ) is undefined. Here, when k = 0, we mean that

λ0h,z(p) =

(Λ0

h,z · U)p

=

[Γ1

h,z ∩ V

(∂h

∂z0

)]

p

=(

[Γ1

h,z

]·

[V

(∂h

∂z0

)])

p

.

(This last equality holds whenever Γ1h,z is one-dimensional at p, for then Γ1

h,z has no embedded

subvarieties by 1.7 and Γ1h,z ∩ V

(∂h∂z0

)is zero-dimensional. See Appendix A.4.)

Note that if λkh,z(p) is defined, then λk

h,z is defined at all points near p and∣∣∣Λk

h,z

∣∣∣ must have

correct dimension at p. Also note that, since Γk+1h,z and Γk

h,z depend only on the choice of the

coordinates z0 through zk, the k-th Le cycle,[Λk

h,z

], depends only on the choice of (z0, . . . , zk).

Finally, note that if h is a polynomial, then since we are taking linear coordinates, we remaininside the algebraic category.

Remark 1.8. We have defined the Le cycles as we did in all of our previous work (see, for instance,[Mas6]). We wish to see that this agrees with the Vogel cycles of Jz(f) := V

(∂h∂z0

, . . . , ∂h∂zn

)under

reasonable hypotheses (see 1.9, below). Note, however, that it follows from the definitions that thesets underlying the Le cycles, |Λi

f,z|, are always equal to the Vogel sets DiJz(f).

Proposition 1.9. If dim Γkf,z is purely k-dimensional for all k 0, then, for all k 0, the k-th

Le cycle of f with respect to z is equal to the k-th Vogel cycle of the Jacobian (n + 1)-tuple of fwith respect to z, i.e., Λk

f,z = ∆kJz(f).

Proof. This follows immediately from the second paragraph of Proposition 1.4, Definition I.2.14,and Proposition I.2.12.

While we shall defer most of our examples until later – when we will have more results to playwith – it is instructive to include at least one at this early stage.

Example 1.10. Let h = y2 − x3 − tx2; this is the Whitney umbrella of Example 0.4, but writtenas a family of nodes degenerating to a cusp. We fix the coordinate system z = (t, x, y) and willsuppress any further reference to it.

We findΣh = V (−x2, −3x2 − 2tx, 2y) = V (x, y).

Thus, the critical locus of h is one-dimensional and consists of the t-axis.Now the critical locus is one-dimensional, while the dimension of every component of V

(∂h∂y

)is

at least two. Hence, V(

∂h∂y

)cannot possibly have any components contained in Σh and, therefore,

ΛΛΛΛ0000

h, z

Figure 1.11. Polar and Lê cycles

ΓΓΓΓ 2h, z

ΓΓΓΓ1

h, z

ΛΛΛΛ1h, z


we begin calculating polar varieties with Γ2h. We have simply

Γ2h = V

(∂h

∂y

)= V (2y) = V (y)

with no components to dispose of.

Next, we have

Γ2h ∩ V

(∂h

∂x

)= V (y) ∩ V (−3x2 − 2tx) = V (y,−3x2 − 2tx).

Applying 1.2 and then I.1.3.iv, we find

Γ1h =

(Γ2

h ∩ V

(∂h

∂x

))¬ Σh = V (y,−3x2 − 2tx) ¬ V (x, y) = V (y,−3x − 2t).

From the definition of the Le cycles (1.11), we obtain

Λ1h =

[V (y,−3x2 − 2tx)

]− [V (y,−3x − 2t)] =

([V (y, x)] + [V (y,−3x − 2t)]

)− [V (y,−3x − 2t)] = [V (y, x)] .

Thus, Λ1h has as its underlying set the t-axis, and this axis occurs with multiplicity 1.

Now we find

Λ0h =

[Γ1

h ∩ V

(∂h

∂t

)]=

[V (y,−3x − 2t) ∩ V (−x2)

]= 2 [V (t, x, y)] = 2[0].

46 DAVID B. MASSEY

Finally, we calculate the Le numbers: λ1h(0) = (V (y, x) · V (t))0 = 1 and clearly λ0

h(0) = 2.

Proposition 1.12. The Le cycles are all non-negative and are contained in the critical set ofh. Every component of

∣∣∣Λk

h,z

∣∣∣ has dimension at least k. If s = dimpΣh then, for all k with

s < k < n + 1, p is not contained in∣∣∣Λk

h,z

∣∣∣, i.e.,

∣∣∣Λk

h,z

∣∣∣ is empty at p; thus, for s < k < n + 1,

λkh,z(p) is defined and equal to 0.

Proof. The first statement follows from 1.2. The second statement follows from the definition ofthe Le cycles and the fact that every component of Γk+1

h,z has dimension at least k + 1. The thirdstatement follows from the first two.

Due to the result of 1.12, we usually only consider λ0h,z(p), . . . , λs

h,z(p).

Proposition 1.13. As sets, for all k, Γk+1h,z ∩ Σh =

⋃ik

Λih,z. In particular, letting k = s :=

dimpΣh, as germs of sets at p, Σh =⋃

isΛi

h,z.

Proof. This follows immediately from I.2.4.

Recall from Remark 1.6 that, if γ1h,z(p) is defined and p ∈ Σh, then

Σ(h|V (z0−p0)) = V (z0 − p0) ∩ Σh

at p. This is especially useful for inductive proofs when combined with the easy:

Proposition 1.14. If s := dimpΣh 1, Λih,z has correct dimension at p for all i s− 1 (by this,

we mean to allow that p may not be contained in some of the Λjh,z’s), and λs

h,z(p) is defined, thendimp(Σh ∩ V (z0 − p0)) = s − 1.

Proof. As we are assuming that Λih,z has correct dimension at p for all i s − 1, it follows from

1.13 that we have only to show that the hyperplane slice V (z0−p0) actually reduces the dimensionof Λs

h,z. But, this must be the case, since

Λsh,z ∩ V (z0 − p0, . . . , zs−1 − ps−1)

is zero-dimensional at p.

Proposition 1.15. Fix an integer k 0. Suppose that p ∈ Σh. If, for all j with 0 j k, Λjh,z

is purely j-dimensional at p, then, for all j such that 1 j k + 1,∣∣Γj

h,z

∣∣ is purely j-dimensional


at p, and the cycles[Γj

h,z ∩ V

(∂h

∂zj−1

)]and

[Γj

h,z

]·[V

(∂h

∂zj−1

)]

are equal at p.In addition, if for all j with 0 j k, Λj

h,z is purely j-dimensional at p, then, for 0 j k+1,

every j-dimensional (isolated) component of the critical locus of h through p is contained in∣∣∣Λj

h,z

∣∣∣.

Proof. Since Γ0h,z is empty, we may inductively apply the final statement in Remark I.2.6 to

conclude that∣∣Γj

h,z

∣∣ is purely j-dimensional for all j k + 1; it follows from 1.3 that Γj

h,z has noembedded subvarieties through p for j k + 1.

Now, for 1 j k + 1,∣∣Γj

h,z

∣∣ is purely j-dimensional at p, and

∣∣Γj

h,z ∩ V(

∂h∂zj−1

) ∣∣ =

∣∣Γj−1

h,z

∣∣ ∪

∣∣Λj−1

h,z

∣∣ is purely (j − 1)-dimensional at p. Thus, V

(∂h

∂zj−1

)contains no components or embedded

subvarieties of Γjh,z, and therefore we may apply Appendix A.4 to conclude that

[Γj

h,z ∩ V

(∂h

∂zj−1

)]and

[Γj

h,z

]·[V

(∂h

∂zj−1

)]

are equal at p.The last statement follows immediately from Proposition I.2.4

In practice, we use the first part of 1.15 to calculate the Le cycles as follows:

Assume for the moment that all the Le cycles have the correct dimension, and let s denote thedimension of the the critical locus of h. Then, in a neighbor hood of the critical locus,

[Γs+1

h,z

]=

[V

(∂h

∂zs+1, . . . ,

∂h

∂zn

)];

[Γs+1

h,z

]·[V

(∂h

∂zs

)]=

[Γs

h,z

]+

[Λs

h,z

];

[Γs

h,z

]·[V

(∂h

∂zs−1

)]=

[Γs−1

h,z

]+

[Λs−1

h,z

];

...[Γ2

h,z

]·[V

(∂h

∂z1

)]=

[Γ1

h,z

]+

[Λ1

h,z

];

[Γ1

h,z

]·[V

(∂h

∂z0

)]=

[Λ0

h,z

].

In each line above, one obtains[Γk

h,z

]from the calculation in the previous line.

Now, to write the above equalities, we have used that the Le cycles have correct dimension –but, in any case, the equalities are true for sets (using intersection and union of sets, of course).

FFFFiiiigggguuuurrrreeee 1111....11117777.... TTTThhhheeee hhhhyyyyppppeeeerrrrssssuuuurrrrffffaaaacccceeee ddddeeeeffffiiiinnnneeeedddd bbbbyyyy hhhh

48 DAVID B. MASSEY

And so, after doing the above calculations, one verifies that the cycles that we have written aboveas the Le cycles do, in fact, have the correct dimension and thus the equalities are correct. Onthe other hand, if the cycles that we have written above as the Le cycles do not have the correctdimension, then the equalities above may be false.

Remark 1.16. As we will see in Example 2.1, in the case of an isolated singularity, λ0h,z is nothing

other than the Milnor number.In the general case, it is tempting to think of λ0

h,z(p) as the local (generic) degree of the Jacobianmap of h at p, i.e., the number of points in

Bε ∩ V

(∂h

∂z0− a0, . . . ,

∂h

∂zn− an

),

whereBε is a small open ball centered at p and a is a generic point with length that is small

compared to ε; unfortunately, there is no such local degree.

Consider the example h = z22 + (z0 − z2

1)2 and let p be the origin.

Then,Bε ∩ V

(∂h

∂z0− a0,

∂h

∂z1− a1,

∂h

∂z2− a2

)=

Bε ∩ V (2(z0 − z2

1) − a0, 2(z0 − z21)(−2z1) − a1, 2z2 − a2).

The solutions to these equations are

z0 =a0

2+

a21

4a20

, z1 = − a1

2a0, z2 =

a2

2.

The number of solutions of these equations inside any small ball does not just depend on pickingsmall, generic a0, a1, and a2, but also depends on the relative sizes of a0 and a1. If a1 is small


relative to a0, then there will be one solution inside the ball; if a0 is small relative to a1, then therewill be no solutions inside the ball.

Do either of these numbers actually agree with λ0h,z(0)? Yes; with these coordinates, λ0

h,z(0) = 1.This can be seen from the above calculations together with the discussion below, which shows how“close” λ0

h,z is to being the generic degree of the Jacobian map of h.

We claim that, if dimpΓ1h,z = 1, then λ0

h,z(p) exists and equals the number of points in

Bε ∩ V

(∂h

∂z0− a0, . . . ,

∂h

∂zn− an

),

whereBε is a small open ball centered at p, a0 = 0 is small compared to ε, and a1, . . . , an are

generic, with length that is small compared to that of a0.To see this, note that this number of points equals the sum of the intersection numbers given

by∑

q

(V

(∂h

∂z1, . . . ,

∂h

∂zn

)· V

(∂h

∂z0− a0

))

q

where the sum is over all q in

Bε ∩ V

(∂h

∂z0− a0,

∂h

∂z1, . . . ,

∂h

∂zn

).

But, for a0 = 0, these points, q, do not occur on the critical locus of h, and so this sum equals

∑

q

(Γ1

h,z · V

(∂h

∂z0− a0

))

q

.

This last sum is none other than

λ0h,z(p) =

(Γ1

h,z · V

(∂h

∂z0

))

p

.

It is also possible to give a more intuitive characterization of λsh,z(p) where s = dimpΣh.

Assuming that λsh,z(p) exists, by moving to a generic point, it is trivial to show that

λsh,z(p) =

∑

ν

nνµν ,

where ν runs over all s-dimensional components of Σh at p, nν is the local degree of the map(z0, . . . , zs−1) restricted to ν at p, and

µν denotes the generic transverse Milnor number of h along

the component ν in a neighborhood of p. In particular, if the coordinate system is generic enoughso that nν is actually the multiplicity of ν at p for all ν, then λs

h,z(p) is merely the multiplicity ofthe Jacobian scheme of h (the scheme defined by the vanishing of the Jacobian ideal) at p.

The next proposition tells us how the Le numbers behave under the taking of hyperplane sections– a fundamental result; the statements concerning cycles could be derived from the Restriction

50 DAVID B. MASSEY

Lemma (I.3.1), but we want to use polar and Le numbers both in the hypotheses and the conclu-sions.

Proposition 1.18. Suppose Σh ∩ V (z0 − p0) = Σ(h|V (z0−p0)), and use the coordinates z =

(z1, . . . , zn) for V (z0 − p0). Let k 1 and suppose that γkh,z(p) and λk

h,z(p) are defined.

Then, γk−1h|V (z0−p0)

,z(p) and λk−1h|V (z0−p0)

,z(p) are defined,

Γkh|V (z0−p0)

,z = Γk+1h,z · V (z0 − p0),

Γk−1h|V (z0−p0)

,z + Λk−1h|V (z0−p0)

,z =(Γk

h,z + Λkh,z

)· V (z0 − p0),

and

γk−1h|V (z0−p0)

,z(p) + λk−1h|V (z0−p0)

,z(p) = γkh,z(p) + λk

h,z(p).

In the special case when k = 1, it follows that if γ1h,z(p) and λ1

h,z(p) are defined, then so isλ0

h|V (z0−p0),z(p), and

λ0h|V (z0−p0)

,z(p) = γ1h,z(p) + λ1

h,z(p).

Moreover, we conclude that if k 1 and γkh,z(p), λk

h,z(p), γk+1h,z (p), and λk+1

h,z (p) are defined,then so are

γk−1h|V (z0−p0)

,z(p), λk−1h|V (z0−p0)

,z(p), γkh|V (z0−p0)

,z(p), and λkh|V (z0−p0)

,z(p),

andΓk

h|V (z0−p0),z = Γk+1

h,z · V (z0 − p0),

Λkh|V (z0−p0)

,z = Λk+1h,z · V (z0 − p0),

and soγk

h|V (z0−p0),z(p) = γk+1

h,z (p),

andλk

h|V (z0−p0),z(p) = λk+1

h,z (p).

Proof. Clearly, it suffices to prove the assertions for p = 0. The assumption that γkh,z(0) and

λkh,z(0) are defined is equivalent to

dim0Γk+1h,z ∩ V

(∂h

∂zk

)∩ V (z0, . . . , zk−1) 0.

Hence, Γk+1h,z is purely (k + 1)-dimensional at the origin and thus has no embedded subvarieties

(Proposition 1.3). Also, Γk+1h,z ∩V

(∂h∂zk

)is purely k-dimensional at the origin and so, by Appendix

A.4, we have an equality of cycles[Γk+1

h,z ∩ V

(∂h

∂zk

)]= Γk+1

h,z · V(

∂h

∂zk

)= Γk

h,z + Λkh,z.


In addition, we see that Γk+1h,z ∩ V

(∂h∂zk

)∩ V (z0) is purely (k − 1)-dimensional at the origin; we

easily conclude thatdim0Γk+1

h,z ∩ Σh ∩ V (z0) k − 1.

Now, let us consider the cycle Γkh|V (z0)

,z. By definition,

Γkh|V (z0)

,z = V

(z0,

∂h

∂zk+1, . . . ,

∂h

∂zn

)¬ Σ(h|V (z0)

).

Using Lemma I.1.3.ii and our hypothesis that Σh ∩ V (z0) = Σ(h|V (z0)), the equality above gives

usΓk

h|V (z0),z =

(V (z0) ∩ Γk+1

h,z

)¬ (Σh ∩ V (z0)) =

(V (z0) ∩ Γk+1

h,z

)¬ Σh.

But, V (z0) ∩ Γk+1h,z is purely k-dimensional at the origin and, as we saw earlier, dim0Γk+1

h,z ∩ Σh ∩V (z0) k − 1; therefore, Σh contains no isolated components of V (z0) ∩ Γk+1

h,z and so, as cycles,

Γkh|V (z0)

,z = Γk+1h,z ∩ V (z0) = Γk+1

h,z · V (z0).

We find

Γk−1h|V (z0)

,z + Λk−1h|V (z0)

,z = Γkh|V (z0)

,z · V(

∂h

∂zk

)=

Γk+1h,z · V (z0) · V

(∂h

∂zk

)=

(Γk

h,z + Λkh,z

)· V (z0).

That γk−1h|V (z0)

,z(0) and λk−1h|V (z0)

,z(0) are defined and that

γk−1h|V (z0)

,z(0) + λk−1h|V (z0)

,z(0) = γkh,z(0) + λk

h,z(0)

follows by intersecting the cycle V (z1, . . . , zn) with each side of the above equality of cycles.The remaining equalities follow easily – we leave them as an exercise.

The following corollary is essentially a converse of the result stated in Remark 1.6.

Corollary 1.19. Let k 0. Suppose

Σh ∩ V (z0 − p0, . . . , zk − pk) = Σ(h|V (z0−p0,...,zk−pk)),

and that γih,z(p) and λi

h,z(p) are defined for all i k. Then, γk+1h,z (p) is defined.

In particular, if s := dimp Σh, λih,z(p) is defined for 0 i s, and, for all k such that

0 k n − 1,Σh ∩ V (z0 − p0, . . . , zk − pk) = Σ(h|V (z0−p0,...,zk−pk)

),

then γih,z(p) is defined for 0 i n.

Proof. The last statement follows immediately from the first by induction, since γ0h,z(p) is always

defined (and is zero).

52 DAVID B. MASSEY

It suffices to prove the first statement when p = 0.

Case 1: If 0 ∈ Σh, then near 0,

Γk+1h,z = V

(∂h

∂zk+1, . . . ,

∂h

∂zn

)

and soΓk+1

h,z ∩ V (z0, . . . zk) = Σ(h|V (z0,...,zk)) = Σh ∩ V (z0, . . . , zk) = ∅.

Hence, γk+1h,z (0) is defined and equal to zero.

Case 2: 0 ∈ Σh. The proof is by induction on k.

For k = 0, the claim is that if 0 ∈ Σh, λ0h,z(0) is defined, and Σh ∩ V (z0) = Σ(h|V (z0)

), thendim0Γ1

h,z ∩ V (z0) 0. As 0 ∈ Σh and λ0h,z(0) is defined, we must have that dim0Γ1

h,z 1. So,if dim0Γ1

h,z ∩ V (z0) 1, then V (z0) must contain a component of Γ1h,z through the origin. But,

since Σh ∩ V (z0) = Σ(h|V (z0)),

Γ1h,z ∩ V (z0) ⊆ V

(z0,

∂h

∂z1, . . . ,

∂h

∂zn

)= V

(z0,

∂h

∂z0, . . . ,

∂h

∂zn

).

Hence, any component of Γ1h,z contained in V (z0) must also be contained in Σh; this contradicts

the definition of Γ1h,z.

Suppose now that the corollary is true up to k − 1, where k 1. Suppose 0 ∈ Σh, Σh ∩V (z0, . . . , zk) = Σ(h|V (z0,...,zk)

), and that γih,z(0) and λi

h,z(0) are defined for all i k. As 0 ∈ Σh

and γih,z(0) is defined for all i k, Remark 1.6 implies that Σh ∩ V (z0, . . . , zi) = Σ(h|V (z0,...,zi)

)for all i k − 1.

In particular, as k 1, Σh ∩ V (z0) = Σ(h|V (z0)). Thus, we may apply Proposition 1.18 to

conclude that γih|V (z0)

,z(0) and λih|V (z0)

,z(0) are defined for all i k − 1 and, as sets,

Γkh|V (z0)

,z = Γk+1h,z ∩ V (z0).

Since Σ(h|V (z0)) ∩ V (z1, . . . , zk) = Σ(h|V (z0,...,zk)

), we are in a position to apply our inductivehypothesis to h|V (z0)

.We conclude that γk

h|V (z0),z(0) is defined, i.e.,

dim0Γkh|V (z0)

,z ∩ V (z1, . . . , zk) 0.

As Γkh|V (z0)

,z = Γk+1h,z ∩ V (z0), the proof is finished.

We shall need the following relation between three intersection numbers. For isolated singular-ities, this formula appears in the proof of Proposition II.1.2 of [Te2] – our argument is essentiallythe same.


Proposition 1.20. Let p ∈ Σh. Then, λ0h,z(p) is defined if and only if dimpΓ1

h,z 1.Moreover, if γ1

h,z(p) is defined, then λ0h,z(p) is defined, the dimension of Γ1

h,z ∩ V (h − h(p)) atp is at most zero, and (

Γ1h,z · V (h − h(p))

)p

= λ0h,z(p) + γ1

h,z(p).

Proof. Γ1h,z consists of those components of V

(∂h∂z1

, . . . , ∂h∂zn

)which are not contained in |Σh|.

Thus, V(

∂h∂z0

)contains no components of Γ1

h,z. Therefore, Γ1h,z is purely one-dimensional at p if

and only if Γ1h,z ∩ V

(∂h∂z0

)is purely zero-dimensional at p, i.e., if and only if λ0

h,z(p) is defined.

If γ1h,z(p) is defined, then Γ1

h,z must be purely one-dimensional at p and so λ0h,z(p) is defined,

by the above.The remainder of the proof is an argument which first appeared in Proposition II.1.2 of [Te2],

and then appeared again in Proposition 1.3 of [Le1]; this argument shows that an easy applicationof the chain rule yields dimpΓ1

h,z ∩ V (h − h(p)) 0, and(Γ1

h,z · V (h − h(p)))p

= λ0h,z(p) + γ1

h,z(p).

For convenience, we assume that p = 0 and that h(0) = 0.Suppose Γ1

h,z =∑

mW

[W ] as cycles. We know that we can calculate the intersection numberof a curve and a hypersurface by parameterizing the curve and looking at the multiplicity of thecomposition of the defining function of the hypersurface with the parameterization. So, for eachcomponent W , pick a local analytic parameterization α(t) of W such that α(0) = 0. We mustshow two things: that h(α(t)) is not identically zero, and that

multth(α(t)) = multt

(∂h

∂z0

)

|α(t)

+ multtz0(α(t)).

As we already know that the righthand side of the above equality is finite, we have only to provethat the equality holds in order to conclude that h(α(t)) is not identically zero. But this is easy:

multth(α(t)) = 1 + multtd

dth(α(t)) = 1 + multt

(∂h

∂z0

)

|α(t)

· α′0(t)

,

where the remaining terms that come from the chain rule are zero since α(t) parameterizes acomponent of the polar curve. Thus,

multth(α(t)) = 1 + multt

(∂h

∂z0

)

|α(t)

+ multtα′0(t)

= multt

(∂h

∂z0

)

|α(t)

+ multtα0(t) = multt

(∂h

∂z0

)

|α(t)

+ multtz0(α(t))

and we are finished.

Of course, what we want to know is that, for a generic choice of coordinates, z, the polarnumbers and the Le numbers are actually defined. Our results in Part I – specifically, I.2.11 and

54 DAVID B. MASSEY

I.2.22 – tell us that, by replacing z by a generic linear reorganization, we can guarantee that thepolar and Le cycles exist and have the correct dimension. However, we must still do some work toknow that the polar and Le numbers are defined generically. To show this, we pick z genericallywith respect to a certain type of stratification of the hypersurface defined by h. Below, we definethe type of stratification that we need, together with a proposition guaranteeing its existence.

Definition 1.21. A good stratification for h at a point p ∈ V (h) is an analytic stratification, G, ofthe hypersurface V (h) in a neighborhood, U , of p such that the smooth part of V (h) is a stratumand so that the stratification satisfies Thom’s ah condition with respect to U − V (h). That is, ifqi is a sequence of points in U − V (h) such that qi → q ∈ S ∈ G and TqiV (h − h(qi)) convergesto some hyperplane T , then TqS ⊆ T .

Proposition 1.22 (Hamm and Le [H-L]). There exists a good stratification for all h : (U ,0) →(C,0) at all p ∈ V (h).

The notion defined below, that of prepolar coordinates, is crucial throughout the remainder ofPart II. It provides a generic condition on linear choices of coordinates which implies that all theLe numbers and polar numbers are defined. Moreover, prepolarity seems to be the right conditionto obtain many topological results. The importance of this definition cannot be overstated.

Definition 1.23. Suppose that Sα is a good stratification for h in a neighborhood, U , of theorigin. Let p ∈ V (h). Then, a hyperplane, H, in C

n+1 through p is a prepolar slice for h at p withrespect to Sα provided that H transversely intersects all the strata of Sα– except perhaps thestratum p itself – in a neighborhood of p.

If H is a prepolar slice for h at p with respect to Sα, then, as germs of sets at p, Σ(h|H ) =(Σh) ∩ H and dimpΣ(h|H ) = (dimpΣh) − 1 provided dimpΣh 1; moreover, H ∩ Sα is a goodstratification for h|H at p (see [H-L]).

By 2.1.3 of [H-L], for a fixed good stratification for h, prepolar slices are generic.

We say simply that H is a prepolar slice for h at p provided that there exists a good stratificationwith respect to which H is a prepolar slice.

Let (z0, . . . , zn) be a linear choice of coordinates for Cn+1, let p ∈ V (h), and let Sα be a good

stratification for h at p.For 0 i n, (z0, . . . , zi) is a prepolar-tuple for h at p with respect to Sα if and only if V (z0−

p0) is a prepolar slice for h at p with respect to Sα and for all j such that 1 j i, V (zj − pj)is a prepolar slice for h|V (z0−p0,...,zj−1−pj−1)

at p with respect to Sα ∩V (z0 − p0, . . . , zj−1 − pj−1).As prepolar slices are generic, a generic linear reorganization of z will produce a prepolar-tuple.Naturally, we say that (z0, . . . , zi) is a prepolar-tuple for h at p provided that there exists a good

stratification for h at p with respect to which (z0, . . . , zi) is a prepolar-tuple.Finally, we say that the coordinates (z0, . . . , zn) are prepolar for h if and only if for all p ∈ V (h),

if s denotes dimpΣh, then (z0, . . . , zs−1) is a prepolar-tuple for h at p (if s = 0 or p ∈ Σh, wemean that there is no condition on the coordinates.)

Note that, as prepolar for h is a condition at all points in Σh, it is not immediate that suchcoordinates exist (we shall, however, prove this in 10.2.)


Remark 1.24. It will be helpful to interpret good stratifications and prepolar-tuples in terms ofconormal geometry and blowing-up the Jacobian tuple Jz(h). Let BlJz(h) U π−→ U denote the blow-up; BlJz(h) U ⊆ U × P

n. Let E denote the corresponding exceptional divisor. We identify theprojectivized cotangent space P(T ∗U) with U × P

n.Then, a good stratification for h at the origin is a stratification Sα of V (h) in a neighborhood

of 0 such that the smooth part of V (h) is a stratum and such that, for all Sα, π−1(Sα) ⊆ P(T ∗Sα

U).The tuple (z0, . . . , zk) is prepolar at the origin with respect to Sα if and only if, for all i with

0 i k, for all Sα with dimSα i + 1,

P(T ∗Sα

U) ∩(V (z0, . . . , zi) × P

i × 0)

= ∅.

We will show that by choosing coordinates which are prepolar, one guarantees the existence ofthe Le and polar numbers. First, we need a lemma. We use the notation from Remark 1.24.

Lemma 1.25. Suppose that (z0, . . . , zk) is a prepolar tuple for h at the origin. Then, over aneighborhood of the origin, for all i with 0 i k,

E ∩(V (z0, . . . , zi−1) × P

i × 0)⊆ 0 × P

i × 0.

(When i = 0, we mean that E ∩(U × P

0 × 0)⊆ 0 × P

0 × 0.)

Proof. If (z0, . . . , zk) is a prepolar tuple, then (z0, . . . , zi) is a prepolar tuple for all i such that0 i k; thus, we only need to prove the claim when i = k. We use our characterizations fromRemark 1.24. Let Sα be a good stratification for h at the origin.

We first show that it suffices to prove the claim with zi−1 replaced by zi; more precisely weshow that:

(†) for all Sα, in a neighborhood of the origin, for all i such that 0 i n,

P(T ∗Sα

U)∩(V (z0, . . . , zi−1)×P

i×0)⊆

(V (z0, . . . , zi)×P

i×0)∪

(V (z0, . . . , zi−1)×P

i−1×0).

(When i = 0, we mean that P(T ∗Sα

U) ∩(U × P

i × 0)⊆

(V (z0) × P

0 × 0).)

Suppose we have an analytic curve

β(t) := (r(t), [a0(t)dz0 + · · · + ai(t)dzi]) ∈ P(T ∗Sα

U) ∩(V (z0, . . . , zi−1) × P

i × 0)

such that r(0) = 0 and such that, for all t = 0, r(t) ∈ Sα.Then, for t = 0, r′(t) = (0, . . . , 0, r′i(t), . . . , r

′n(t)) ∈ Tr(t)Sα and

0 ≡ (a0(t)dz0 + · · · + ai(t)dzi)(r′(t)) = ai(t)r′i(t).

Thus, either ai(t) ≡ 0 or r′i(t) ≡ 0. Since r(0) = 0, if r′i(t) ≡ 0, then ri(t) ≡ 0. This proves (†).

Now, over a neighborhood of the origin,

(∗) E ∩(V (z0, . . . , zi−1) × P

i × 0)⊆

⋃

Sα⊆Σh

(π−1(Sα) ∩

(V (z0, . . . , zi−1) × P

i × 0))

⊆

56 DAVID B. MASSEY

⋃

Sα⊆Σh

(P(T ∗

SαU) ∩

(V (z0, . . . , zi−1) × P

i × 0))

.

We proceed by induction on k.

When k = i = 0, (∗) and (†) combined yield that

E ∩(U × P

0 × 0)⊆

⋃

Sα⊆Σh

(P(T ∗

SαU) ∩

(V (z0) × P

0 × 0))

over a neighborhood of the origin. However, the characterization of z0 being prepolar that we gavein Remark 1.24 tells us that this last quantity equals

P(T ∗0U) ∩

(V (z0) × P

0 × 0)

= 0 × P0 × 0.

This proves the claim when k = 0.

Now, suppose that the lemma is true for k; we wish to see that it is also true for k + 1 = i + 1.Combining (∗) and (†) again, over a neighborhood of the origin, we find

E ∩(V (z0, . . . , zk) × P

k+1 × 0)⊆

⋃

Sα⊆Σh

(P(T ∗

SαU) ∩

(V (z0, . . . , zk) × P

k+1 × 0))

⊆

⋃

Sα⊆Σh

(P(T ∗

SαU)∩

(V (z0, . . . , zk+1)×P

k+1×0))

∪⋃

Sα⊆Σh

(P(T ∗

SαU)∩

(V (z0, . . . , zk)×P

k×0))

.

Prepolarity, as described in Remark 1.24, implies that the image under π of this last quantity iscontained in

(V (z0, . . . , zk+1) ∩

⋃

Sα⊆Σhdim Sαk+1

Sα

)∪

(V (z0, . . . , zk) ∩

⋃

Sα⊆Σhdim Sαk

Sα

).

As (z0, . . . , zk+1) is prepolar, near the origin, both of the above intersections are contained in 0.Thus, we conclude that, over a neighborhood of the origin,

E ∩(V (z0, . . . , zk) × P

k+1 × 0)⊆ 0 × P

k+1 × 0.

In the following theorem, we continue to use the notation from Remark 1.24. The characteri-zation here of the Le cycles in terms of blowing-up was first shown to us by T. Gaffney (withoutthe description of how generic the coordinates must be). We generalize this result in IV.1.10.

Theorem 1.26. Let p ∈ V (h) and let s = dimpΣh.

If (z0, . . . , zk) is a prepolar tuple for h at p, then there exists a neighborhood Ω of p such that,for all i such that 0 i k, the exceptional divisor E properly intersects Ω× P

i × 0 in Ω× Pn.

Hence, if z0 is prepolar for h at p, then γ1h,z0

(p) is defined.


Moreover, if (z0, . . . , zn) is a prepolar tuple for h at p, then, for all i, the Le numbers and polarnumbers λi

h,z(p) and γih,z(p) exist and, in a neighborhood of Σh,

Γi+1h,z = π∗(BlJz(h) U · (Ω × P

i × 0))

andΛi

h,z = π∗(E · (Ω × Pi × 0)),

where the intersection takes place in Ω × Pn and π∗ denotes the proper push-forward.

Proof. For convenience, we will assume that p = 0. Fix a good stratification Sα for h at 0.

We show the first statement in the theorem by induction on k. We need to show that if(z0, . . . , zk) is a prepolar tuple, then there is a neighborhood of the origin over which E ∩ (U ×P

i × 0) is purely i-dimensional for all i k. As (z0, . . . , zk) being a prepolar tuple implies that(z0, . . . , zi) is prepolar for all i k, it suffices to show that if (z0, . . . , zk) is a prepolar tuple, thenthere is a neighborhood of the origin over which E ∩ (U × P

k × 0) is purely k-dimensional.

The lemma implies that, near 0,

E ∩ (U × P0 × 0) ⊆ P(T ∗

0U) ∩ (V (z0) × P

0 × 0) = 0 × P0 × 0.

This proves the desired result when k = 0.Suppose now that E properly intersects U × P

k × 0 over the origin, but does not properlyintersect U × P

k+1 × 0 over the origin. Let C be a component of E ∩(U × P

k+1 × 0)

whichhas dimension at least k + 2 and such that 0 ∈ π(C). We will use w0, . . . , wn as homogeneouscoordinates on P

n.Our inductive hypothesis implies that 0 ∈ π(C ∩ V (wk+1)). We shall derive a contradiction by

using our other hypotheses to prove that 0 ∈ π(C ∩ V (wk+1)).Certainly, 0 ∈ π

(C ∩ (V (z0, . . . , zk) × P

k+1 × 0)), and the lemma implies that

C ∩ (V (z0, . . . , zk) × Pk+1 × 0) ⊆ 0 × P

k+1 × 0.

Therefore, as each component of C ∩ (V (z0, . . . , zk) × Pk+1 × 0) has dimension at least 1, it

follows that each component must intersect V (wk+1). This is a contradiction, and establishes thefirst statement of the theorem.

If z0 is prepolar for h at 0, then Σh ∩ V (z0) = Σ(h|V (z0)), and we have just shown that E

properly intersects U ×P0 ×0 over a neighborhood of the origin. By Remark I.2.21, this implies

that Γ1h,z is purely 1-dimensional at 0, and now Corollary 1.19 allows us to conclude that γ1

h,z0(p)

is defined.

If (z0, . . . , zn) is prepolar for h at the origin, then the Segre-Vogel Relation (Corollary I.2.22)allows us to conclude that there is a neighborhood, Ω, of the origin such that

Γi+1h,z = π∗(BlJz(h) U · (Ω × P

i × 0))

andΛi

h,z = π∗(E · (Ω × Pi × 0)).

Lemma 1.25 tells us that, as germs of sets at the origin,

Λih,z ∩ V (z0, . . . , zi−1) = π(E ∩ (V (z0, . . . , zi−1) × P

i × 0)) ⊆ 0.

58 DAVID B. MASSEY

Therefore, all of the Le numbers are defined, and Corollary 1.19 implies that all of the polarnumbers are also defined.

Remark 1.27. While it is true that prepolar coordinates occur generically and guarantee theexistence of the Le numbers, it is not true that all sets of prepolar coordinates yield the same Lenumbers.

If dimpΣh = 1, then V (h)−Σh, Σh− p,p is a good stratification for h in a neighborhood ofp; hence, V (z0 − p0) is a prepolar slice if and only if dimpΣ(h|V (z0−p0)

) = 0. This is the case if andonly if γ1

h,z(p) and λ1h,z(p) are defined.

Now, consider the example from Remark 1.16. The coordinates z = (z0, z1, z2) are prepolar forh = z2

2 + (z0 − z21)2 at the origin, and λ0

h,z(0) = 1 and λ1h,z(0) = 2. However, the coordinates z are

really not very generic, as Σh is smooth at the origin, but V (z0) intersects Σh with multiplicity 2at 0. The generic values of λ0

h and λ1h (that is, the values with respect to generic coordinates) are

0 and 1, respectively.Note that the alternating sum of the Le numbers is the same for the non-generic and generic

coordinates. As we shall see in Chapter 3, this is a general fact: as long as the coordinates areprepolar, the alternating sum of the Le numbers is independent of the coordinates and is, in fact,equal to the reduced Euler characteristic of the Milnor fibre. We know of no algebraic way to provethis independence.

It is reasonable to ask why we do not strengthen our notion of prepolar in order to disallowexamples such as the one above, where the Le numbers do not have their generic values. Theanswer is that later (in Proposition 10.2) we shall show that, given h and a point p ∈ V (h), onemay pick generic coordinates which are prepolar for h at every point in a neighborhood of p.This result allows us to give another characterization of the Le cycles in Chapter 10. Example 2.4shows that this result would be false if we were to strengthen the notion of prepolar to require theLe numbers to obtain their generic values at each point in this open neighborhood of p.

Finally, note that there are generic values for the Le numbers; this follows easily from thecharacterization of the Le cycles given in Theorem 1.26, combined with Kleiman’s TransversalityLemma [Kl].

We conclude this chapter with four results which do not seem to be of fundamental importance,but which are fairly surprising.

Proposition 1.28. Suppose that dimpΣh = 1 and V (z0 − p0) is a prepolar slice for h at p. IfV (z0 − p0) does not transversely intersect the set |Σh| at p (in particular, if |Σh| is not smooth atp), then λ0

h,z(p) = 0.

Proof. Despite the different appearance of the statement, this is precisely what Le proves in[Le12].

Proposition 1.29. Let k 1. Suppose that Λ0h,z, . . . ,Λ

k−1h,z have correct dimension at p. Suppose,

for all pairs of distinct irreducible germs, V and W , of Σh through p, that dimp(V ∩W ) k − 1.Finally, suppose that λk

h,z(p) = 0. Then, λjh,z(p) = 0 for all j k.


Proof. One applies 2.3 of [La] to the case where the irreducible normal variety is Cn+1 and the

subvariety locally defined by n − k equations is V(

∂h∂zk+1

, . . . , ∂h∂zn

), which equals Γk+1

h,z ∪ Σh as aset.

Let V be an irreducible component of Σh at p and let Wii be the remaining irreduciblecomponents of Σh at p.

Then, the lemma of Lazarsfeld says that if

p ∈(Γk+1

h,z ∩ V)∪

(( ⋃

i

Wi

)∩ V

)

,

then

dimp

(Γk+1

h,z ∩ V)∪

(( ⋃

i

Wi

)∩ V

)

k.

Now the proposition follows easily from 1.13.

Proposition 1.30. Let s = dimpΣh and suppose that λjh,z(p) and γj

h,z(p) exist for all j s.Suppose that the critical locus of h at p is itself singular and denote the singular set of the criticallocus by ΣΣh. Then, every (s − 1)-dimensional component of ΣΣh through p is contained in theset |Λs−1

h,z |.

Proof. Let C be an (s− 1)-dimensional component of ΣΣh at p. As all the Le and polar numbersexist, we may inductively apply Proposition 1.18, together with Remark 1.6 and Proposition 1.14,to conclude that p is a singular point of the one-dimensional critical locus of h|V (z0−p0,...,zs−2−ps−2)

.Using (zs−1, . . . , zn) as coordinates for h|V (z0−p0,...,zs−2−ps−2)

, we also conclude from Proposition1.18 that, at p,

λ1h|V (z0−p0,...,zs−2−ps−2)

and γ1h|V (z0−p0,...,zs−2−ps−2)

exist – which, for a one-dimensional critical locus, is equivalent to V (zs−1 − ps−1) being prepolarfor h|V (z0−p0,...,zs−2−ps−2)

at p.Therefore, by Proposition 1.28, λ0

h|V (z0−p0,...,zs−2−ps−2)(p) = 0. This is equivalent to saying that

p ∈ Γ1h|V (z0−p0,...,zs−2−ps−2)

and now, by applying Proposition 1.18 once more, we find that p ∈ Γsh,z.

As we may apply this same argument at each point of C near p, we find that C ⊆ Γsh,z ∩ Σh.

Finally, as the Le numbers are defined, each of the Le cycles has correct dimension at p and so theresult follows from Proposition 1.13.

Proposition 1.30. Let s = dimpΣh, suppose that λjh,z(p) and γj

h,z(p) exist for all j s, andsuppose that λs−1

h,z (p) = 0. Then, λjh,z(p) = 0 for all j s − 1.

Proof. The result follows from Proposition 1.29, using i = s − 1, since the preceding propositionproves: if there exist two irreducible components, V and W , of Σh at p such that dimp(V ∩W ) =s − 1, then p ∈ Λs−1

h,z and so λs−1h,z (p) = 0.

Figure 2.3. Generalization of nodes degenerating to a cusp

60 DAVID B. MASSEY

Chapter 2. ELEMENTARY EXAMPLES

Example 2.1. If 0 is an isolated singularity of h, then regardless of the coordinate system z, itfollows from Proposition 1.11 that the only possibly non-zero Le number is λ0

h,z(0). Moreover,

as V(

∂h∂z0

, ∂h∂z1

, . . . , ∂h∂zn

)is zero-dimensional, V

(∂h∂z1

, . . . , ∂h∂zn

)is one-dimensional with no compo-

nents contained in Σh and with no embedded subvarieties. Therefore,

Γ1h,z = V

(∂h

∂z1, . . . ,

∂h

∂zn

)

and so

λ0h,z(0) =

(Γ1

h,z · V(

∂h

∂z0

))

0

=(

V

(∂h

∂z1, . . . ,

∂h

∂zn

)· V

(∂h

∂z0

))

0

=

[V

(∂h

∂z0, . . . ,

∂h

∂zn

)]

0

= the Milnor number of h at 0.

Example 2.2. Here, we generalize Example 1.12. Let h = y2 − xa − txb, where a > b > 1. We fixthe coordinate system z = (t, x, y) and will suppress any further reference to it.

We find:

Σh = V (−xb, −axa−1 − btxb−1, 2y) = V (x, y).

Γ2h = V

(∂h

∂y

)= V (2y) = V (y).

Γ2h · V

(∂h

∂x

)= V (y) · V (−axa−1 − btxb−1) =

V (y) · (V (−axa−b − bt) + V (xb−1)) = V (−axa−b − bt, y) + (b − 1)V (x, y)


= Γ1h + Λ1

h.

Γ1h · V

(∂h

∂t

)= V (−axa−b − bt, y) · V (−xb) = bV (t, x, y) = b[0] = Λ0

h.

Thus, λ0h(0) = b and λ1

h(0) = b − 1.Notice that the exponent a does not appear; this is because h = y2−xa−txb = y2−xb(xa−b−t)

which, after an analytic coordinate change at the origin, equals y2 − xbu.

Example 2.4 (The FM Cone). Let h = y2−x3−(u2+v2+w2)x2 and fix the coordinates (u, v, w, x, y).

Σh = V (−2ux2, −2vx2, −2wx2, −3x2 − 2x(u2 + v2 + w2), 2y) = V (x, y).

As Σh is three-dimensional, we begin our calculation with Γ4h.

Γ4h = V (−2y) = V (y).

Γ4h · V

(∂h

∂x

)= V (y) · V (−3x2 − 2x(u2 + v2 + w2)) =

V (−3x − 2(u2 + v2 + w2), y) + V (x, y) = Γ3h + Λ3

h.

Γ3h · V

(∂h

∂w

)= V (−3x − 2(u2 + v2 + w2), y) · V (−2wx2) =

V (−3x − 2(u2 + v2), w, y) + 2V (u2 + v2 + w2, x, y) = Γ2h + Λ2

h.

Γ2h · V

(∂h

∂v

)= V (−3x − 2(u2 + v2), w, y) · V (−2vx2) =

V (−3x − 2u2, v, w, y) + 2V (u2 + v2, w, x, y) = Γ1h + Λ1

h.

Γ1h · V

(∂h

∂u

)= V (−3x − 2u2, v, w, y) · V (−2ux2) =

V (u, v, w, x, y) + 2V (u2, v, w, x, y) = 5[0] = Λ0h.

Hence, Λ3h = V (x, y), Λ2

h = 2V (u2 +v2 +w2, x, y) = a cone (as a set), Λ1h = 2V (u2 +v2, w, x, y),

and Λ0h = 5[0]. Thus, at the origin, λ3

h = 1, λ2h = 4, λ1

h = 4, and λ0h = 5.

Σh

Λ2

h, z

Λ1

h, z

Figure 2.5. The critical locus of h

62 DAVID B. MASSEY

Though λ1h is independent of a generic choice of coordinates, Λ1

h depends on the coordinates –for, by symmetry, if we re-ordered u, v, and w, then Λ1

h would change correspondingly. Moreover,one can check that this is a generic problem.

Such “non-fixed” Le cycles arise from the absolute polar varieties ([L-T2], [Te4], [Te5]) of thehigher dimensional Le cycles (this follows from two results in Part IV: Theorems 1.10 and 3.2).For instance, in the present case, Λ2

h is a cone, and its one-dimensional polar variety varies withthe choice of coordinates, but generically always consists of two lines; this is the case for Λ1

h aswell.

Example 2.6. Let h = xyz, so that V (h) consists of the coordinate planes in C3. (See Example

0.2.) Then, Σh = V (x, y) ∪ V (x, z) ∪ V (y, z) = union of the three coordinate axes.

The coordinates (x, y, z) are extremely non-generic, so choose some other generic coordinatesz = (z0, z1, z2). Then, the set |Λ1

h,z| = Σh. Hence,

λ1h,z(0) =

(Λ1

h,z · V (z0))0

=∑

p

(Λ1

h,z · V (z0 − ξ))p

=∑

p

λ1h,z(p),

where the sum is over all p ∈Bε ∩ Λ1

h,z ∩ V (z0 − ξ) for small ε and 0 < ξ ε; this set consistsof three points and, by symmetry, λ1 must be the same at each of these three points. We wish touse Proposition 1.27 to calculate λ1

h,z(p).As each p ∈ Σh, it follows from 1.10 that γ1

h,z is supported only at Λ0h,z, which is generically

zero-dimensional. Thus, our points p are such that γ1h,z(p) = 0, and it follows from 1.27 that

λ1h,z(p) = λ0

h|V (z0−p0),z

(p), where z denotes the restriction of the coordinates z to V (z0 − p0).

Now, h|V (z0−p0)has an isolated singularity at each of our three points p, and so λ0

h|V (z0−p0),z

(p) =

the Milnor number of h|V (z0−p0)at p, and this is easily seen to equal 1. It follows, finally, that

λ1h,z(0) = 3.

The generic value of λ0h,z(0) is somewhat messier to calculate, and is just as easy to treat in the

more general case given in Example 2.8. (However, the answer is that λ0h,z(0) = 2.)


Example 2.7. Let U be an open subset of Cn+1, let z = (z0, . . . , zn) be the coordinates for C

n+1,and h : U → C be any analytic function. The coordinates z may be non-generic for h. We wish tosee how to calculate λ0

h,z for a generic linear choice of z.So, let z be a generic linear choice of coordinates for C

n+1, and let aij denote ∂zi

∂zj.

Now,

Γ1h,z = V

(∂h

∂z1, . . . ,

∂h

∂zn

)¬ Σh =

V

(a01

∂h

∂z0+ · · · + an1

∂h

∂zn, . . . , a0n

∂h

∂z0+ · · · + ann

∂h

∂zn

)¬ Σh.

By performing elementary row operations, we find that the ideal⟨

a01∂h

∂z0+ · · · + an1

∂h

∂zn, . . . , a0n

∂h

∂z0+ · · · + ann

∂h

∂zn

⟩

is generated by∂h

∂z0+ b0

∂h

∂zn,

∂h

∂z1+ b1

∂h

∂zn, . . . ,

∂h

∂zn−1+ bn−1

∂h

∂zn,

where b0, . . . , bn−1 are generic = 0.Thus,

Γ1h,z = V

(∂h

∂z0+ b0

∂h

∂zn,

∂h

∂z1+ b1

∂h

∂zn, . . . ,

∂h

∂zn−1+ bn−1

∂h

∂zn

)¬ Σh,

and Λ0h,z is given by intersecting this with a00

∂h∂z0

+ · · · + an0∂h∂zn

.It is important to note that we are not claiming that the cycle Γ1

h,z can be calculated byconsidering the cycle

V

(∂h

∂z0+ b0

∂h

∂zn

)· V

(∂h

∂z1+ b1

∂h

∂zn

)· . . . · V

(∂h

∂zn−1+ bn−1

∂h

∂zn

)

and then disposing of any portions of the cycle which are contained in Σh. There could easily bea problem with embedded subvarieties.

Example 2.8. We can use the above example to calculate the generic value of λ0 in Example 2.6.Actually, we can just as easily do a more general calculation.

Let h = z0z1 . . . zn, so that V (h) is the union of the coordinate planes in Cn+1 and Σh =⋃

i=jV (zi, zj) = the union of intersections of pairs of the different coordinate planes. We wish to

show, for a generic choice of coordinates, z, that λ0h,z(0) = n.

By the above, we find that Γ1h,z equals

V(z1z2 . . . zn−1(zn + b0z0), z0z2 . . . zn−1(zn + b1z1),

. . . , z0z1 . . . zn−2(zn + bn−1zn−1))¬ Σh.

Applying 1.2.iii repeatedly, we conclude that

Γ1h,z = V (zn + b0z0, zn + b1z1, . . . , zn + bn−1zn−1).

Finally, by intersecting this with

V

(∂h

∂z0

)= V (a00z1z2 . . . zn + · · · + an0z0z1 . . . zn−1)

64 DAVID B. MASSEY

we obtain the desired result that λ0h,z(0) = n.

We shall obtain this same result, but by inductive methods, in Example 5.2.

Example 2.9. Let h be an analytic map in the variables x and y, and suppose that h = P∏

Qαii ,

where P and∏

Qαii are relatively prime and αi 2, i.e., h gives a non-reduced curve singularity.

We wish to calculate the Le numbers of h at the origin.Let z0 = ax + by, where a = 0, and let z1 = y. Then,

[V

(∂h

∂z1

)]=

[V

(∂h

∂x

(−b

a

)+

∂h

∂y

)]=

∑ [V

(Qαi−1

i

)]+

[

V

(∂h∂x

(−ba

)+ ∂h

∂y∏Qαi−1

i

)]

= Λ1h,z + Γ1

h,z.

Thus, whenever [

V

(∂h∂x

(−ba

)+ ∂h

∂y∏Qαi−1

i

)]

has no components contained in the critical locus of h (an easy argument shows that this is thecase for a generic choice of (a, b) ), we have that

λ1h,z =

∑(αi − 1) (V (Qi) · V (ax + by))0

and

λ0h,z =

(

V

(∂h∂x

(−ba

)+ ∂h

∂y∏Qαi−1

i

)

· V(

∂h

∂x

))

0

,

where we have used that V(

∂h∂z0

)= V

(∂h∂x

).

Note that the formula

λ1h,z =

∑(αi − 1) (V (Qi) · V (ax + by))0

agrees with our earlier formula from the end of Remark 1.19,

λ1h,z =

∑

ν

nνµν ,

since we clearly have nν = (V (Qi) · V (ax + by))0 andµν = αi − 1.

Example 2.10. In this example, we show that – unlike the Milnor number – the Le numbers in afamily need not be upper-semicontinuous. While this may seem to be mildly disturbing at first,the example makes it clear what can happen; if a high-dimensional Le number jumps up, then thelower-dimensional Le numbers are free to jump up or down.

Let ft(x, y, z) = z2 − y3 − txy2. The coordinates (x, y, z) are prepolar at the origin for ft for allt; we fix this set of coordinates and will suppress further reference to them.

For t0 = 0, we are back in the situation of Example 2.2, with a = 3 and b = 2; therefore,λ0

ft0(0) = 2 and λ1

ft0(0) = 1.


On the other hand, the hypersurface defined by f0 is a cross-product singularity; hence, λ0f0

(0) =0, and one trivially finds that λ1

f0(0) = 2.

Thus, at t = 0, λ1 jumps up to 2 from its generic value of 1; this allows the behavior of λ0ft

(0)to be about as “bad” as possible; the generic value of λ0 is 2, while the special value is 0.

The situation is not completely uncontrolled – as we shall see in Corollary 4.16, if we have afamily ft, then the tuple of Le numbers

(λs

ft,z(0), λs−1ft,z

(0), . . . , λ0ft,z(0)

)

is lexigraphically upper-semicontinuous in the t variable.

66 DAVID B. MASSEY

Chapter 3. A HANDLE DECOMPOSITION OF THEMILNOR FIBRE

In this chapter, we give a handle decomposition of the Milnor fibre of an analytic function witha critical locus of arbitrary dimension. This decomposition is more refined than that obtained byiteratively applying Le’s attaching result (Theorem 0.9).

Throughout this chapter, h : U → C will be an analytic function on an open subset of Cn+1. If

p ∈ V (h), then we let Fh,p

denote the Milnor fibre of h at p (more generally, for all p ∈ U , Fh,p

denotes the Milnor fibre of f − f(p) at p).

Our main tool is a proposition based on the argument of Le and Perron [L-P], which is thesame argument that is used in [Ti], [Va1], and [Va2].

In what follows, if we have a pair of topological spaces X ⊆ Y , then we say that Y is obtainedfrom X by canceling k m-handles provided that X has a handle decomposition in which thehandles of highest index are of index m and Y is obtained from X by attaching k (m + 1)-handleseach of which cancels with an m-handle of X (in terms of Morse functions, this says that Y isobtained from X by passing through k critical points – all of index m + 1 – and these criticalpoints cancel with k critical points in X each of index m. See [Mi1] and [Sm].) In particular, thisimplies that the cohomology groups of X and Y are identical except in degree m, where we haveHm(X) ∼= Z

k ⊕ Hm(Y ).

Proposition 3.1. Let p ∈ V (h). If V (z0 − p0) is a prepolar slice for h at p, and n = 2, thenthe Milnor fibre of h at p is obtained – up to diffeomorphism – from the product of a disk with theMilnor fibre of h|V (z0−p0)

at p by first attaching γ1h,z(p) n-handles, which cancel against γ1

h,z(p)

(n − 1)-handles ofD × F

h|V (z0−p0),p, and then attaching λ0

h,z(p) more n-handles.

If n = 2, we have the same conclusion except that the canceling is only up to homotopy.

Proof. Essentially, this is Proposition 4.2 of [Mas7], except that here we have weakened thehypothesis on the genericity of the hyperplane slice. We use the coodinates (z0, . . . , zn) for ourambient space. Clearly, it suffices to prove the claim for p = 0. We will follow the argument of[Va2].

By Proposition C.6.iii, we may use neighborhoods of the form Dδ × Bε, 0 < δ ε, to definethe Milnor fibre of h at the origin up to homotopy. Choose ε and δ such that Bε is a Milnor ballfor h|V (z0)

and Γ1h,z0

∩ (Dδ × ∂Bε) = ∅ – we may accomplish this last equality since Theorem 1.26implies that dim0(Γ1

h,z0∩V (z0)) 0. Choose η such that (Bε, Dη) is a Milnor pair for h|V (z0)

at theorigin (see Appendix C.5). Let Ψ := (h, z0) and let ∆ denote Ψ(Γ1

h,z0) in C

2 (∆ is the Cerf diagramof h with respect to z0). ∆ is given its fitting ideal structure, which is possibly non-reduced (see[Loo]).

Choose (α, β) ∈ C2 − ∆ sufficiently small and let hβ := h|V (z0−β)

. Let D be a small disc inDη ×β centered at (α, β), and let A be the region in Dη ×β formed by joining to D small discscentered at each of the points of ∆ ∩ V (z0 − β), where the joining is via thickened paths whichavoid (0, β) (see Figure 3.2). Note that, counted with multiplicity, there are (Γ1

h,z0·V (z0))0 points

in ∆ ∩ V (z0 − β).

ββββ

((((αααα,,,,ββββ))))D

Dη × β∆

Figure 3.2. The Cerf diagram, and the sets A and C


Then, the argument of Le and Perron [L-P] and Vannier [Va1] shows that the Milnor fibre of h

is obtained from W := h−1β (A)∩ (β×Bε) by attaching λ0

h,z0(0) =

(Γ1

h,z0· V

(∂h∂z0

))

0n-handles

– this part of their argument does not depend on the dimension of Σh. (Though, as we are notassuming that the polar curve is reduced, the details of the generalization of the isotopy resultused by Le and Perron need to be checked – this is done in [Ti].)

The problem is thus to show that W is obtained from the product of a disc with the Milnorfibre of h|V (z0)

by canceling γ1h,z0

(0) = (Γ1h,z0

· V (z0))0 (n− 1)-handles, even in the case where thecritical locus of h does not have dimension one.

Let C ⊆ Dη × β be formed by taking a small disc around (0, β) and joining it to D witha thickened path (see Figure 3.2). Let U := h−1

β (C) ∩ (β × Bε). Then, as A ∪ C is a strongdeformation retract of Dη, U ∪ V is homotopy-equivalent to h−1

β (Dη) ∩ (β × Bε), which is inturn diffeomorphic to a real 2n-ball, B2n. Moreover, U ∩ W is diffeomorphic to h−1

β (D) which isdiffeomorphic to the product of a disc and the Milnor fibre of h|V (z0)

.

Now, U ∪ W is obtained from U by attaching γ1h,z0

(0) n-handles (that is, one handle of indexn for each point of ∆∩ V (z0 − β) in A, counted with multiplicity). As U ∪W is contractible, thisimplies that U has the homotopy-type of a bouquet of γ1

h,z0(0) (n − 1)-spheres.

Consider the Mayer-Vietoris sequence of U and W . As U ∪ W is contractible, we have thatHk(U ∩ W ) ∼= Hk(U) ⊕ Hk(W ) for all k 1, where we know that U has the homotopy-type of abouquet of (n − 1)-spheres, and U ∩ W is diffeomorphic to the product of a disc and the Milnorfibre of h|V (z0)

, which has the homotopy-type of an (n − 1)-dimensional CW complex. Thus, wesee that W has the homotopy-type of an (n − 1)-dimensional CW complex.

But now, since h−1β (D) is diffeomorphic to the product of a disc with the Milnor fibre of h|V (z0)

,and since W itself is obtained by attaching γ1

h,z0(0) n-handles to h−1

β (D), we see that up tohomotopy these γ1

h,z0(0) n-handles must be canceling γ1

h,z0(0) (n− 1)-handles. But, if n 3, then

the real dimension of W is greater than or equal to 6, and so this handle cancellation is up todiffeomorphism [Mi1], [Sm].

Finally, if n = 1, then – by the classification of surfaces – we have that the handle cancellationis up to diffeomorphism.

68 DAVID B. MASSEY

By an inductive application of Proposition 3.1 to each hyperplane slice, we arrive at Theorem4.3 of [Mas7]. This theorem describes a handle decomposition of the Milnor fibre.

Theorem 3.3. Let U be an open subset of Cn+1, let h : U → C be an analytic map, let p ∈ V (h),

let s denote dimpΣh, and let z = (z0, . . . , zs−1) be prepolar for h at p.

If s n − 2, then Fh,p

is obtained up to diffeomorphism from a real 2n-ball by successivelyattaching λn−k

h,z (p) k-handles, where n − s k n;

if s = n − 1, then Fh,p

is obtained up to diffeomorphism from a real 2n-manifold with thehomotopy-type of a bouquet of λn−1

h,z (p) circles by successively attaching λn−kh,z (p) k-handles, where

2 k n

Hence, the reduced Euler characteristic of the Milnor fibre of h at p is given by

χ(Fh,p

) =s∑

i=0

(−1)n−iλih,z(p)

and the reduced Betti numbers, bi(Fh,p), satisfy Morse inequalities with respect to the Le numbers,

i.e., for all k with n − s k n,

(−1)kk∑

i=n−s

(−1)ibi(Fh,p) (−1)k

k∑

i=n−s

(−1)iλn−ih,z (p)

and

(−1)kn∑

i=k

(−1)ibi(Fh,p) (−1)k

n∑

i=k

(−1)iλn−ih,z (p)

Proof. By induction on s. When s = 0, the result follows from Milnor’s work. Now, assume theresult for s − 1. As before, we consider only the case where p = 0.

As V (z0) is prepolar, we may apply Proposition 3.1 to conclude that the Milnor fibre of h at0 is obtained from the product of a disk with the Milnor fibre of h|V (z0)

at 0 by first attaching

γ1h,z(0) n-handles, which cancel against γ1

h,z(0) (n−1)-handles ofD×F

h|V (z0),0, and then attaching

λ0h,z(0) more n-handles – up to diffeomorphism if n = 2 and up to homotopy otherwise.But, z = (z1, . . . , zs−1) is prepolar for h|V (z0)

at the origin and, by our inductive hypothesis, theMilnor fibre of h|V (z0)

at the origin is obtained by successively attaching λn−1−kh|V (z0)

,z(0) k-handles for

(n− 1)− (s− 1) k n− 1. By Proposition 1.18, if n− 1− k = 0, then λn−1−kh|V (z0)

,z(0) = λn−kh,z (0),

and λ0h|V (z0)

,z(0) = γ1h,z(0) + λ1

h,z(0). The conclusions concerning handles follow.

The Morse inequalities follow formally.

Siersma’s main result in [Si2] allows us to improve this result in a special case.

Corollary 3.4 Let U be an open subset of Cn+1, let h : U → C be an analytic map, let p ∈ V (h),

and let s denote dimpΣh. Suppose that (z0, . . . , zs−1) is prepolar for h at p, and suppose thatλs

h,z(p) = 1.


Then, eitherλ0

h,z(p) = λ1h,z(p) = · · · = λs−1

h,z (p) = 0,

or the single (n− s)-handle in the handle decomposition of the previous theorem gets canceled – upto homotopy – by the attaching of one of the λs−1

h,z (p) (n − s + 1)-handles.

Proof. The proof is exactly the inductive proof of 3.3, except in the first step one applies the resultof [Si].

The function h|V (z0−p0,...,zs−2−ps−2)has a one-dimensional critical locus at p. Using (zs−1, . . . , zn)

as coordinates for V (z0 − p0, . . . , zs−2 − ps−2), we conclude from 1.18 that

λ1h|V (z0−p0,...,zs−2−ps−2)

(p) = λsh,z(p) = 1

andλ0

h|V (z0−p0,...,zs−2−ps−2)(p) = γs−1

h,z (p) + λs−1h,z (p).

Hence, h|V (z0−p0,...,zs−2−ps−2)is an isolated line singularity in the sense of Siersma [Si2], and

his result is that either λ0h|V (z0−p0,...,zs−2−ps−2)

(p) = 0 or that one only has homology in middle

dimension, i.e., the one possible (n − s)-handle must get canceled up to homotopy.The equality

λ0h|V (z0−p0,...,zs−2−ps−2)

(p) = γs−1h,z (p) + λs−1

h,z (p) = 0

corresponds to the case where

λ0h,z(p) = λ1

h,z(p) = · · · = λs−1h,z (p) = 0,

since λs−1h,z (p) = 0 implies that p ∈ Γs−1

h,z and, by 1.13, Γs−1h,z ∩ Σh =

⋃is−2

Λih,z. Hence, if

p ∈ Γs−1h,z , then it follows that all the lower Le numbers are also zero.

One might question whether the above result can possibly be correct. What about the casewhere λs

h,z(p) = 1, λs−1h,z (p) = 0, and one of the lower λ’s is not zero? In such a case, there would

be no way to cancel the (n − s)-handle. Note, however, that 1.30 rules out the possibility of theexistence of this case.

70 DAVID B. MASSEY

Chapter 4. GENERALIZED LE-IOMDINE FORMULAS

In this chapter, we generalize the formula of Le and Iomdine (see [Le4], [Io], [Mas8], [Mas11],[M-S], and [Si3]) to functions with an arbitrary-dimensional critical locus; on the level of cycles,this will be a special case of the Le-Iomdine-Vogel formulas from I.3.4.

The Le-Iomdine formulas that we present here tell us how the Le numbers of a hypersurfacesingularity are related to the Le numbers of a certain “sequence of hypersurface singularities” – asequence which “approaches” the original singularity, but such that the critical loci of the terms inthe sequence are of one dimension smaller than the original. These formulas have a large numberof applications.

The statement that we give here has an improvement in a certain bound over what we provedin [Mas14]; in the case of a one-dimensional critical locus, this is the form of the statement as itappears in [M-S] and [Si3]. To give this improved bound, we need a definition. Throughout thischapter, we concentrate our attention at the origin.

We are about to introduce the polar ratios. These quantities first appeared in Proposition 3.5.2of [Te4], and the fact that they are invariants of the “equi-singularity type” appears in Theorem6 of [Te 5]. In [Te 5], Teissier investigates a number of questions of “Iomdine/Sebastiani-Thomtype” in the case of isolated hypersurface singularities.

Definition 4.1. Suppose that Γ1h,z0

is purely one-dimensional at the origin. Let η be an irreduciblecomponent of Γ1

h,z0(with its reduced structure) such that η∩V (z0) is zero-dimensional at the origin.

Then, the polar ratio of η (for h at 0 with respect to z0) is

(η · V (h))0(η · V (z0))0

=

(η · V

(∂h∂z0

))

0

+ (η · V (z0))0

(η · V (z0))0=

(η · V(

∂h∂z0

))0

(η · V (z0))0+ 1.

(The equalities follow from our proof of Proposition 1.20.)

If η ∩ V (z0) is not zero-dimensional at the origin (i.e., if η ⊆ V (z0)), then we say that the polarratio of η equals 1.

A polar ratio (of h at 0 with respect to z0) is any one of the polar ratios of any component ofthe polar curve (if the polar curve is empty at 0, we say that the maximum polar ratio equals 1).

If 0 ∈ Σ and all of the Le cycles have the correct dimension at 0 (in particular, if the Le numbersare defined at 0), this definition is (up to adding 1) a particular case of I.3.2.

Let f :=(

∂h∂z0

, . . . , ∂h∂zn

)and g := z0. Then, Γ1

h,z0= Π1

f , and there is an equality of sets∣∣Π1

f

∣∣ =

∣∣Π1

f

∣∣ by the Dimensionality Lemma (I.2.5). Let η be an irreducible component of Γ1

h,z0.

Looking at I.3.2 (and using d := n + 1 and k := n), we see that the polar ratio of η (for h) isprecisely the gap ratio of η (for f) plus one.

Remark 4.2. The case where h is a homogeneous polynomial of degree d is particularly easy toanalyze. Provided that Γ1

h,z0is one-dimensional at the origin, each component of the polar curve

is a line, and so the polar ratios are all 1 or d.


We are going to consider functions of the form h+azj0, where a is a non-zero complex number and

j is suitably large. Clearly, however, the coordinate z0 is extremely non-generic for h+azj0. Hence,

if we are using the coordinates (z0, z1, . . . , zn) for h, we use the coordinates (z1, z2, . . . , zn, z0) forh+azj

0. The purpose of this “rotation” of the coordinate system is merely to get the z0 coordinateout of the way. Normally, if h has an s-dimensional critical locus at the origin, then h + azj

0 willhave an (s−1)-dimensional critical locus at the origin; thus, it is only the choice of the coordinatesz0, . . . , zs−1 that we care about for h, and the coordinates z1, . . . , zs−1 for h + azj

0.

Lemma 4.3. Let j 2. Let h : (U ,0) → (C, 0) be an analytic function, let s denote dim0Σh, andassume that s 1. Let z = (z0, . . . , zn) be a linear choice of coordinates such that λi

h,z(0) is definedfor all i s. Let a be a non-zero complex number, and use the coordinates z = (z1, . . . , zn, z0) forh + azj

0.

If j is greater than or equal to the maximum polar ratio for h then, for all but a finite numberof complex a,

i) dim0Γ1h,z ∩ V

(∂h∂z0

+ jazj−10

)= 0;

ii) λ0h,z(0) =

(Γ1

h,z · V(

∂h∂z0

+ jazj−10

))

0

;

iii) Σ(h + azj0) is (s − 1)-dimensional at the origin and equal to Σh ∩ V (z0) as germs of sets at

0;

iv) if i 1, then we have an equality of cycles

Γih+azj

0,z= Γi+1

h,z · V(

∂h

∂z0+ jazj−1

0

)

near the origin;

v)(Γ1

h+awj ,w−z0· V (h + awj)

)

0= jλ0

h,z(0), where w is a variable disjoint from those of h.

Moreover, if we have the strict inequality that j is greater than the maximum polar ratio for h,then the above equalities hold for all non-zero a; in particular, this is the case if j 2 + λ0

h,z(0).

Proof. Parts i), ii), iii), and iv) follow immediately by applying Lemma I.3.3, where the X, f , g,a, j, and p of Lemma I.3.3 are replaced by U ,

(∂h∂z0

, . . . , ∂h∂zn

), z0, j · a, j − 1, and 0, respectively.

We will prove v) by applying Lemma I.3.6.

Let f :=(

∂h∂z0

, . . . , ∂h∂zn

), and let g :=

(wj−1, ∂h

∂z1, . . . , ∂h

∂zn, ∂h

∂z0+ jawj−1

). Then, I.3.6 tells us

that the Vogel sets of g have correct dimension at 0, for all i n + 1, C × Πif properly intersects

V(

∂h∂z0

+ jawj−1), and there is an equality of germs of cycles at 0 given by

(†) Πig =

(C × Πi

f

)· V

( ∂h

∂z0+ jawj−1

).

As the Vogel sets of f and g have correct dimension at 0, we may use the Dimensionality Lemmaand Proposition I.2.9 to conclude that we may replace each of the inductive gap varieties in (†) bythe ordinary gap varieties.

72 DAVID B. MASSEY

Now, let L = w − z0 and use (L, z0, . . . , zn) as coordinates for C × U . Then,

Γ1h+awj ,w−z0

= Γ1h+a(L+z0)j ,L =

V

(∂h

∂z0+ ja(L + z0)j−1,

∂h

∂z1, . . . ,

∂h

∂zn

)¬ Σ(h + a(L + z0)j)

which, back in (w, z0, . . . , zn) coordinates, is equal to

V

(∂h

∂z1, . . . ,

∂h

∂zn,

∂h

∂z0+ jawj−1

)¬ Σ(h + awj).

Thus, we see that Γ1h+awj ,w−z0

= Π1g, and so (†) tells us that

(∗) Γ1h+awj ,w−z0

=(C × Γ1

h,z

)· V

( ∂h

∂z0+ jawj−1

).

Now, let us assume for the moment that γ1h+awj ,w−z0

(0) exists. Then, we may apply PropositionI.1.20 to conclude that (

Γ1h+awj ,w−z0

· V (h + awj))

0=

(Γ1

h+awj ,w−z0· V

(∂(h + awj)∂(w − z0)

))

0

+(Γ1

h+awj ,w−z0· V (w − z0)

)

0=

(Γ1

h+awj ,w−z0· V (jawj−1)

)

0+

(Γ1

h+awj ,w−z0· V (w − z0)

)

0.

By (∗), this is equal to

(j − 1)λ0h,z

(0) +(

V (w − z0) ·(C ×

(Γ1

h,z· V

( ∂h

∂z0+ jazj−1

0

)))

0

,

which, by i) and ii), is equal to (j − 1)λ0h,z

(0) + λ0h,z

(0) = jλ0h,z

(0).

Thus, we would be finished if we could show that γ1h+awj ,w−z0

(0) exists, but, as we saw above,Γ1

h+awj ,w−z0∩ V (w − z0) is zero-dimensional at the origin.

Our next result will be to obtain the generalized Le-Iomdine formulas; these formulas are astunningly useful tool for reducing questions on general hypersurface singularities to the mucheasier case of isolated hypersurface singularities. The formulas tell how the Le numbers of hchange when a large power of one of the variables is added. By 4.3.iii, this modification of thefunction h will have a critical locus of dimension one smaller than that of h itself. Proceedinginductively, one arrives at the case of an isolated singularity.

Figure 4.4. The effect of adding a large power of a variable


Theorem 4.5 (Le-Iomdine formulas). Let j 2, let h : (U ,0) → (C, 0) be an analytic function,let s denote dim0Σh, and assume that s 1. Let z = (z0, . . . , zn) be a linear choice of coordinatessuch that λi

h,z(0) is defined for all i s. Let a be a non-zero complex number, and use thecoordinates z = (z1, . . . , zn, z0) for h + azj

0.

If j is greater than or equal to the maximum polar ratio for h then, for all but a finite numberof complex a, Σ(h + azj

0) = Σh∩V (z0) as germs of sets at 0, dim0Σ(h + azj0) = s− 1, λi

h+azj0,z

(0)exists for all i s − 1, and

λ0h+azj

0,z(0) = λ0

h,z(0) + (j − 1)λ1h,z(0),

and, for 1 i s − 1,λi

h+azj0,z

(0) = (j − 1)λi+1h,z (0).

Moreover, if we have the strict inequality that j is greater than the maximum polar ratio for h,then the above equalities hold for all non-zero a; in particular, this is the case if j 2 + λ0

h,z(0).

Proof. This follows immediately from the Le-Iomdine-Vogel formulas (I.3.4) by letting X, f , g, a,j, and p of I.3.4 be replaced by U ,

(∂h∂z0

, . . . , ∂h∂zn

), z0, j · a, j − 1, and 0, respectively.

By applying the Le-Iomdine formulas inductively, we immediately conclude

Corollary 4.6. Let h : (U ,0) → (C, 0) be an analytic function, let s denote dim0Σh, and letz = (z0, . . . , zn) be a linear choice of coordinates such that λi

h,z(0) is defined for all i s. Then,for 0 j0 j1 · · · js−1,

h + zj00 + zj1

1 + · · · + zjs−1s−1

74 DAVID B. MASSEY

has an isolated singularity at the origin, and its Milnor number is given by

µ(h + zj00 + zj1

1 + · · · + zjs−1s−1 ) =

s∑

i=0

(

λih,z(0)

i−1∏

k=0

(jk − 1)

)

=

λ0h,z(0) + (j0 − 1)λ1

h,z(0) + (j1 − 1)(j0 − 1)λ2h,z(0) + . . .

+(js−1 − 1) . . . (j1 − 1)(j0 − 1)λsh,z(0).

As another quick application of the Le-Iomdine formulas, we have the following Plucker formula.

Corollary 4.7. Let h be a homogeneous polynomial of degree d in n+1 variables, let s = dim0Σh,and suppose that λi

h,z(0) exists for all i s. Then,

s∑

i=0

(d − 1)iλih,z(0) = (d − 1)n+1.

Proof. By Remark 4.2, the maximum polar ratio is d. By an inductive application of the Le-Iomdineformulas, we arrive at a function,

f := h + a0zd0 + a1z

d1 + · · · + as−1z

ds−1,

with an isolated singularity at the origin and such that the Milnor number of f = λ0f (0) =∑s

i=0(d − 1)iλih,z(0). Now, by [M-O], this Milnor number is precisely (d − 1)n+1.

In Chapter 9, we will see that the Le cycles are actually Segre cycles. Knowing this, the abovePlucker formula is a special case of a much more general result of Van Gastel [Gas1, 1.2.c].

Remark 4.8. In general, Corollary 4.7 makes it slightly easier to calculate the Euler characteristicof the Milnor fibre of a homogeneous polynomial. In the case of a one-dimensional critical locus,4.7 tells us that, if we know the degree and λ1, then we know λ0 and, hence, the Euler characteristic(see also [M-S] and [Si3]).

Being able to calculate the Euler characteristic of the Milnor fibre of a homogeneous singularityimplies in many cases that we can also calculate the Euler characteristic of the Milnor fibre of aweighted-homogeneous polynomial.

To see this, let f : Cn+1 → C be a weighted homogeneous polynomial. Then, there exist positive

integers r0, . . . , rn such that, if π : Cn+1 → C

n+1 is given by

π(z0, . . . , zn) = (zr00 , . . . , zrn

n ),

then h := f π is homogeneous.


We may define, up to diffeomorphism, the Milnor fibre of h at the origin by using the “weighted”ball

B′

ε :=(z0, . . . , zn)

∣∣ |z0|2r0 + · · · + |zn|2rn < ε2

;

that is, for 0 |ξ| ε 1, we define Fh,0

to beB′

ε ∩ h−1(ξ). That this yields the samediffeomorphism-type as the standard ball is well-known; see, for instance, [G-M2, II.2]. Clearly,the restriction of π induces a map from F

h,0to the Milnor fibre (using standard balls), F

f,0, of f

at the origin; denote this map by π : Fh,0

→ Ff,0

.Now, consider the stratification of C

n+1 derived from the hyperplane arrangement given by allthe coordinate hyperplanes. That is, let I denote the indexing set 0, . . . , n, and for each J ⊆ I,let w

Jdenote the intersection of hyperplanes (a.k.a. the flat) given by

wJ

:= V (zj | j ∈ J),

and let SJ

denote the Whitney stratum

SJ

:= wJ−

⋃

J K

wK

.

Near a given point, a representative of the Milnor fibre of a given function will transverselyintersect all strata of any fixed Whitney stratification; this follows from the fact that, locally,the stratified critical values of an analytic function are isolated. Thus, the stratification S

J

determines Whitney stratifications SJ∩ F

h,0 and S

J∩ F

f,0 of F

h,0and F

f,0, respectively, and

with these stratifications, π becomes a stratified map. Moreover, the restriction of π to a mapfrom S

J∩ F

h,0to S

J∩ F

f,0is a topological covering map with fibre equal to

∏i∈J

ri points.Hence,

χ(Fh,0

) =∑

J

χ(SJ∩ F

h,0) =

∑

J

∏

i ∈J

ri

χ(SJ∩ F

f,0).

Some elementary combinatorics shows that this last quantity is equal to

∑

J

cJ

∑

J⊆K

χ(SK∩ F

f,0)

,

where

cJ

:= (−1)|J| ∑

L⊆J

(−1)|L| ∏

i ∈L

ri

.

The advantage of this last form is that

∑

J⊆K

χ(SK∩ F

f,0) = χ(F

f|wJ

,0).

Therefore, we have thatχ(F

h,0) =

∑

J

cJχ(F

f|wJ

,0),

where cJ

is as above.

76 DAVID B. MASSEY

It follows that

χ(Fh,0

) = (r0 . . . rn)χ(Ff,0

) +∑

J =∅c

Jχ(F

f|wJ

,0),

and so, finally, we arrive at the formula

χ(Ff,0

) =χ(F

h,0) −

∑J =∅

cJχ(F

f|wJ

,0)

r0 . . . rn.

This formula is inductively useful since, if J = ∅, then f|wJ

is a weighted-homogeneous polynomialin fewer variables (compare with [Di]). Note that, in this formula, we need not consider the termwhere J = 0, . . . , n, for then w

J= 0 and hence χ(F

f|wJ

,0) = 0.

This is particularly useful in the case where f is a weighted-homogeneous polynomial with aone-dimensional critical locus and each restriction to a flat, f|w

J, also has a one-dimensional (or

zero-dimensional) critical locus.

Example 4.9. For instance, consider the case of a possibly non-reduced, weighted-homogeneousplane curve singularity. Suppose that the irreducible factorization of f(z0, z1) is za

0zb1

∏fmi

i , wherewe allow for the case where a or b equals 0. Let π(z0, z1) = (zr0

0 , zr11 ), and let h denote the homo-

geneous polynomial f π. Let hi denote the homogeneous polynomial fi π. Let d be the degreeof h and let di be the degree of hi.

Then, the formula of 4.7 becomes

χ(Ff,0

) =χ(F

h,0) + (r0 − 1)r1χ(F

f|V (z0),0

) + (r1 − 1)r0χ(Ff|V (z1)

,0)

r0r1.

Now, χ(Ff|V (zk)

,0) = 0 if f|V (zk)

≡ 0 and simply equals the multiplicity of f|V (zk)otherwise. In

addition, as h is homogeneous, we may calculate χ(Fh,0

) from 4.6 by knowing only λ1h(0), which

we may calculate as in 2.9.We find easily that

r0r1χ(Ff,0

) =

−d∑

di, if a = 0, b = 0d(r0 −

∑di), if a = 0, b = 0

d(r1 −∑

di), if a = 0, b = 0d(r0 + r1 −

∑di), if a = b = 0.

Example 4.10. We can also apply the formula of 4.8 in harder cases. Consider the swallowtailsingularity; this is given as the zero locus of

f = 256z30 − 27z4

1 − 128z20z2

2 + 144z0z21z2 + 16z0z

42 − 4z2

1z32

(see, for instance, [Te3]).

Figure 4.11. The swallowtail singularity


If π(z0, z1, z2) = (z40 , z3

1 , z22), then h = f π is homogeneous of degree 12.

Using the notation of 4.8, we find

c0 = (−1)(4 · 3 · 2 − 3 · 2) = −18.

f|w0= −27z4

1 − 4z21z3

2 = z21(−27z2

1 − 4z32).

Hence, from 4.9, we find thatχ(F

f|w0,0

) = −8.

Similarly,c1 = (−1)(4 · 3 · 2 − 4 · 2) = −16.

f|w1= 256z3

0 − 128z20z2

2 + 16z0z42 = 16z0(16z2

0 − 8z0z22 + z4

2) =

16z0(4z0 − z22)2.

χ(Ff|w1

,0) = −3.

andc2 = (−1)(4 · 3 · 2 − 4 · 3) = −12.

f|w2= 256z3

0 − 27z41 .

χ(Ff|w2

,0) = −5.

We also findf|w0,1

≡ 0,

c0,2 = 4 · 3 · 2 − 3 · 2 − 4 · 3 + 3 = 9,

f|w0,2= −27z4

1 ,

χ(Ff|w0,2

,0) = 4,

78 DAVID B. MASSEY

and

c1,2 = 4 · 3 · 2 − 4 · 2 − 4 · 3 + 4 = 8,

f|w1,2= 256z3

0 ,

χ(Ff|w0,2

,0) = 3.

Having made these calculations, it still remains for us to calculate χ(Fh,0

). We can do thisusing 4.7, provided that we know that h has a one-dimensional critical locus and provided that wecan calculate λ1

h(0). While this calculation can be made by hand, it is rather tedious; a computeralgebra program – such as Macaulay, a public domain program written by Michael Stillman andDave Bayer – can tell us that not only does h have a one-dimensional critical locus, but that themultiplicity of the Jacobian scheme at the origin is 83, i.e., λ1

h(0) = 83. Therefore,

χ(Fh,0

) = λ0h(0) − λ1

h(0) + 1 = dλ1h(0) − (d − 1)n+1 + 1 =

12 · 83 − 113 + 1 = 336.

Finally,

χ(Ff,0

) =336 − (−18)(−8) − (−16)(−3) − (−12)(−5) − 9 · 4 − 8 · 3

4 · 3 · 2 = 1.

Remark 4.12. In the above example, we resorted to a computer calculation at one point. If weare willing to use a computer algebra program at each step, then there is a much easier way tocalculate the Euler characteristic of the Milnor fibre in the case of a one-dimensional critical locus– whether the function, h, is a weighted-homogeneous polynomial or not. This is the method thatwe describe in [M-S].

Any computer program which can calculate the multiplicities of ideals in a polynomial ring,given a set of generators, can calculate the Le numbers of a polynomial. (A number of programshave this capability, but by far the most efficient that we know of is Macaulay.)

Given such a program and a polynomial, h, with a one-dimensional singular set, one proceedsas follows to calculate the Le numbers, λ0 and λ1, at the origin with respect to a generic set ofcoordinates.

As we saw in 1.16, λ1 is nothing other than the multiplicity of the Jacobian scheme of h at theorigin. So, one can have the program calculate it.

Now, we need a hyperplane that is generic enough so that its intersection number (at the origin)with the (reduced) singular set is, in fact, equal to the multiplicity of the singular set. Usually,one knows the singular set (as a set) well enough to know such a hyperplane. (Alternatively, thereare programs which can find the singular set for you – though how they present the answer isnot always helpful.) We shall assume now, in addition to having λ1, that we also have such ahyperplane, V (L), for some linear form, L .

By the work of Iomdine [Io] and Le [Le4] (or our generalization in 4.5 or [M-S]), we havethat: for all k sufficiently large, h + Lk has an isolated singularity at the origin and the Milnornumber µ(h + Lk) equals λ0 + (k − 1)λ1. But, the Milnor number is again nothing other thanthe multiplicity of the Jacobian scheme at the origin, and so we may use our program to calculate


it. Thus, we can find λ0 – provided that we have an effective method for knowing when we havechosen k large enough so that the formula of Iomdine and Le holds.

However, we have such a method. If h + Lk has an isolated singularity, let µk denote its Milnornumber. (Given a particular k, one must either check by hand whether h + Lk has an isolatedsingularity or have a program do it. Macaulay will tell you the dimension of the singular set in thecourse of calculating the multiplicity of the Jacobian scheme at the origin.) A quick look at theproof of the Iomdine-Le formula in 4.5 shows that the formula holds provided that

µk − (k − 1)λ1 k − 2.

Therefore, to find λ0, one starts with a relatively small k and checks whether µk k− 2+(k−1)λ1. If the inequality is false, pick a larger k. Eventually, the inequality will hold and then

λ0 = µk − (k − 1)λ1.

This is the same argument that we used in Remark I.3.5 in the more general setting of the Le-Iomdine-Vogel formulas.

As we also mentioned in I.3.5, there is an alternative method for calculating not only the Lenumbers but also the maximum polar ratio of h. Here, one needs to make sure that the linear formL has been chosen generically enough so that there are no exceptional pairs (see I.3.5). As we areassuming the use of a computer, all that one needs to do is select the linear form “randomly”.

Then, to find the maximum polar ratio, one calculates µk for successive values of k – lookingfor a difference of λ1. Once this occurs, k is at least the maximum polar ratio and, as before, weconclude that

λ0 = µk − (k − 1)λ1.

Note, moreover, that this method works whether λ1 equals the multiplicity of the Jacobianscheme at the origin or not. As λ1 at least the multiplicity of the Jacobian scheme at the origin,with equality being the generic case, it follows that if µk+1 − µk equals the multiplicity of theJacobian scheme at the origin, then λ1 equals the multiplicity of the Jacobian scheme at the originand λ0 = µk − (k − 1)λ1.

While this method requires one to calculate at least two Milnor numbers, µk, it will still bea more efficient way of calculating λ0 – provided that the maximum polar ratio is significantlysmaller than λ0 itself. This would be the case, for instance, if the polar curve had a large numberof components.

Consider again the swallowtail of Example 4.10 defined by

h = 256z30 − 27z4

1 − 128z20z2

2 + 144z0z21z2 + 16z0z

42 − 4z2

1z32 .

We use Macaulay to find that the multiplicity of the Jacobian scheme at the origin equals 5.(Alternatively, we know that the singularities of the swallowtail consist of a smooth curve ofordinary double points plus a multiplicity two curve of cusps; hence, the multiplicity of the Jacobianscheme at the origin = 1 + 2 · 2 = 5.) Now, using the notation above and letting L = z2 (this is“random” enough this time, but the reader is invited to check this by picking a “more random”linear form), we find

k = 2 3 4 5 6 7µk = 6 12 18 24 30 35

80 DAVID B. MASSEY

From the table, we see that the maximum polar ratio is at most 6, λ1 is, in fact, equal to5, and λ0 = 30 − (6 − 1)5 = 5. Hence, the Euler characteristic of the Milnor fibre of h equalsλ0 − λ1 + 1 = 1, which agrees with our previous calculation.

As the swallowtail is such an important singularity, one might wonder how close the Morseinequalities of Theorem 3.3 are to being equalities in this example. The answer is: not very. A.Suciu informs us that the degree 1 and 2 homology groups of the Milnor fibre of the swallowtailat the origin are both free Abelian of rank 2.

Why are the Le numbers off by so much from the Betti numbers? It is because the Le numbersrecord information that the Betti numbers do not.

In the case of the swallowtail, λ1 records the information that there is an entire cusp of cuspsingularities coming into the origin plus a line of quadratic singularities. Thus, λ1 equals ( themultiplicity of the cusp )( the Milnor number of the cusp ) plus ( the Milnor number of thequadratic singularity ) = (2)(2)+1 = 5. Now, as λ1 is forced to be 5, λ0 also has to be 5 – in orderto make λ0 − λ1 + 1 come out to equal the Euler characteristic.

Remark 4.13. For a projective hypersurface, X, defined by a homogeneous polynomial, h, in theprojective coordinates (z0 : · · · : zn), one may ask if there is some reasonable notion of global Lenumbers.

It is easy to see that if one takes affine patches on X, calculates λ0 at points of each patchwith respect to generic coordinates, and adds together the finite number of non-zero results thatone gets, then the answer is precisely λ1

h(0) (in the ordinary, affine sense) with respect to genericcoordinates. It seems reasonable, then, to define the global λi

Xto be λi+1

h (0).If one makes this definition, it might initially look as though λ0

h(0) should provide a new inter-esting invariant of X. However, Corollary 4.7 tells us that λ0

h(0) can be calculated from the higherLe numbers together with the degree of X.

We shall now prove a uniform version of the generalized Le-Iomdine formulas for one-parameter

families of germs of hypersurface singularities at the origin, i.e.,D will be an open disc about the

origin in C, U will be an open neighborhood of the origin in Cn+1 and f : (

D×U ,

D× 0) → (C, 0)

will be an analytic function; naturally, we write ft for the function defined by ft(z) := f(t, z).

First, we need a lemma

Lemma 4.14. For all i and for all p = (t0, z0, . . . , zn) near the origin such that t0 = 0,

Γift0 ,z = Γi+1

f,(t,z)∩ V (t − t0) = Γi+1

f,(t,z)· V (t − t0)

as cycles at p, regardless of how generic (t, z0, . . . , zn) may be.

Proof. Fix any good stratification, G, for f at the origin. The stratified critical values of thefunction t are isolated; hence, in a neighborhood of the origin, the map t restricted to each of thestrata of G can have only 0 as a critical value (see [Mas11, Prop. 1.3]). Therefore, for all smallt0 = 0, V (t − t0) is a prepolar slice of f at p. In particular, Σf ∩ V (t − t0) = Σ(f|V (t−t0)

).Thus,

Γift0 ,z = V

(t − t0,

∂f

∂zi, . . . ,

∂f

∂zn

)¬ Σ(ft0) =


V

(t − t0,

∂f

∂zi, . . . ,

∂f

∂zn

)¬ Σf ∩ V (t − t0) =

V

(t − t0,

∂f

∂zi, . . . ,

∂f

∂zn

)¬ Σf =

(Γi+1

f,(t,z)∩ V (t − t0)

)¬ Σf,

where the last equality uses 1.2.i.But, we claim that this equals Γi+1

f,(t,z)∩ V (t − t0) up to embedded subvariety for small t0 = 0.

For otherwise, Γi+1f,(t,z)

∩ V (t − t0) would have a component contained in Σf for an infinite numberof small t0 = 0, which would imply that Γi+1

f,(t,z)has a component contained in Σf – a contradiction

of the definition of Γi+1f,(t,z)

. Therefore, Γift0 ,z = Γi+1

f,(t,z)∩ V (t − t0) up to embedded subvariety, and

the conclusion follows easily.

As in Theorem 4.5, when we use the coordinates z = (z0, . . . , zn) for ft, we use the rotatedcoordinates z = (z1, z2, . . . , zn, z0) for ft + zj

0.

Theorem 4.15 (Uniform Le-Iomdine formulas). Let s := dim0Σf0, and suppose that s 1.Suppose that λi

ft,z(0) is defined for all i s and for all small t. Then, there exist τ > 0 and j0

such that, for all j j0 and for all t ∈Dτ , dim0Σ(f0 + zj

0) = s − 1, λift+zj

0,z(0) is defined for all

i s − 1, and

i) λ0ft+zj

0,z(0) = λ0

ft,z(0) + (j − 1)λ1

ft,z(0);

ii) λift+zj

0,z(0) = (j − 1)λi+1

ft,z(0), for 1 i s − 1;

iii) Σ(ft + zj0) = Σft ∩ V (z0) near 0.

Proof. Given 4.5, all that we must show is that λ0ft0 ,z(0)t0 is bounded for small t0. Clearly, it

suffices to show that λ0ft0 ,z(0)t0 is bounded for small t0 = 0. Of course, what we actually show

is that, for small t0 = 0, λ0ft0 ,z(0) is independent of t0.

For small t0 = 0, we may apply the lemma to conclude

Λ0ft0 ,z = Γ1

ft0 ,z · V(

∂f

∂z0

)=

Γ2f,(t,z)

· V (t − t0) · V(

∂f

∂z0

)=

(Γ1

f,(t,z)+ Λ1

f,(t,z)

)· V (t − t0).

Thus, Γ1f,(t,z)

+ Λ1f,(t,z)

has a one-dimensional component, nν [ν], which coincides with C× 0 near 0(and so, must actually be a component of Λ1

f,(t,z)) and such that λ0

ft0 ,z(0) =(nν [ν]·V (t−t0)

)(t0,0)

=nν for all small non-zero t0. The conclusion follows.

As we saw in Example 2.10, the Le numbers in a family are not individually upper-semicontin-uous. However, we do have the following.

82 DAVID B. MASSEY

Corollary 4.16. Using the notation of the theorem, the tuple of Le numbers

(λs

ft,z(0), λs−1ft,z

(0), . . . , λ0ft,z(0)

)

is lexigraphically upper-semicontinuous in the t variable, i.e., for all t small, either

λsf0,z(0) > λs

ft,z(0)

orλs

f0,z(0) = λsft,z(0) and λs−1

f0,z(0) > λs−1ft,z

(0)

or...

orλs

f0,z(0) = λsft,z(0), λs−1

f0,z(0) = λs−1ft,z

(0), . . . , λ1f0,z(0) = λ1

ft,z(0),

and λ0f0,z(0) λ0

ft,z(0).

Proof. By applying 4.15 inductively, as in 4.6, we find that, if 0 j0 j1 · · · js−1, then,for all small t, ft + zj0

0 + zj11 + · · · + z

js−1s−1 has an isolated singularity at the origin, and its Milnor

number is given byµ(ft + zj0

0 + zj11 + · · · + z

js−1s−1 ) =

λ0ft,z(0) + (j0 − 1)λ1

ft,z(0) + (j1 − 1)(j0 − 1)λ2ft,z(0) + . . .

+(js−1 − 1) . . . (j1 − 1)(j0 − 1)λsft,z(0).

Now, as the Milnor number is upper-semicontinuous, the conclusion is immediate.

Before we leave this chapter, we want to see how adding a large power of z0 affects the prepolaritycondition.

Proposition 4.17. Let G be a good stratification for h at 0 and let V (z0) be a prepolar slice withrespect to G at the origin. Suppose a = 0 and that j is such that Σ(h + azj

0) = Σh∩ V (z0) as sets.Then,

G′ = V (h + azj0) − Σh ∩ V (z0) ∪ G ∩ V (z0) | G is a singular stratum of G

is a good stratification for h + azj0 at 0.

Proof. Suppose we have pi ∈ Σh ∩ V (z0) such that pi → p ∈ G ∩ V (z0), where G is a singularstratum, and such that Tpi

V (h + azj0 − (h + azj

0)|pi) → T . We wish to show that Tp(G∩V (z0)) =

TpG ∩ TpV (z0) ⊆ T .


If T = TpV (z0) of G = 0, then we are finished. So suppose that T = TpV (z0)and G = 0.Then, for all but a finite number of i, Tpi

V (h+azj0−(h+azj

0)|pi) = Tpi

V (z0−z0(pi)) and pi ∈ Σh.Hence,

TpiV (h − h(pi), z0 − z0(pi)) = Tpi

V (h + azj0 − (h + azj

0)|pi, z0 − z0(pi)) =

TpiV (h + azj

0 − (h + azj0)|pi

) ∩ TpiV (z0 − z0(pi)) → T ∩ TpV (z0)

where, by taking a subsequence, we may assume that TpiV (h−h(pi)) approaches some hyperplane,T . As G is a good stratification for h, TpG ⊆ T . Moreover, as V (z0) transversely intersects G,

TpiV (h − h(pi), z0 − z0(pi)) → T ∩ TpV (z0)

and thus T ∩ TpV (z0) = T ∩ TpV (z0). Therefore,

TpG ∩ TpV (z0) ⊆ T ∩ TpV (z0) = T ∩ TpV (z0) ⊆ T .

Corollary 4.18. Let k 0 and suppose (z0, . . . , zk) is prepolar for h at the origin. If j is suchthat

(*) dim0Γi+1h,z ∩ V

(∂h

∂z0+ jazj−1

0

)∩ V (z1, . . . , zi) 0

for all i with 0 i k, then (z1, . . . , zk) is prepolar for h + azj0 at 0.

Proof. When i = 0, (∗) yields dim0Γi+1h,z ∩ V

(∂h∂z0

+ jazj−10

)= 0 and so, as sets,

Σ(h + azj0) = V

(∂h

∂z0+ jazj−1

0

)∩ V

(∂h

∂z1, . . . ,

∂h

∂zn

)=

V

(∂h

∂z0+ jazj−1

0

)∩

(Σh ∪ Γ1

h,z

)=

(Σh ∩ V (z0)

)∪

(Γ1

h,z ∩ V

(∂h

∂z0+ jazj−1

0

))= Σh ∩ V (z0).

Thus, the hypothesis of 4.17 is satisfied and we apply it; this leaves us with only the problem ofshowing that each successive hyperplane slice transversely intersects the smooth part, i.e., as germsof sets at the origin, for all i with 0 i k,

Σ(h + azj0|V (z1,...,zi)

) = V

(z1, . . . , zi,

∂h

∂z0+ jazj−1

0

)∩ V

(∂h

∂zi+1, . . . ,

∂h

∂zn

)=

V

(z1, . . . , zi,

∂h

∂z0+ jazj−1

0

)∩

(Σh ∪ Γi+1

h,z

)=

(Σh ∩ V (z0, . . . , zi)) ∪(

Γi+1h,z ∩ V

(∂h

∂z0+ jazj−1

0

)∩ V (z1, . . . , zi)

)

which, by (∗), equals Σh ∩ V (z0, . . . , zi).

84 DAVID B. MASSEY

Proposition 4.19. Let k 0 and suppose (z0, . . . , zk) is prepolar for h at the origin. Then, forall large j,

dim0Γi+1h,z ∩ V

(∂h

∂z0+ jazj−1

0

)∩ V (z1, . . . , zi) 0

for all i with 0 i k and so (z1, . . . , zk) is prepolar for h + azj0 at 0.

Proof. As (z0, . . . , zk) is prepolar for h, we may apply 1.26 to conclude that γi+1h,z (0) exists for all

i with 0 i k, i.e.,dim0Γi+1

h,z ∩ V (z0, z1, . . . , zi) 0.

It follows immediately thatdim0Γi+1

h,z ∩ V (z1, . . . , zi) 1.

Therefore,

dim0Γi+1h,z ∩ V

(∂h

∂z0+ jazj−1

0

)∩ V (z1, . . . , zi) 0

if and only if V(

∂h∂z0

+ jazj−10

)contains a component of Γi+1

h,z ∩ V (z1, . . . , zi) through the ori-

gin. But, if a component W of Γi+1h,z ∩ V (z1, . . . , zi) through the origin were contained in both

V(

∂h∂z0

+ j1azj1−10

)and V

(∂h∂z0

+ j2azj2−10

)for j1 = j2, then z0 would have to equal 0 along that

component – a contradiction, as dim0W ∩ V (z0) 0. The conclusion follows.


Chapter 5. LE NUMBERS AND HYPERPLANEARRANGEMENTS

The Plucker formula of Corollary 4.7 states: Let h be a homogeneous polynomial of degreed in n + 1 variables, let s = dim0Σh, and suppose that λi

h,z(0) exists for all i s. Then,∑si=0(d − 1)iλi

h,z(0) = (d − 1)n+1.This formula allows us to calculate the Le numbers for a central hyperplane arrangement in a

purely combinatorial manner from the lattice of flats of the arrangement (see [O-T] and below). Itwas experimentally observed by D. Welsh and G. Ziegler that there was a fairly trivial relationshipbetween the Le numbers of the arrangement and the Mobius function (again, see [O-T] and below).This relationship generalizes to matroid-based polynomial identities (see [MSSVWZ]).

In this chapter, we give the combinatorial characterization of the Le numbers for central hyper-plane arrangements and prove the relation between the Le numbers and the Mobius function.

A central hyperplane arrangement in Cn+1 is simply the zero-locus of an analytic function

h : Cn+1 → C where h is a product of d linear forms on C

n+1 (here, we are not necessarilyassuming that the forms are distinct). Though this may appear to be fairly trivial as a hypersurfacesingularity, this apparent simplicity is deceiving – the study of hyperplane arrangements is quitecomplex and touches on many areas of mathematics (see, for instance, [O-R], [O-S],[O-T]).

Example 5.1. Suppose we have such an h. In this case, V (h) equals the union of hyperplanes,Hii∈I

, where I is the indexing set 1, . . . , d′, each Hi occurs with some multiplicity mi :=multHi, and

∑mi = d (in particular, if h is reduced, then each mi = 1 and d′ = d).

There is an obvious good, Whitney stratification of V (h) obtained from the “flats” of thehyperplane arrangement; the collection of flats is given by w

J

J⊆I, where

wJ

:=⋂

i∈J

Hi.

If we now take the stratification SJ

J⊆I, where

SJ

= wJ−

⋃

J K

wK

,

then clearly h is analytically trivial along the strata, and therefore one has trivially a Whitneystratification. In words, the strata are intersections of the hyperplanes minus smaller intersectionsof hyperplanes.

We wish to calculate the Le numbers of h at the origin with respect to generic coordinates z.As h is analytically trivial along the strata, it is easy to see that, as sets, the Le cycles are givenby the unions of the flats of correct dimension. Hence, as cycles, for all k,

Λkh,z =

∑

dim SJ=k

aJ[w

J]

for some aJ. By 1.18, a

Jmay be calculated by taking any p ∈ S

Jand a normal slice N to S

Jin

Cn+1 at p, and then a

J= λ0

h|N(p), where we use generic coordinates. After a translation to make

the point p the origin, we see that h|N at p is again (up to multiplication by units) a product oflinear forms of degree e

J:=

∑i∈J

mi.

86 DAVID B. MASSEY

Therefore, we may use 4.7 to calculate the Le numbers of h at the origin by a downward inductionon the dimension of the flats. (In the following, it looks nicer if we suppress the subscripts.) Wedenote a hyperplane in the arrangement by H, a flat by w or v, and define

e(w) :=∑

w⊆H

multH.

Next, we define the vanishing Mobius function, η, by downward induction on the dimension ofthe flats. For a hyperplane, H, in the arrangement, define

η(H) := multH − 1;

for a smaller dimensional flat, w, 4.7 tells us that we need

η(w) := (e(w) − 1)n+1−dim w −∑

vw

η(v) · (e(w) − 1)codimv w.

This equality is equivalent to∑

v⊇w

(e(w) − 1)dim vη(v) = (e(w) − 1)n+1.

Finally, having calculated the vanishing Mobius function, one has that, for all i,

λih,z(0) =

∑

dim w=i

η(w).

By 3.3, knowing the Le numbers of the hyperplane arrangement gives us the Euler characteristicof the Milnor fibre together with Morse inequalities on the Betti numbers. (Another method forcomputing the Euler characteristic of the Milnor fibre from the data provided by the containmentrelations among the flats, i.e., by knowing the intersection lattice, is given in [O-T].)

Example 5.2. We wish to see what the above method gives us in the case of a generic centralarrangement of d hyperplanes in C

n+1 (see [O-R]). Here, “generic” means as generic as possibleconsidering that all the hyperplanes pass through the origin – that is, each hyperplane occurs withmultiplicity 1, and if w is a flat of dimension k, and k = 0, then w is the intersection of preciselyn+1−k hyperplanes of the arrangement; in terms of the above discussion, this says that if w = 0,then e(w) = n + 1 − k. We assume that d > n + 1 for, otherwise, after a change of coordinates,h = z0z1 . . . zd−1 and the Milnor fibre is diffeomorphic to the (d − 1)-fold product of C

∗’s.For a generic arrangement, it is easy to see that for all j-dimensional flats w = 0, the number of

k-dimensional flats containing w is given by(n+1−j

k−j

), provided that k j. One also knows that,

if k 1, then the number of k-dimensional flats containing the origin is given by(

dn+1−k

). This is

all the information that one needs to calculate the vanishing Mobius function, η, from the formula

η(w) := (e(w) − 1)n+1−dim w −∑

vw

η(v) · (e(w) − 1)codimv w

together with the fact that for all hyperplanes, H, in the arrangement we have η(H) = 0.It is an amusing exercise to prove that this implies that, if dimw = j = 0, then η(w) = n − j.

Alternatively, this also follows from Example 2.8. (The above is the inductive proof of the formulaof 2.8 that is referred to in that example.)


Therefore, for a generic central arrangement of d hyperplanes in Cn+1, we have with respect to

generic coordinatesλn

h(0) = 0,

λn−1h (0) =

∑

dim w=n−1

η(w) =(

d

2

)(1),

...

λih,z(0) =

∑

dim w=i

η(w) =(

d

n + 1 − i

)(n − i),

...

λ1h(0) =

∑

dim w=1

η(w) =(

d

n

)(n − 1).

So, finally,

λ0h(0) = (d − 1)n+1 −

n∑

i=1

(d − 1)iλih,z(0) =

(d − 1)n+1 −n∑

i=1

(d − 1)i

(d

n + 1 − i

)(n − i) =

(d − 1)(

d − 1n

),

where the last equality is an exercise in combinatorics.

Now, by our earlier work, since we know the Le numbers, we know the Euler characteristic ofthe Milnor fibre, F

h,0, together with Morse inequalities on the Betti numbers, bi(Fh,0

). But, inthis special case, it is not difficult to obtain the Betti numbers precisely.

By an observation of D. Cohen [Co1], if d > n+1, a generic central arrangement of d hyperplanesin C

n+1 is obtained by taking repeated hyperplane sections of a generic hyperplane arrangementof d hyperplanes in C

d. It follows that for i n − 1, bi(Fh,0) =

(d−1

i

). Therefore, we have only

to calculate bn(Fh,0

); but, since we know the Euler characteristic, this is easy, and we find – aftersome more combinatorics- that

bn(Fh,0

) = (d − n)(

d − 1n

),

which agrees with the results of [Co1] and [O-R].Note that the Morse inequalities of 3.3 can be far from equalities; for instance, the two easiest

inequalities are

(d − n)(

d − 1n

)= bn(F

h,0) λ0

h,z(0) = (d − 1)(

d − 1n

)

and

d − 1 = b1(Fh,0) λn−1

h,z (0) =(

d

2

)=

d(d − 1)2

.

88 DAVID B. MASSEY

Now, we wish to describe the relation between the Le numbers of a central arrangement andthe Mobius function – this is the result which is generalized in [MSSVWZ].

Let h be the product of d distinct linear forms on Cn+1, so that each hyperplane in the ar-

rangement V (h) occurs with multiplicity 1. Let A denote the collection of hyperplanes which arecomponents of V (h). We use the variable H to denote hyperplanes in A. We use the letters vand w to denote flats of arbitrary dimension. Finally, in agreement with our notation in 5.1, leteA(v) = the number of hyperplanes of A which contain the flat v.

As we saw in 5.1 and 5.2, the Le numbers of a central hyperplane arrangement can be describedin terms of a function ηA defined inductively on the flats by: for all H ∈ A, ηA(H) = 0, and forall flats w, ∑

w⊆v

(eA(w) − 1)dim vηA(v) = (eA(w) − 1)n+1.

The Mobius function, µA , on A (see [O-T]) is defined inductively on the flats by: µA(Cn+1) = 1and for all flats v w, ∑

flats uv⊆u⊆w

µA(u) = 0.

Here, we subscript by η, e, and µ by A because our proof is by induction on the ambientdimension, and the inductive step requires slicing A by hyperplanes, N , not contained in A. Thiswill produce new arrangements inside the ambient space N . So it is important that we indicatewhich arrangement is under consideration.

More notation now, related to the slicing. We will be taking two kinds of hyperplane slices. Nwill denote a prepolar hyperplane slice through the origin in C

n+1, i.e., a hyperplane slice whichcontains no flats of A other than the origin. We will also use normal slices to the one-flats; if vis a one-dimensional flat and pv ∈ v − 0, Nv will denote a normal slice to v at pv – that is, Nv isa hyperplane in C

n+1 which transversely intersects v at pv. We use A ∩ N to denote the obviousinduced arrangement in N (which is identified with C

n). The arrangement A ∩ Nv is consideredas a central arrangement where pv becomes the origin and all hyperplanes not containing pv areignored. Note that the number of hyperplanes in the arrangement A ∩ Nv is eA(v).

An arrangement is essential provided that the origin is a flat of the arrangement (hence, thearrangement is not trivially a product).

What we want to show is that, if A is a an essential, central hyperplane arrangement, then

ηA(0) = (d − 1)(−1)n+1µA(0) = (d − 1)|µA(0)|.

To induct, we will first need the following three easy lemmas on η, µ, which describe the effectsof slicing. We leave the first two as exercises using the inductive definitions of ηA andµA givenabove. However, we prove the third.

Lemma 5.3.

ηA∩N(0) =

ηA(0)d − 1

+∑

dim v=1

ηA(v)

and, if v is a one-dimensional flat,ηA∩Nv

(pv) = ηA(v).


Lemma 5.4.µA∩N

(0) = −∑

dim v≥2

µA(v).

and, if v is a one-dimensional flat,µA∩Nv

(pv) = µA(v).

Lemma 5.5.dµA(0) +

∑

dim v=1

(d − eA(v))µA(v) = 0.

Proof. By one of Weisner’s formulas (see Lemma 2.40 of [O-T]), for all H ∈ A,

∑

v∩H=0

µA(v) = 0.

Hence,

0 =∑

H

µA(0) +

∑

dim v=1v ⊆H

µA(v)

= dµA(0) +

∑

dim v=1

(d − e(v))µA(v).

Now, we can prove

Theorem 5.6. If A is a an essential, central hyperplane arrangement consisting of d hyperplanesin C

n+1, thenηA(0) = (d − 1)(−1)n+1µA(0) = (d − 1)|µA(0)|.

Proof. The proof is by induction on the ambient dimension. The formula is stupidly true whenn = 1.

Now, suppose the formula is true for ambient dimension n. Then,

ηA∩N(0) = (d − 1)(−1)nµA∩N

(0).

Hence, by Lemmas 5.3 and 5.4,

ηA(0)d − 1

+∑

dim v=1

ηA(v) = (d − 1)(−1)n

−∑

dim v2

µA(v)

.

This gives usηA(0)d − 1

+∑

dim v=1

ηA∩Nv(pv) = (d − 1)(−1)n+1

∑

dim v2

µA(v).

90 DAVID B. MASSEY

Using our inductive hypothesis again, we have

ηA(0)d − 1

+∑

dim v=1

(eA(v) − 1)(−1)nµA∩Nv(pv) = (d − 1)(−1)n+1

∑

dim v2

µA(v).

Therefore,

ηA(0)d − 1

+∑

dim v=1

(eA(v) − 1)(−1)nµA(v) = (d − 1)(−1)n+1∑

dim v≥2

µA(v)

and so

ηA(0) = (d − 1)(−1)n+1

(d − 1)∑

dim v≥2

µA(v) +∑

dim v=1

(eA(v) − 1)µA(v)

=

(d − 1)(−1)n+1

[

(d − 1)(− µA(0) −

∑

dim v=1

µA(v))

+∑

dim v=1

(eA(v) − 1)µA(v)

]

= (d − 1)(−1)n+1

[

µA(0) − dµA(0) −∑

dim v=1

(d − eA(v))µA(v)

]

.

Now apply Lemma 5.5.

Our inductive proof given above is somewhat unsatisfactory, for it gives us no geometric insightas to why the theorem is true. Should there be a geometric explanation for the identity in 5.6?Probably so. The result of Orlik and Solomon [O-S] is that |µA(0)| is the (n + 1)-st Betti numberof the complement of the arrangement A in C

n+1. How this could be used to prove 5.6 still escapesus.

V(f )

axis

level surfacesof f

limiting tangentplane

Figure 6.2. Failure of a along an axisf


Chapter 6. THOM’S af CONDITION

In this chapter, we will use the Le numbers to provide conditions under which a submanifold ofaffine space satisfies Thom’s af condition with respect to the ambient stratum. Let us recall thedefinition of the af condition (see [Mat]).

Definition 6.1. Let U be an open subset of some affine space, let f : U → C be an analyticfunction, and let M ⊆ V (f) be a submanifold of U . Thom’s af condition is satisfied betweenU − Σf and M (or along M , or by the pair (U − Σf, M)) if, whenever pi ∈ U − Σf , pi → p ∈ M ,and TpiV (f − f(pi)) → T , then TpM ⊆ T .

We are about to prove five different results – all of the form: if the Le numbers are constant,then Thom’s af condition holds. The order in which we must prove these results is interesting.

First, we give a proof of Le and Saito’s result that a constant Milnor number at the origin in aone-parameter family implies the Thom condition along the parameter axis. We then use Le andSaito’s result, combined with the generalized Le-Iomdine formulas, to prove that the constancyof the Le numbers at the origin in a one-parameter family implies the Thom condition along theparameter axis [Mas14]. We use this parameterized version to prove a non-parameterized version:if the Le numbers of a single function are constant along a submanifold, then Thom’s conditionis satisfied along the submanifold. This non-parameterized version allows us to prove a multi-parameter version of Le and Saito’s result: if we have a family of isolated hypersurface singularitieswith constant Milnor number parameterized along a submanifold, then that submanifold satisfiesThom’s af condition with respect to the ambient stratum. Finally, we use this last result to proveour best result – the multi-parameter version of the Le number result above: if we have a familyof hypersurface singularities with constant Le numbers parameterized along a submanifold, thenthat submanifold satisfies Thom’s af condition with respect to the ambient stratum.

92 DAVID B. MASSEY

In all of the results described above, it is extremely important that our assumptions on thegenericity of the coordinate system will be solely that the Le numbers exist. This is adimensional requirement which is very easy to check. This should be contrasted with the resultsof [H-M] and [HMS].

First, we need a well-known lemma.

Lemma 6.3. LetD be an open disc about the origin in C, let U be an open neighborhood of the

origin in Cn+1, and let f : (

D × U ,

D × 0) → (C, 0) be an analytic function; we write ft for the

function defined by ft(z) := f(t, z). Suppose that dim0Σf0 = 0.Then, for all small t, the Milnor number of ft at the origin is independent of t if and only if

there exists an open neighborhood, W, of the origin inD × U such that W ∩ V

(∂f∂z0

, . . . , ∂f∂zn

)=

W ∩ (C × 0).

Proof. There are many proofs of this fact. We shall use intersection numbers.

The Milnor number of f0 at the origin, µ0(f0), equals the multiplicity of the origin in the cycle[V

(∂f0∂z0

, . . . , ∂f0∂zn

)]. Because f0 has an isolated critical point at the origin, V

(t, ∂f

∂z0, . . . , ∂f

∂zn

)is

a local complete intersection, and so we have an equality of cycles[V

(∂f0

∂z0, . . . ,

∂f0

∂zn

)]=

[V

(t,

∂f

∂z0, . . . ,

∂f

∂zn

)]= [V (t)] ·

[V

(∂f

∂z0, . . . ,

∂f

∂zn

)].

Therefore,µ0(f0) =

(V (t) · V

(∂f

∂z0, . . . ,

∂f

∂zn

))

0

=∑

p∈Bε

(V (t − η) · V

(∂f

∂z0, . . . ,

∂f

∂zn

))

p

= µ0(fη) + R,

whereBε is a sufficiently small open ball around the origin in C

n+2, η is chosen small with respectto ε – that is, there exists δε > 0 such that we may use any η satisfying 0 < |η| < δε – and R denotesthe sum of the remaining terms, i.e., the terms coming from points p which are not in C×0. Note

that the sum is actually finite since we are really summing over p ∈Bε∩V (t−η)∩V

(∂f∂z0

, . . . , ∂f∂zn

).

As all the intersection numbers are non-negative, R being zero is equivalent to there being no

remaining terms, i.e., equivalent to (η,0) being the only point inBε ∩V (t− η)∩V

(∂f∂z0

, . . . , ∂f∂zn

).

The desired conclusion follows immediately, where the set W in the statement can be taken to

beBε ∩

( Dδε

× U).

Recall now the result of Le and Saito [Le-Sa] as stated in the introduction.

Theorem 6.4 (Le-Saito [Le-Sa]). LetD be an open disc about the origin in C, let U be an open


neighborhood of the origin in Cn+1, and let f : (

D×U ,

D×0) → (C, 0) be an analytic function; we

write ft for the function defined by ft(z) := f(t, z).Suppose that dim0Σf0 = 0 and that, for all small t, the Milnor number of ft at the origin is

independent of t. Then,D × 0 satisfies Thom’s af condition at the origin with respect to the

ambient stratum, i.e., if pi is a sequence of points inD × U − Σf such that pi → 0 and such that

TpiV (f − f(pi)) converges to some T , then C × 0 = T0(D × 0) ⊆ T .

Proof. We begin by noting that the existence of good stratifications as given in Proposition 1.22

implies that Thom’s af is satisfied, near the origin, byD × 0 − 0 with respect to the ambient

stratum.

Now, consider the blow-up ofD × U by the Jacobian ideal of f :

BlJ(f)

( D × U

)⊆

( D × U

)× P

n+1

π1 π2

D × U P

n+1

We first wish to show that the fibre π−11 (0) has dimension at most n.

The point q := [1 : 0 : · · · : 0] ∈ Pn+1 corresponds to the hyperplane V (t). As µ0(ft) is

independent of t, the lemma implies that, in a neighborhood of the origin, π1(π−12 (q)) ⊆

D × 0.

However, as we noted above, the af condition holds generically onD × 0. Therefore, near 0,

either π1(π−12 (q)) is empty or consists only of the origin. But, the dimension of every component

of π−12 (q) is at least dim BlJ(f)

( D × U

)− dim P

n+1 = n + 2 − (n + 1) = 1. Thus, 0 ∈ π1(π−12 (q)),

i.e., q ∈ π2(π−11 (0)). It follows that π−1

1 (0) is a proper subset of Pn+1 and, hence, has dimension

at most n.But, every component of the exceptional divisor E := π−1

1 (Σf) has dimension n+1. Therefore,above an open neighborhood of the origin, E equals the topological closure of E−π−1

1 (0), which is

contained in (D×0)×(0×P

n) since the af condition holds generically on the t-axis. It follows

that π2(π−11 (0)) ⊆ 0 × P

n, i.e., that the af condition holds alongD × 0 at the origin.

Our first generalization of the result of Le and Saito is:

Theorem 6.5. LetD be an open disc about the origin in C, let U be an open neighborhood of the

origin in Cn+1, and let f : (

D × U ,

D × 0) → (C, 0) be an analytic function; we write ft for the

function defined by ft(z) := f(t, z).Let s = dim0Σf0. Suppose that, for all small t, for all i with 0 i s, λi

ft,z(0) is defined and

is independent of t. Then,D × 0 satisfies Thom’s af condition at the origin with respect to the

ambient stratum, i.e., if pi is a sequence of points inD × U − Σf such that pi → 0 and such that

TpiV (f − f(pi)) converges to some T , then C × 0 = T0(D × 0) ⊆ T .

94 DAVID B. MASSEY

Proof. The proof is by induction on s. For s = 0, the theorem is exactly that of Le and Saito in6.4.

Now, suppose that s 1 and that, for all small t, for all i with 0 i s, λift,z

(0) is defined and

is independent of t, but that there exists a sequence pi of points inD×U −Σf such that pi → 0,

such that TpiV (f − f(pi)) converges to some T , and T0(

D × 0) ⊆ T .

As the collection of such limiting T is analytic, we may apply the curve selection lemma (see[Loo]) to conclude that there exists a real analytic curve

α : [0, ε) → 0 ∪ (D × U − Σf)

such that α(u) = 0 if and only if u = 0 and such that

limu→0

Tα(u)V (f − f(α(u))) = T .

As α is real analytic, it is trivial to show that, for all large j,

limu→0

grad(f + zj0)|α(u)

| grad(f + zj0)|α(u)

|= lim

u→0

grad(f)|α(u)

| grad(f)|α(u)| .

Therefore, for all large j, the family ft + zj0 also has T as a limit to level hypersurfaces, i.e.,

D× 0

does not satisfy the af+zj0

condition at the origin with respect to the ambient stratum.However, λ0

ft,z(0) is independent of t and, applying Theorem 4.5, if j 2 + λ0

ft,z(0), then the

family ft + zj0 has Le numbers independent of t, and f0 + zj

0 has a critical locus of dimension s− 1.Thus, our inductive hypothesis contradicts the previous paragraph.

Corollary 6.6. Let h : U → C be an analytic function on an open subset of Cn+1, let z =

(z0, . . . , zn) be a linear choice of coordinates for Cn+1, let M be an analytic submanifold of V (h),

let q ∈ M , and let s denote dimqΣh.If, for each i such that 0 i s, λi

h,z(p) is defined and is independent of p, for all p ∈ M nearq, then M satisfies Thom’s ah condition at q with respect to the ambient stratum; that is, if qi isa sequence of points in U −Σh such that qi → q and such that Tqi

V (h−h(qi)) converges to someT , then TqM ⊆ T .

Proof. This follows from 6.5 by a fairly standard trick. Let c(t) be a smooth analytic path in Msuch that c(0) = q. If we can show that any limiting tangent plane, T , contains the tangent tothe image of c at q, then we will be finished.

So, take such a c, and suppose that we have a sequence of points, qi, in U−Σh such that qi → qand such that Tqi

V (h − h(qi)) → T .Define f(t, z) := h(z + c(t)), and consider the sequence of points (0,qi − q). If one now applies

the theorem, the result is that c′(0) ⊆ T ; we leave the details to the reader.

Remark 6.7. It is important to note that, in 6.6, we only require that the coordinates are genericenough so that the Le numbers are defined; we are not requiring that the coordinates are prepolar.

On the other hand, Corollary 6.6 tells us how we can obtain good stratifications: if we have ananalytic stratification of V (h) such that the Le numbers are defined and constant along the strata,


then the stratification is actually a good stratification. However, there is no guarantee that thecoordinates used to define the Le numbers are prepolar with respect to this good stratification.

Now we can prove the multi-parameter version of the result of Le and Saito.

Theorem 6.8. Let M be an open neighborhood of the origin in Ck, let U be an open neighborhood

of the origin in Cn+1, and let f : (M × U , M × 0) → (C, 0) be an analytic function; we write ft

for the function defined by ft(z) := f(t, z), where t ∈ M and z ∈ U .Suppose that dim0Σf0 = 0 and that, for all t near the origin, the Milnor number of ft at the

origin is independent of t. Then, M × 0 satisfies Thom’s af condition at the origin with respectto the ambient stratum, i.e., if pi is a sequence of points in M × U − Σf such that pi → 0 andsuch that Tpi

V (f − f(pi)) converges to some T , then T0(M × 0) ⊆ T .

Proof. If the constant value of the Milnor number is 0, then there is nothing to prove. Note,though, that if the constant value of the Milnor number is non-zero, then it follows from Sard’stheorem that M ×0 ⊆ Σf , i.e., the critical points of the ft are not merely a result of critical pointsof the map t restricted to the smooth part of V (f).

Let a be an element of M near the origin. The Milnor number of fa at the origin satisfies theequality

µ0(fa) =[V

(t0 − a0, . . . , tk−1 − ak−1,

∂f

∂z0, . . . ,

∂f

∂zn

)]

(a,0)

.

In particular,

dim(a,0)V

(t0 − a0, . . . , tk−1 − ak−1,

∂f

∂z0, . . . ,

∂f

∂zn

)= 0.

This immediately implies that dim(a,0)Σf k. Hence,

[V

(∂f

∂z0, . . . ,

∂f

∂zn

)]= Γk

f,(t,z) + Λkf,(t,z),

both γkf,(t,z)(a,0) and λk

f,(t,z)(a,0) exist, and

µ0(fa) = γkf,(t,z)(a,0) + λk

f,(t,z)(a,0).

As µ0(fa) is independent of a, and both γkf,(t,z)(a,0) and λk

f,(t,z)(a,0) are upper-semicontinuousas functions of a, we conclude that both γk

f,(t,z)(a,0) and λkf,(t,z)(a,0) are independent of a.

This implies that γkf,(t,z)(a,0) is independent of a for (a,0) in a k-dimensional component of

Σf . But, Γkf,(t,z) cannot contain a component of Σf . Therefore, the constant value of γk

f,(t,z)(a,0)for a ∈ M must be 0; that is, Γk

f,(t,z) does not intersect M × 0 near the origin.But, all the lower-dimensional relative polar cycles are contained in Γk

f,(t,z); thus, none of themhit M × 0. This implies that λi

f,(t,z)(a,0) = 0 for all a ∈ M and all i with 0 i k − 1. As wealready saw that λk

f,(t,z)(a,0) is independent of a ∈ M , we see that all the Le numbers of f areconstant along M × 0.

Now, apply Corollary 6.6.

96 DAVID B. MASSEY

Finally, we have the multi-parameter generalization of the result of Le and Saito, where thecritical loci may have arbitrary dimension.

Theorem 6.9. Let M be an open neighborhood of the origin in Ck, let U be an open neighborhood

of the origin in Cn+1, and let f : (M × U , M × 0) → (C, 0) be an analytic function; we write ft

for the function defined by ft(z) := f(t, z), where t ∈ M and z ∈ U .Let s = dim0Σf0. Suppose that, for all small t, for all i with 0 i s, λi

ft,z(0) is defined and

is independent of t. Then, M × 0 satisfies Thom’s af condition at the origin with respect to theambient stratum, i.e., if pi is a sequence of points in M ×U −Σf such that pi → 0 and such thatTpi

V (f − f(pi)) converges to some T , then T0(M × 0) ⊆ T .

Proof. The proof is by induction on s. To obtain 6.9 from 6.8, one follows word for word ourderivation of 6.5 from 6.4.


Chapter 7. ALIGNED SINGULARITIES

In this chapter, we once again consider analytic functions h : U → C. We wish to investigatethose h for which the critical locus, Σh, is of a particularly nice form – a form which generalizesisolated singularities, smooth one-dimensional singularities (line singularities), and the singularitiesfound in hyperplane arrangements (see Chapter 5).

The obvious generalization of merely requiring the irreducible components of Σh to be smoothappears to be too general to yield nice results; what one would like is to put some restrictions onthe subset of Σh where h fails to be “equisingular”. For instance, in the case where Σh is smoothand 2-dimensional, any reasonable notion of equisingularity could fail on a subset of dimension atmost one; a reasonable condition to impose is that this one-dimensional subset itself be smooth.Essentially this is what we require of an aligned singularity.

For convenience, throughout this section, we concentrate our attention on hypersurface germsat the origin.

Definition 7.1. If h : (U ,0) → (C, 0) is an analytic function, then an aligned good stratificationfor h at the origin is a good stratification for h at the origin in which the closure of each stratumof the singular set is smooth at the origin.

If such an aligned good stratification exists, we say that h has an aligned singularity at theorigin.

If Sα is an aligned good stratification for h at the origin, then we say that a linear choice ofcoordinates, z, is an aligning set of coordinates for Sα provided that for each i, V (z0, . . . , zi−1)transversely intersects the closure of each stratum of dimension at least i at the origin. Naturally,we say simply that a set of coordinates, z, is aligning for h at the origin provided that there existsan aligned good stratification for h at the origin with respect to which z is aligning.

Note that, given an aligned singularity, aligning sets of coordinates are generic (in the IPZ-topology) and prepolar. It is important that, in fact, aligning coordinates are prepolar at allpoints in an entire neighborhood of the origin; the importance of this fact stems from the followingresult (in which we are not assuming that h has an aligned singularity).

Theorem 7.2. If the coordinates z are prepolar at all points in a neighborhood Ω of a point p withrespect to a good stratification Sα for h at p, then, inside Ω, for all i,

|Λih,z

| ⊆⋃

dim0Sαi

Sα.

Proof. When i = 0, we must show that if p is in Λ0h,z

, then p is also a stratum. Suppose not.Then, p is in some stratum S of dimension at least 1 and, as p ∈ Λ0

h,z, we must have p ∈ Γ1

h,z.

As our coordinates are prepolar, V (z0 − p0) transversely intersects S at p. This, however, is acontradiction, since p ∈ Γ1

h,zimplies that there is a sequence of limiting tangent planes to level

hypersurfaces which converges to 0 ×Cn at p and, hence, TpS should be contained in 0 ×C

n

since S is a good stratum.

98 DAVID B. MASSEY

Now, suppose that we have a point q ∈ Ω such that

q ∈ |Λih,z

| and q ∈⋃

dim0Sαi

Sα.

Then, q ∈ |Λih,z

| ∩ Sβ for some good stratum Sβ of dimension at least i + 1. As z is prepolar at q,

S ′ := Sα ∩ V (z0 − q0, . . . , zi−1 − qi−1)α

is a good stratification for h|V (z0−p0,...,zi−1−pi−1)at q; in addition, V (z0 − p0, . . . , zi−1 − pi−1)

transversely intersects Sβ at q in a set of dimension at least 1. Thus, S ′ does not contain q asa stratum.

Now, let z := (zi, . . . , zn). Then, z is prepolar with respect to S ′ and so, we conclude from thei = 0 case (at the beginning of the proof) that

q ∈ Λ0h|V (z0−p0,...,zi−1−pi−1)

,z.

By repeated applications of Theorem 1.26 and Proposition 1.18, it follows that q ∈ Λih,z

; this is acontradiction.

Closely related to the notion of aligning sets of coordinates is

Definition 7.3. If h : (U ,0) → (C, 0) is an analytic function on an open subset of Cn+1, then a

linear choice of coordinates, z, for Cn+1 is pre-aligning for h at the origin provided that for each Le

cycle, Λih,z

, and for each irreducible component, C, of Λih,z

passing through the origin, the followingconditions are satisfied:

i) dim0C = i;

ii) C is smooth at the origin;

iii) V (z0, z1, . . . , zi−1) transversely intersects C at the origin.

Proposition 7.4. If h has an aligned singularity at the origin, then for a generic linear reorgani-zation of the coordinates z, z is pre-aligning for h at the origin, and for all p near the origin, thereduced Euler characteristic of the Milnor fibre of h at p is given by

χ(Fh,p

) =s∑

i=0

(−1)n−iλih,p(0).

Proof. One may simply choose z to be aligning. Theorem 7.2 then implies that z is pre-aligning.The Euler characteristic statement follows at once from Theorem 3.3.

Remark 7.5. It is tempting to think that if z is a set of pre-aligning coordinates for h at the origin,then we can produce an aligned good stratification by considering the components of

Λih,z

−⋃

ji−1

Λjh,z

.


This might seem reasonable in light of Corollary 6.6 and Remark 6.7. However, we see no reasonfor the higher Le numbers to be constant along these proposed “strata”.

We could define a more restricted class of singularities – super aligned singularities – by requiringthe existence of an aligned good stratification in which, for each i with 0 i dim0Σh =: s, thereis at most one connected stratum, say Si, of dimension i and

S0 ⊆ S1 ⊆ · · · ⊆ Ss−1 ⊆ Ss .

It is easy to see that this is equivalent to the existence of a set of pre-aligning coordinates, z,for h at the origin such that each Λi

h,zhas a single smooth component at the origin and, as germs

of sets at the origin,Λ0

h,z⊆ Λ1

h,z⊆ · · · ⊆ Λs

h,z.

For such super aligned singularities, we obtain a good stratification for h by taking the stratification

Λj+1h,z

− Λjh,z

.

Our main interest in aligned singularities is due to

Proposition 7.6. Suppose that h has an aligned s-dimensional singularity at the origin andthat the coordinates z are aligning. Then, the Le cycles and Le numbers can be characterizedtopologically in the following inductive manner:

As a set, Λsh,z

equals the union of the s-dimensional components of the singular set of h. Todetermine the Le cycle, to each s-dimensional component, C of Σh, we assign the multiplicitymC = (−1)n−sχ(F

h,p) for generic p ∈ C, where F

h,pdenotes the Milnor fibre of h at p and χ is

the reduced Euler characteristic. Moreover, for all p ∈ |Λsh,z

|, λsh,z

(p) =∑

p∈C mC .

Now, suppose that we have defined the Le numbers, Λih,z

(p) for all i k + 1 and for all p nearthe origin.

Then, as a set, Λkh,z

equals the closure of the k-dimensional components of the set of pointsp ∈ V (h) where

χ(Fh,p

) =s∑

i=k+1

(−1)n−iλih,z(p).

The Le cycle is defined by assigning to each irreducible component C of this set the multiplicity

mC = (−1)n−k

(

χ(Fh,p

) −s∑

i=k+1

(−1)n−iλih,z(p)

)

,

for generic p ∈ C. Finally, for all p ∈ |Λkh,z

|, we have λkh,z

(p) =∑

p∈C mC .

Proof. As the aligning coordinates are prepolar at each point near the origin, this essentially followsfrom Theorem 3.3. However, in writing that

λkh,z

(p) =∑

p∈CmC ,

100 DAVID B. MASSEY

we are crucially using that the components of the Le cycle are smooth and tranversely inter-sected by V (z0 − p0, . . . , zk−1 − pk−1). If this were not the case, the intersection multiplicities ofV (z0 − p0, . . . , zk−1 − pk−1) with the components of the Le cycles would enter the picture, and thecharacterization would no longer be purely topological.

The following two corollaries are immediate:

Corollary 7.7. If h has an aligned singularity at the origin, then all aligning coordinates zdetermine the same Le cycles and Le numbers.

Corollary 7.8. Let f and g be reduced, analytic germs with aligned singularities at the origin inC

n+1. Let z and z be aligning sets of coordinates for f and g, respectively. If H is a local, ambienthomeomorphism from the germ of V (f) at the origin to the germ of V (g) at the origin, then asgerms of sets at the origin,

H(Λif,z

) = Λig,z

,

for all i, and for all p near the origin in Cn+1,

λif,z

(p) = λig,z

(H(p)),

for all i.

Now, we will give an amusing application of the results of this section. In [Z], Zariski conjecturesthat the multiplicity of a hypersurface at a point is an invariant of the local, ambient topologicaltype of the hypersurface. A number of people have concentrated on the case of a one-parameterfamily of isolated singularities, but even this case has not been settled (however, for families ofquasi-homogeneous isolated singularities, the proof of the conjecture has been given by Greuel[Gr1] and O’Shea [O’S]).

In our paper [Mas13], we prove a result which perhaps supplies a better place to look forcounterexamples to the Zariski Multiplicity Conjecture; we prove, for families of hypersurfaces ofdimension unequal to 2, that the Zariski Multiplicity Conjecture is true for families of hypersurfaceswith isolated singularities if and only if it is true for families of hypersurfaces with smooth one-dimensional critical loci. The results of this section allow us to generalize this.

Theorem 7.9. The following are equivalent:

i) for all n 3, the Zariski Multiplicity Conjecture is true for families of reduced analytichypersurfaces ft : (U ,0) → (C, 0), where U is an open subset of C

n+1 and dim0Σft = 0;

ii) for all n 3, there exists a k such that the Zariski Multiplicity Conjecture is true for familiesof reduced analytic hypersurfaces ft : (U ,0) → (C, 0) with aligned singularities, where U is an opensubset of C

n+1 and dim0Σft = k;

iii) for all n 3, for all k, the Zariski Multiplicity Conjecture is true for families of reducedanalytic hypersurfaces ft : (U ,0) → (C, 0) with aligned singularities, where U is an open subset ofC

n+1 and dim0Σft = k.


Proof. Certainly, iii) implies ii). We will show that ii) implies i) and that i) implies iii).Suppose that ii) is true for some k 1. Let ft : (U ,0) → (C, 0) be an analytic family, where U

is an open subset of Cn+1, n 3, dim0Σft = 0, and such that the local ambient topological type

of the hypersurfaces V (ft) at the origin is independent of t. Then clearly, the family

ft : (U × Ck,0) → (C, 0)

defined by ft(z,w) := ft(z) is a family of aligned singularities of dimension k with constanttopological type.

Hence, by ii), mult0 ft is independent of t. Now, as mult0 ft clearly equals mult0 ft, we arefinished with the implication that ii) implies i).

The interesting implication is, of course, that i) implies iii). Ideally, we would like to be ableto select linear coordinates, z, for C

n+1 which are aligning for ft at the origin for all small t;however, a proof that this is possible seems problematic. We will avoid needing such a result bybeing somewhat devious and applying the Baire Category Theorem.

Suppose that i) is true, and that we have a family of reduced analytic hypersurfaces ft : (U ,0) →(C, 0) with aligned singularities, where U is an open subset of C

n+1, n 3, dim0Σft = k, and suchthat the local ambient topological type of the hypersurfaces V (ft) at the origin is independent oft.

Let tm be an infinite sequence in C which approaches 0, e.g., tm = 1m . For each tm, there exists

a generic subset of PGL(Cn+1) representing aligned coordinates for ftm. We may apply the Baire

Category Theorem to conclude that there exists a choice of coordinates, z, which is aligning for f0

and for ftm for all m. Let us fix such a choice of coordinates.Then, by 7.8, the Le numbers λi

f0,z(0) are equal to the Le numbers λi

ftm ,z(0) for all large m.

By 4.6, if we take 0 j0 j1 · · · js−1, then f0 + zj00 + zj1

1 + · · · + zjs−1s−1 has an isolated

singularity at the origin; this implies that, for all small t, ft + zj00 + zj1

1 + · · ·+ zjs−1s−1 has, at worst,

an isolated singularity at the origin. In addition, 4.6 tells us that f0 + zj00 + zj1

1 + · · · + zjk−1k−1 has

the same Milnor number at the origin as ftm + zj00 + zj1

1 + · · ·+ zjk−1k−1 for all large m. As the Milnor

number at the origin in the family ft + zj00 + zj1

1 + · · · + zjk−1k−1 is upper-semicontinuous, it follows

that, in fact, the Milnor number in this family is independent of t for all small t.Hence, by [L-R], the local, ambient topological type is independent of t in the family ft + zj0

0 +zj11 + · · ·+ z

jk−1k−1 . Since we are assuming i), this implies that the multiplicity is independent of t in

the family ft + zj00 + zj1

1 + · · · + zjk−1k−1 . Finally, as the jm’s are arbitrarily large, this implies that

the multiplicity is independent of t in the family ft.

Figure 8.2. The fibres over the disc

102 DAVID B. MASSEY

Chapter 8. SUSPENDING SINGULARITIES

In [Ok], [Sak], and [Se-Th], the general question is addressed of how the structure of theMilnor fibre of f(z) + g(w) (where z and w are disjoint sets of variables) depends on the Milnorfibres of f and g. However, in each of these papers, f and g have isolated singularities or arequasi-homogeneous. Sakamoto remarks at the end of his paper that, by using Le’s notion of agood stratification, he can prove his main lemmas without the isolated singularity assumptions.As we crucially need this result in the special case of f(z) + wj , we will use our results in theappendix to indicate how one needs to modify Sakamoto’s proof.

After we describe the homotopy-type of the Milnor fibre of f(z)+wj , we will use this descriptionto give a new generalization of the formula of Le and Iomdine [Le4] – a different generalizationthan the formulas of Chapter 4. This new generalization appears in [Mas12].

Proposition 8.1. If j 2, then up to homotopy, the Milnor fibre of h(w, z) := h(z) + wj at theorigin is the one-point union (wedge) of j − 1 copies of the suspension of the Milnor fibre of h atthe origin.

Proof. By Proposition C.14, we may use neighborhoods of the form Dω ×Bε, 0 < ω ε, to definethe Milnor fibre of h at the origin.

Now, for 0 < |ξ| ω ε, consider the map

(Dω × Bε

)∩ V (h + wj − ξ) w−−−→ Dω.

This map is a proper, stratified submersion above all points of Dω −V (wj − ξ), i.e., except at the jroots of ξ. Thus, except above these j points, the fibre is the same as that above 0, which is clearlynothing more than the Milnor fibre of h at the origin. In addition, above each point of V (wj − ξ),the fibre is Bε ∩ V (h), which is contractible.

In fact, around each point α1, . . . , αj in V (wj − ξ) ⊆ Dω, there is an arbitrarily small discDαi

⊆ Dω above which the total space is contractible. We choose the Dαidisjoint. Connect all of

the Dαito the origin by disjoint paths. Let P denote the subset of Dω consisting of the paths; so,

P is a contractible set which has exactly one point in common with each of the Dαi , and the fibreabove each point of P has the homotopy-type of the Milnor fibre of h at the origin.


The result now follows easily. For the details, we refer the reader to Sakamoto [Sak] – theremainder of our proof now follows his exactly.

We shall now use 8.1 to give our second generalization of the formula of Le and Iomdine.

Let U be an open neighborhood of the origin in Cn+1, let h : (U ,0) → (C, 0) be an analytic

function, and suppose that the linear from L : Cn+1 → C is prepolar with respect to h at the

origin.The formula of Le and Iomdine says that, if dim0Σh = 1, then, for all large j, h + Lj has an

isolated singularity at the origin and

bn(h + Lj) = µ(h + Lj) = bn(h) − bn−1(h) + j∑

ν

mνδν(h),

where bi() denotes the i-th Betti number of the Milnor fibre of a function at the origin, µ denotesthe Milnor number of the isolated singularity at the origin, the summation is over all components,ν, of Σh, mν is the local degree of L restricted to ν at the origin, and δν(h) is the Milnor numberof a generic hyperplane slice of h at a point p ∈ ν − 0 sufficiently close to the origin.

This formula has, at least, two possible generalizations. One generalization is in terms of Lenumbers, as given in Chapter 4. But, while there are Morse inequalities between the Le numbersand the Betti numbers of the Milnor fibre, the Le numbers are not themselves (generally) Bettinumbers of the Milnor fibre. So, one might ask for a generalization of the formula of Le andIomdine which generalizes the Betti number information.

In remainder of this chapter, we prove that, if dim0Σh = s 1, then, for all large j, dim0Σ(h+Lj) = s − 1 and

bn(h + Lj) = bn(h) − bn−1(h) + j(bn−1(h|V (L)

) − γ1h,L

(0)).

In the case where h has a one-dimensional critical locus at the origin, it is easy to show that thisnew formula reduces to that of Le and Iomdine.

We consider this new Le-Iomdine formula interesting because it implies that

bn−1(h|V (L)) γ1

h,L(0).

In terms of deformations, this says that the top possible non-zero Betti number of the Milnor fibreof h|V (L)

is greater than or equal to γ1h,L

(0) for all h which have V (L) as a prepolar slice. Thus, ifwe define h to be a prepolar deformation of h|V (L)

precisely when V (L) is a prepolar slice of h, weobtain a class of deformations of h|V (L)

which give lower bounds on the top Betti number of theMilnor fibre. This also suggests that it might be useful to study prepolar deformations for whichγ1

h,L(0) obtains its maximum value.

We will need

Proposition 8.3. If j 2 and S is a good stratification for h at the origin, then

V (h + wj) − Σ(h + wj) ∪ 0 × S | S is a singular stratum of S

104 DAVID B. MASSEY

is a good stratification for h + wj at the origin.

Proof. As j 2, Σ(h + wj) = 0 × Σh.

Let p = (0,q) ∈ 0 × Σh, where S ∈ S, and let pi = (ui,qi) be a sequence of points inC × U − 0 × Σh such that pi → p and

TpiV (h + wj − (h + wj)|pi

) → T .

We wish to show that Tp(0 × S) = 0 × TqS ⊆ T .If T = TpV (w) = 0 × C

n+1, then we are finished. So, suppose otherwise. Then, by taking asubsequence, we may assume that qi ∈ Σh and that

TqiV (h − h(qi)) → η.

As T transversely intersects TpV (w), TpiV (h+wj−(h+wj)|pi) transversely intersects TpiV (w−

wi) for all pi close to p. Thus,

T ∩ (0 × Cn+1) = lim TpiV (h + wj − (h + wj)|pi

) ∩ TpiV (w − wi) =

lim TpiV (h + wj − (h + wj)|pi

, w − wi) = lim TpiV (h − h(qi), w − wi) = 0 × η.

Now, as S is a good stratum for h, Tq(S) ⊆ η and the proposition follows.

Corollary 8.4. If V (z0) is a prepolar slice for h at 0 then, for all j 2 + λ0h,z(0), V (z0 −w) is a

prepolar slice for h + wj at 0.

Proof. In light of the proposition, all that we must show is that, for all j 2 + λ0h,z(0), Σ(h +

wj) ∩ V (z0 − w) = Σ(h + wj|V (z0−w)

).Now,

Σ(h + wj) ∩ V (z0 − w) = (0 × Σh) ∩ V (z0 − w) = 0 × (Σh ∩ V (z0)).

On the other hand,

Σ(h + wj|V (z0−w)

) = (C × Σ(h + zj0)) ∩ V (z0 − w).

But, near the origin and for j 2 + λ0h,z(0), Σ(h + zj

0) = Σh ∩ V (z0) by 4.3.iii. The conclusionfollows.

Theorem 8.5. If V (z0) is a prepolar slice of h at 0 then, for all j 2 + λ0h,z(0),

bn(h + zj0) = bn(h) − bn−1(h) + j

(bn−1(h|V (z0)

) − γ1h,z0

(0)),

where bi() denotes the i-th Betti number of the Milnor fibre at the origin.In particular, bn−1(h|V (z0)

) γ1h,z0

(0).


Proof.After applying Proposition 3.1 to h+wj and the slice V (z0−w), and considering the long exact

sequence of the pair, we have

bn+1(h + wj) − bn(h + wj) + bn(h + zj0) =

(Γ1

h+wj,z0−w· V (h + wj)

)

0

which, by 4.3.v, equals jλ0h,z(0).

Now, as the Milnor fibre of h+wj has the homotopy-type of the one-point union of j − 1 copiesof the suspension of the Milnor fibre of h, we obtain

(j − 1)bn(h) − (j − 1)bn−1(h) + bn(h + zj0) = jλ0

h,z(0).

Using 3.1 on h itself and rearranging, we get

bn(h + zj0) = bn(h) − bn−1(h) + j

[λ0

h,z(0) −((

Γ1f,z0

· V (f))0− bn−1(h|V (z0)

))]

.

Finally, using the formula of Proposition 1.20 that(Γ1

h,z0· V (h)

)

0= γ1

h,z0(0) + λ0

h,z0(0),

we obtain the desired result.

The result of Theorem 8.5 is best thought of in terms of prepolar deformations: every prepolardeformation, h, of a fixed h0 yields a lower bound on the top Betti number of the Milnor fibre ofh0.

One might hope that, by considering a prepolar deformation, h, for which γ1h,z

(0) obtains itsmaximal value, one would actually obtain the top Betti number of the Milnor fibre of h0. This seemsunlikely however; certain singularities seem to be “rigid” with respect to prepolar deformations, inthe weak sense that any prepolar deformation, h, has no polar curve at the origin.

Nonetheless, the lower bounds provided by prepolar deformations give new data which helpsdescribe the Milnor fibre of a completely general affine hypersurface singularity; these data do notappear to follow from our Morse inequalities between the Betti numbers of the Milnor fibre andthe Le numbers of the hypersurface, as given in Theorem 3.3. As part of these Morse inequalities,we showed that λ0

h0,z(0), provides an upper-bound on the top Betti number of the Milnor fibre ofh0. Also, it follows from 1.18 that if h is a prepolar deformation of h0, then

λ0h0,z(0) = λ1

h,z(0) + γ1h,z

(0).

Thus, given a prepolar deformation, h, of h0, we have bounded the top Betti number of theMilnor fibre of h0:

γ1h,z

(0) bn−1(h0) λ1h,z(0) + γ1

h,z(0).

As λ0h0,z(0) is fixed, a prepolar deformation, h, with maximal γ1

h,z(0) will have minimal λ1

h,z(0).We prefer to call such a deformation a minimal prepolar deformation, instead of a maximal one.Note that a minimal prepolar deformation will not only have the maximal possible lower boundon the top Betti number of the Milnor fibre, it also provides the smallest difference between ourgeneral upper and lower bounds. One might hope that it is always possible to find a prepolardeformation, h, for which λ1

h,z(0) = 0, for then we would have bn−1(h0) = γ1h,z0

(0); unfortunately,Proposition 1.29 implies that it is usually impossible to find such a deformation.

106 DAVID B. MASSEY

Chapter 9. CONSTANCY OF THE MILNOR FIBRATIONS

In this chapter, we prove what is perhaps our most important result, and certainly the resultwhich requires the most machinery – we generalize the result of Le and Ramanujam [L-R] as statedin Theorem 0.10 in the introduction. Basically, we prove that if the Le numbers are constant ina one-parameter family, then the Milnor fibrations are constant in the family, regardless of thedimension of the critical loci.

Unfortunately, we do not obtain the result that the local, ambient topological-type of the hy-persurfaces remains constant in the family. It is an open question whether the constancy of the Lenumbers is strong enough to imply this topological constancy.

While the idea behind our proof of this generalized Le-Ramanujam is simple, the technicaldetails are horrendous. It is this chapter alone which is responsible for the existence of AppendixC of this book; we have relegated most of the technical details to the appendix. Before we provethe main result, there remain only two lemmas which we need (besides the results which appearin the appendix). Also, we will restate one of the results from the appendix in a form which iscomprehensible without reading the entire appendix.

First, however, we wish to sketch the proof of the main theorem, so that the reader can see thatthe idea really is fairly easy. On the other hand, the proof is not straightforward – instead, it usesa trick which gives one very little insight as to why the result should be true.

Throughout this chapter, U will denote an open neighborhood of the origin in Cn+1 and ft :

(U ,0) → (C, 0) will be an analytic family in the variables z = (z0, . . . , zn). Let s = dim0Σf0.

A sketch of the proof is as follows:

The result of Proposition 8.1 is that the Milnor fibre of ft + wj at the origin is homotopy-equivalent to the one-point union of j − 1 copies of the suspension of the Milnor fibre of ft at theorigin. So, it certainly seems reasonable to expect that the Milnor fibrations are independent of tin the family ft if and only if the Milnor fibrations are independent of t in the family ft + wj . Butwhy should the family ft + wj be any easier to study than the family ft itself?

It is easier because we have very nice hyperplanes defined by L = w−z0 such that, when we takethe sections (ft +wj)|V (L)

, we get the family ft +zj0 which, for generic z0 and for large j, is a family

of singularities of one less dimension (by the results of Chapter 4); that is, dim0Σ(f0 + zj0) = s−1.

Moreover, Le’s attaching result (Theorem 0.9) tells us how the Milnor fibre of ft + wj is obtainedfrom the Milnor fibre of a generic hyperplane section. The Milnor fibre of ft +wj is obtained fromthe Milnor fibre of ft + zj

0 by attaching(Γ1

ft+wj ,w−z0· V (ft + wj)

)

0(n + 1)-handles.

By induction on s, we may require the Milnor fibrations of ft + zj0 to be independent of t. If we

also require the number of attached (n + 1)-handles to be independent of t, it seems reasonable toexpect that the Milnor fibrations of the family ft + wj should be independent of t and, thus, thatthe Milnor fibrations of ft are independent of t.

The Le numbers enter the picture because Lemma 4.3 says that, for large j, the intersectionnumber

(Γ1

ft+wj ,w−z0· V (ft + wj)

)

0= jλ0

ft,z(0). Combining this with the Le-Iomdine formulas

of Theorem 4.5, we find that the inductive requirement that the Milnor fibrations of ft + zj0 are

independent of t amounts to requiring all the Le numbers of ft to be independent of t.


We first wish to prove a result which will tell us that the main theorem of this chapter is notvacuously true.

Lemma 9.1. For all i with 0 i n, for a generic linear reorganization of the coordinates(z0, . . . , zi), (z0, . . . , zi) is prepolar at the origin for ft for all small t.

Proof. Fix a good stratification, G, for f in a neighborhood, V, of the origin. By refining ifnecessary, we may also assume that G satisfies Whitney’s condition a). We will also assume that

S := V ∩ t-axis − 0 = V ∩ (C × 0) − 0

and 0 are strata of G. As the function t has isolated stratified critical values, V (t−t0) transverselyintersects all strata of G near (t0,0); hence, V (t − t0) ∩ G provides a good stratification for ft0 at0 which still satisfies Whitney’s condition a).

By induction on i, we will prove that: for all i with 0 i n, for a generic linear reorganizationof the coordinates (z0, . . . , zi), (z0, . . . , zi) is prepolar at the origin for ft0 with respect to the goodstratification V (t − t0) ∩ G for all small, non-zero t0.

i = 0: Using the terminology of Goresky and MacPherson [G-M2], the set of degenerate conormalcovectors to S is a complex analytic subvariety of codimension at least 1 inside the total space ofthe conormal bundle of S inside C

n+2 (see [G-M2, Prop. 1.8, p.44]). Projectivizing and dualizing,this says that the set Ω, defined by

(p, H) ∈ S × Gn(Cn+1) | Tp(C × H) contains a generalized tangent plane of G at p,

has dimension at most n. Hence, Ω ⊆ (V ∩ (C× 0))×Gn(Cn+1) has dimension at most n and thefibre over 0, call it Ψ, must therefore have dimension at most n− 1. Thus, Gn(Cn+1)−Ψ is openand dense in Gn(Cn+1). We claim that, for all H in Gn(Cn+1)−Ψ, H is a prepolar slice for ft at0 for all small, non-zero t.

Certainly, if H ∈ Ψ, then, for all small, non-zero t0, T(t0,0)(C × H) contains no generalizedtangent plane from G at (t0,0). Also, Whitney’s condition a) guarantees that all limiting tangentplanes from strata of G at (t0,0) actually contains T(t0,0)S = C×0. Combining these two facts, itfollows easily that T0H contains no generalized tangent plane from V (t− t0)∩G at 0. Thus, H isprepolar for ft0 at 0 with respect to the good stratification V (t − t0) ∩ G. (Actually, this impliesmuch more – it implies that H is polar, as defined in [Mas8].)

i 1: Now, assume that we have already chosen (z0, . . . , zi−1) generically (in the IPZ-topology) sothat (z0, . . . , zi−1) is prepolar at the origin for ft0 with respect to the good stratification V (t−t0)∩G

for all small, non-zero t0.Then, there exists a good stratification, G′, for f|V (z0,...,zi−1)

at the origin which satisfies Whit-ney’s condition a). Though G′ may not necessarily be chosen to equal G ∩ V (z0, . . . , zi−1), afterrefining G′ using Proposition C.2, we may certainly assume that each stratum of G′ is containedin a stratum of G and that, in some neighborhood of the origin, (C×0)−0 and 0 are strata of G′.

By the i = 0 case, for a generic choice of zi, V (zi) is a prepolar slice for ft0 |V (z0,...,zi−1)at

the origin with respect to V (t − t0) ∩ G′ for all small non-zero t0. But, as each stratum of G′ is

108 DAVID B. MASSEY

contained in a stratum of G, this last statement is stronger than saying that V (zi) is prepolar withrespect to V (t − t0) ∩ G. This concludes the induction.

Finally, to finish the proof, one simply chooses coordinates as generic as given above and gener-ically enough so that the coordinates are prepolar for f0 at the origin.

We shall also need the following uniform version of Proposition 4.19:

Proposition 9.2. If (z0, . . . , zi) is prepolar at the origin for ft for all small t, then, for all largej, (z1, . . . , zi) is prepolar for ft + zj

0 for all small t.

Proof. In light of Corollary 4.18, what we need to show is that, for all large j, for all k with0 k i and for all small t0,

(∗) dim0Γk+1ft0 ,z ∩ V

(t − t0,

∂f

∂z0+ jzj−1

0

)∩ V (z1, . . . , zk) 0.

In fact, we only have to show that (∗) holds for small non-zero t0, for then – by 4.19 – we mayimpose the extra largeness condition on j so that (∗) also holds for t0 = 0.

By Lemma 4.14, for all small non-zero t0, Γk+1ft0 ,z = Γk+2

f,(t,z) ∩V (t− t0) as sets, in a neighborhood

of the origin. In addition, by Theorem 1.26, γk+1ft0 ,z(0) exists for all small t0; thus, dim0Γk+1

ft0 ,z ∩V (z0, z1, . . . , zk) 0. Putting these two facts together, we find that, for all small non-zero t0,

dim0V (t − t0) ∩ Γk+2f,(t,z) ∩ V (z0, z1, . . . , zk) 0.

Let W denote the union of those irreducible analytic components, C, of Γk+2f,(t,z) ∩ V (z1, . . . , zk),

at the origin, which satisfy the property that the t-axis is contained in C∩V (z0). By the above, W∩V (z0) is at most one-dimensional at the origin and, for all small non-zero t0, Γk+1

ft0 ,z∩V (z1, . . . , zk) =W ∩ V (t − t0) as germs of sets at (t0,0).

It follows that each irreducible component of W at the origin is at most 2-dimensional. Wewould like to show that, for all large j, W ∩ V

(∂f∂z0

+ jzj−10

)is at most 1-dimensional at the

origin, for then – by intersecting with V (t − t0) – we obtain (∗).But, this is easy. For if a 2-dimensional component C of W is contained in V

(∂f∂z0

+ j0zj0−10

)

and V(

∂f∂z0

+ j1zj1−10

)for j0 = j1, then C ⊆ V (z0). However, we know that W ∩ V (z0) is at most

1-dimensional. Hence, there are only a finite number of “bad” values for j.

Proposition 9.3 Suppose that V (z0) is a prepolar slice for ft at the origin for all small t, andthat V (t) does not occur as the limit of tangent spaces to level hypersurfaces of f or f|V (z0)

atthe origin. Then, for all small non-zero t0, there is a natural inclusion of pairs of Milnor fibres(Fft0 ,0, Fft0 ,0 ∩ V (z0)) → (Ff0,0, Ff0,0 ∩ V (z0)).

If we also assume that the intersection number(Γ1

ft,z0· V (ft)

)

0is independent of t for all small

t, then we have the following three results:

i) if the inclusion Fft0 ,0 ∩ V (z0) → Ff0,0 ∩ V (z0) induces isomorphisms on integral homologygroups, then so does the inclusion Fft0 ,0 → Ff0,0;


ii) if s n − 2 and the inclusion Fft0 ,0 ∩ V (z0) → Ff0,0 ∩ V (z0) is a homotopy-equivalence,then so is the inclusion Fft0 ,0 → Ff0,0, and the fibre homotopy-type of the Milnor fibrations isindependent of t for all small t; and

iii) if s n − 3 and the inclusion Fft0 ,0 ∩ V (z0) → Ff0,0 ∩ V (z0) is a homotopy-equivalence,then the inclusion Fft0 ,0 → Ff0,0 is a diffeomorphism and, moreover, the diffeomorphism-type ofthe Milnor fibrations is independent of t for all small t.

Proof. This is primarily a restatement of Theorem C.13 from the appendix.

The condition on V (t) is that V (t) is not in the Thom set of either f or f|V (z0)at the origin (see

Definition C.8). Therefore, by Proposition C.9, the families ft and ft|V (z0)satisfy the universal

conormal condition at the origin (see Definition C.10) and this produces the inclusion of pairs ofMilnor fibres. Now, i) and ii) follow from Theorem C.13, and iii) follows from ii) together withProposition C.12.

We are now able to prove the main result of this chapter: our generalization of part of the resultof Le and Ramanujam. Essentially, we prove that the constancy of the Le numbers in a familyimplies the constancy of the Milnor fibrations in the family.

Theorem 9.4. Let s := dim0Σf0. Suppose that, for all t small, (z0, . . . , zs−1) is prepolar for ft

at 0 and that the Le numbers, λift,z

(0), are independent of t for each i with 0 i s. Then,

i) the homology of the Milnor fibre of ft at the origin is independent of t for all t small;

if s n − 2,

ii) the fibre homotopy-type of the Milnor fibrations of ft at the origin is independent of t for allt small;

and, if s n − 3,

iii) the diffeomorphism-type of the Milnor fibrations of ft at the origin is independent of t for allt small.

Proof. By induction on s.

For s = 0, this is the result of Le and Ramanujam [L-R]. Now, assume that s 1 and that weknow the result for families of hypersurfaces with critical loci of dimension s − 1.

Let j be so large that the uniform Le-Iomdine formulas of Theorem 4.15 hold and so largethat Proposition 9.2 holds. Finally, using that λ0

ft,z(0) is independent of t, let j be so large that

j 2 + λ0ft,z

(0) for all small t, so that we may apply Lemma 4.3 and Corollary 8.4.Consider the family ft + wj , where w is a variable disjoint from the z’s. The dimension of the

critical locus at t = 0 is still equal to s. As the Le numbers of ft are independent of t, we may applyTheorem 6.5 to conclude that V (t) is not the limit of tangent spaces to level hypersurfaces of f

110 DAVID B. MASSEY

at the origin. It follows trivially that V (t) is not the limit of tangent spaces to level hypersurfacesof f + wj at the origin. Moreover, using the uniform Le-Iomdines formulas of Theorem 4.15, wefind that the Le numbers of (ft + wj)|V (z0−w)

= ft + zj0 at 0 with respect to (z1, . . . , zs−1) are

independent of t, and that dim0Σ(f0 + zj0) = s − 1.

Therefore, by our inductive hypothesis, we already have the constancy results for the family(ft+wj)|V (z0−w)

. In addition, we may use Theorem 6.5 again to conclude that V (t) is not the limit oftangent spaces to level hypersurfaces of (f+wj)|V (z0−w)

at the origin. By 8.4, V (z0−w) is a prepolar

slice for ft +wj at the origin for all small t. Also, by Lemma 4.3.v,(Γ1

ft+wj ,w−z0, ·V (ft + wj)

)

0=

jλ0ft,z

(0), which is independent of t for all small t. Thus, we are in a position to apply Proposition9.3 to the family ft + wj and we conclude the desired constancy results for this family.

Finally, now that we know the results for ft + wj , we apply Proposition C.16 to conclude thatthe result actually holds for ft itself.

Remark 9.5. While we are not very fond of discussing it, as we mentioned in Remark 1.27, for afixed function h and a fixed point p, there exist generic Le numbers of h at p – that is, as one variesthe linear choice of coordinates, z, through coordinates for which the Le numbers are defined, onefinds a generic value for each of the Le numbers, λi

h,z(p); let us denote this generic value simplyby λi

h(p).We are not fond of discussing these generic Le numbers because the Le numbers are intended

to be effectively calculable, and we know of no effective way of knowing when a coordinate choiceis sufficiently generic to give λi

h(p). We do know, by Corollary 4.16, that the tuple of generic Lenumbers (λs

h(p), . . . , λ0h(p)) is minimal with respect to the lexigraphic ordering.

We mention all this here because the tuple (λsh(p), . . . , λ0

h(p)) is an analytic invariant (thisfollows from the relationship between the Le numbers and the polar multiplicities; see [Mas11] orPart IV, Theorems 1.10 and 3.2) , and so the reader may wonder whether Theorem 9.4 stills holdsunder the assumption that the generic Le numbers of ft are independent of t at the origin. Theanswer to this question is easily seen to be: yes. This follows quickly from 9.4 itself.

Suppose that the generic Le numbers of ft at the origin are independent of t. Choose coordinatesz that are so generic that z is prepolar for ft at the origin for all small t and so that z is genericenough to give the generic Le numbers of f0 at the origin. Then, for all small t, we have

(λsft

(0), . . . , λ0ft

(0)) (λsft,z(0), . . . , λ0

ft,z(0))

(λsf0,z(0), . . . , λ0

f0,z(0)) = (λsf0

(0), . . . , λ0f0

(0)),

where we are using the lexigraphic ordering. Since

(λsft

(0), . . . , λ0ft

(0)) = (λsf0

(0), . . . , λ0f0

(0)),

it follows that (λsft,z

(0), . . . , λ0ft,z

(0)) equals the tuple at t = 0, and so we may apply Theorem 9.4.


Chapter 10. ANOTHER CHARACTERIZATION OF THELE CYCLES

In this chapter, we give an alternative characterization of the Le cycles and Le numbers of ahypersurface singularity. This alternative characterization is generalized in Part IV, Chapter 3,where we see that the case of the Le numbers of a function f is just the case where the underlyingcomplex of sheaves is the sheaf of vanishing cycles along f .

As a consequence of Theorem 3.3, the Le cycles can be characterized formally by requiring thatthe alternating sum of the Le numbers yields the reduced Euler characteristic of the Milnor fibre ateach point – provided that we know that a fixed choice of coordinates is prepolar at every point.We will show this now.

Lemma 10.1. Let X be an analytic subset containing the origin in some CN and let Sα be an

analytic stratification of X with connected strata. Let p : CN → C

k be a linear map such that p|Sα

is a submersion if dimSα k and dim0

(p−1(0) ∩ Sα

)= 0 if dimSα k − 1.

Then, for generic linear π : Ck → C

k−1, there exists a refinement Rβ of Sα in someneighborhood of the origin which preserves the strata of dimension greater than or equal to k andsuch that π p|Rβ

is a submersion if dimRβ k − 1 and dim0

((π p)−1(0) ∩ Rβ

)= 0 if

dimRβ k − 2.

Proof. Let X ′ denote the union of those strata Sα such that dimSα is less than k. Then, asdim0(p−1(0) ∩ X ′) = 0, there exists an open neighborhood, U , of the origin in C

N and an openneighborhood of the origin, V, in C

k such that the restriction of p to a map from U ∩ X ′ to V isfinite. Therefore, as the conclusion of the lemma is purely local in nature, we may reduce ourselvesto considering only the case where p|X′ is a finite map.

Thus, p(X ′) is an analytic subset of Ck of dimension at most k − 1 and so, for generic lines,

L, through 0 in Ck, we have that L ∩ p(X ′) = 0 near the origin. Hence, for generic linear

π : Ck → C

k−1, we must have that π−1(0) ∩ p(X ′) = 0, and so

(π p)−1(0) ∩ X ′ = p−1(π−1(0) ∩ p(X ′)

)∩ X ′ = p−1(0) ∩ X ′.

But, by hypothesis, p−1(0) ∩ X ′ = 0, and thus – for such a generic π – any refinement of X ′

will give a refinement of Sα which preserves the strata of dimension greater than or equal to kand realizes the desired intersection condition.

Now, we have the restriction π p : X ′ → Ck−1. By the above, if we once again take sufficiently

small neighborhoods around the origins, π p restricts to a finite map on X ′. Assume then that(π p)|X′ is finite. If dimX ′ = k − 1, then π p is a submersion on X ′ − W , where W is aclosed subset of X ′ of dimension at most k − 2. Thus, we may refine the stratification so thatX ′ ∩ (π p)−1(π p)(W ) is a union of strata, where the strata have dimension at most k − 2since π p restricted to X ′ is a finite map. This yields the desired refinement Rβ.

Proposition 10.2. Let U be an open subset of Cn+1 containing the origin, and let h : (U ,0) →

(C,0) be an analytic map. Then, for a generic linear reorganization, z, of coordinate systems for

112 DAVID B. MASSEY

Cn+1, there exists an open neighborhood, U ′, of the origin such that z is prepolar for h|U′ , i.e., for

all p ∈ U ′ ∩ V (h), if we let s denote dimpΣh, then (z0, . . . , zs−1) is prepolar for h at p.

Proof. Begin by applying the lemma to X = V (h), endowed with some good stratification, and p =id : C

n+1 → Cn+1. By an inductive application of the lemma, for a generic linear reorganization

of coordinates, z, for Cn+1, we arrive at a stratification Sα, which is a refinement of the original

good stratification (in a possibly smaller neighborhood of the origin), such that for each i, themap (z0, . . . , zi) is a submersion when restricted to a stratum of dimension greater than or equalto i + 1. This would say precisely that (z0, . . . , zn) is prepolar at each point of V (h) with respectto the stratification Sα – provided that Sα is actually a good stratification for h.

But, any refinement of a good stratification is another good stratification – except for the factthat we may have refined the smooth part of V (h), which we required to be a stratum of any goodstratification. However,

V (h) − Σh ∪ Sα | Sα ⊆ Σhis certainly a good stratification for h, and what remains for us to show is that, for genericallyreorganized coordinates, z, for each p ∈ Σh, V (z0 −p0, . . . , zs−1 −ps−1) transversely intersects thesmooth part of V (h) near p, where again s = dimpΣh.

Assume then that we have chosen the coordinates z generic as above and also generic enoughso that (z0, . . . , zs−1) is prepolar for h at 0, where s = dim0Σh. By Theorem 1.26, γs

h,z(0) exists,and so γs

h,z(p) exists for all p near 0. But, by Remark 1.6, this implies that if p ∈ Σh, then

Σ(h|V (z0−p0,...,zk−1−pk−1)) = V (z0 − p0, . . . , zk−1 − pk−1) ∩ Σh

at p, i.e., V (z0 − p0, . . . , zs−1 − ps−1) transversely intersects the smooth part of V (h) near p (and,of course, s = dim0Σh dimpΣh for all p near 0).

In Part IV, Chapter 3, we will generalize the result below to the case where the underlying spaceis arbitrary.

Theorem 10.3. For a generic linear reorganization of coordinates, z, for Cn+1, the Le cycles

are a collection of analytic cycle germs, Λih,z

, in Σh at the origin such that each Λih,z

is purelyi-dimensional and properly intersects V (z0, . . . , zi−1) at the origin, and

χ(Fh,p

) =s∑

i=0

(−1)n−i(Λi

h,z· V (z0 − p0, . . . , zi−1 − pi−1)

)

p,

for all p ∈ Σh near 0, where χ(Fh,p

) is the reduced Euler characteristic of the Milnor fibre of h atp; specifically, this is the case if z is prepolar for h in a neighborhood of the origin.

Moreover, if z is any linear coordinate system such that such cycles exist, then they are unique.

Proof. The first statement follows immediately from the previous proposition and Theorem 3.3.As for the uniqueness assertion, this is a fairly standard argument for constructible functions.

Suppose that we had two such collections, Λih,z

and Ωih,z

. Let s denote dim0Σh. We will showthat Λi

h,zand Ωi

h,zagree by downward induction on i.


For a generic point, p, in an s-dimensional component, ν, of the support of Λsh,z

, p will be asmooth point of ν, V (z0 − p0, . . . , zs−1 − ps−1) will transversely intersect ν at p, and p will not bein the support of any of the lower-dimensional Λi

h,zor Ωi

h,z. Thus, at such a p,

(Ωs

h,z· V (z0 − p0, . . . , zs−1 − pi−1)

)

p=

(Λs

h,z· V (z0 − p0, . . . , zs−1 − pi−1)

)

p.

Of course, the same conclusion would have followed it we had chosen a generic point, p, in ans-dimensional component, ν, of the support of Ωs

h,z, It follows that Λs

h,z= Ωs

h,z.

Now, suppose that we have shown that Λih,z

= Ωih,z

for all i greater than some k. Then,

k∑

i=0

(−1)n−i(Λi

h,z· V (z0 − p0, . . . , zi−1 − pi−1)

)

p=

k∑

i=0

(−1)n−i(Ωi

h,z· V (z0 − p0, . . . , zi−1 − pi−1)

)

p,

and we repeat the argument above with k in place of s. The conclusion follows.

Theorem 10.3 leaves us with a very strange set of affairs; it tells us that, for generic z, wecould have defined the Le cycles by their characterization in the theorem. This means that theLe cycles and hence, the Le numbers, are determined by the choice of z and the data of the Eulercharacteristic of the Milnor fibre at each point. But, had we defined the Le cycles and numbersthis way, then we would have produced the Morse inequalities on the Betti numbers of the Milnorfibres in Theorem 3.3 from seemingly much less data. This phenomenon occurs in a much moregeneral setting; the explanation appears in [Mas11].

114 DAVID B. MASSEY

Part III. ISOLATED CRITICAL POINTS OF FUNCTIONSON SINGULAR SPACES


In the introduction to Part II, we discussed known results for functions, or families of functions,with isolated critical points. The remainder of Part II was devoted to the development of the Lecycles and Le numbers – a generalization to functions with non-isolated critical loci of the dataprovided by the Milnor number of an isolated critical point.

All of the functions considered in Part II had open subsets of affine space as their domains.Here, in Part III, we will begin our generalization of the Le numbers to the case where the func-tions considered have arbitrary analytic spaces as their domains. However, since the Le numbersgeneralize the Milnor number of an isolated critical point (on affine space), if we want to have Lenumbers for functions with arbitrary domains, then we must first develop some sort of “Milnornumber” theory for functions with “isolated critical points” with arbitrary domains.

This immediately leads us to two fundamental questions for functions with arbitrary domains:

• What is the proper notion of the “critical locus” of such a function?

• What is the proper notion of the “Milnor number” of such a function when its “critical locus”consists of an isolated point?

In the remainder of this introduction, we will discuss possible definitions of the “critical locus”for functions; we will first discuss the non-controversial case of functions on affine space, and thenmove on to the much more complicated general case. There is a slight bit of overlap here withsome of the material presented in the introduction to Part II, but we feel that this minor repetitionaids the exposition.

Let U be an open subset of Cn+1, let z := (z0, z1, . . . , zn) be coordinates for Cn+1, and supposethat f : U → C is an analytic function. Then, all conceivable definitions of the critical locus, Σf ,of f agree: one can consider the points, x, where the derivative vanishes, i.e., dxf = 0, or one canconsider the points, x, where the Taylor series of f at x has no linear term, i.e., f − f(x) ∈ m2

U,x

(where mU,x is the maximal ideal in the coordinate ring of U at x), or one can consider the points,x, where the Milnor fibre of f at x, Ff ,x, is not trivial (where, here, “trivial” could mean even upto analytic isomorphism).

Now, suppose that X is an analytic subset of U , and let f := f|X . Then, what should be meantby “the critical locus of f”? It is not clear what the relationship is between points, x, wheref − f(x) ∈ m2

X,xand points where the Milnor fibre, Ff,x, is not trivial (with any definition of

trivial); moreover, the derivative dxf does not even exist.

We are guided by the successes of Morse Theory and stratified Morse Theory to choosing theMilnor fibre definition as our primary notion of critical locus, for we believe that critical pointsshould coincide with changes in the topology of the level hypersurfaces of f . Therefore, we makethe following definition:

115

116 DAVID B. MASSEY

Definition 0.1. The C-critical locus of f , ΣCf , is given by

ΣCf := x ∈ X | H∗(Ff,x; C) 6= H∗(point; C).

(The reasons for using field coefficients, rather than Z, are technical: we want Lemma 4.1 to betrue.)

In Chapter 1, we will compare and contrast the C-critical locus with other possible notions ofcritical locus, including the ones mentioned above and the stratified critical locus.

After Chapter 1, the remainder of Part III is dedicated to showing that Definition 0.1 reallyyields a useful, calculable definition of the critical locus. We show this by looking at the case of ageneralized isolated singularity, i.e., an isolated point of ΣCf , and showing that, at such a point,there is a workable definition of the Milnor number(s) of f ; we show that the Betti numbers ofthe Milnor fibre can be calculated (3.7.ii), and we give a generalization of the result of Le andSaito [L-S] that constant Milnor number throughout a family implies Thom’s af condition holds.Specifically, in Corollary 6.14, we prove (with slightly weaker hypotheses) that:

Theorem 0.2. Let W be a (not necessarily purely) d-dimensional analytic subset of an open subset

of Cn. Let Z be a d-dimensional irreducible component of W . Let X :=D×W be the product of

an open disk about the origin with W , and let Y :=D× Z.

Let f : (X,D× 0) → (C, 0) be an analytic function, and let ft(z) := f(t, z). Suppose that f0

is in the square of the maximal ideal of Z at 0.Suppose that 0 is an isolated point of ΣC(f0), and that the reduced Betti number bd−1(Ffa,(a,0))

is independent of a for all small a.

Then, bd−1(Ffa,(a,0)) 6= 0 and, near 0, Σ(f|Yreg) ⊆

D×0 and the pair (Yreg−Σ(f|Yreg

),D×0)

satisfies Thom’s af condition at 0.

Thom’s af is important for several reasons, but perhaps the best reason is because it is anhypothesis of Thom’s Second Isotopy Lemma. General results on the af condition have beenproved by many researchers: Hironaka, Le, Saito, Henry, Merle, Sabbah, Briancon, Maisonobe,Parusinski, etc., and the above theorem is closely related to the recent results contained in [BMM]and [P2]. However, the reader should contrast the hypotheses of Theorem 0.2 with those of themain theorem of [BMM] (Theorem 5.2.1); our main hypothesis is that a single number is constantthroughout the family, while the main hypothesis of Theorem 4.2.1 of [BMM] is a conditionwhich requires one to check an infinite amount of data: the property of local stratified triviality.Moreover, the Betti numbers that we require to be constant are actually calculable.

While much of this part is fairly technical in nature, there are three new, key ideas that guideus throughout.

The first of these fundamental precepts is: controlling the vanishing cycles in a family offunctions is enough to control Thom’s af condition and, perhaps, the topology throughout thefamily. While this may seem like an obvious principle – given the results of Le and Saito in [L-S]and of Le and Ramanujam in [L-R] – in fact, in the general setting, most of the known results seemto require the constancy of much stronger data, e.g., the constancy of the polar multiplicities [Te6]

PART III. ISOLATED CRITICAL POINTS 117

or that one has the local stratified triviality property [BMM]. In a very precise sense, controllingthe polar multiplicities corresponds to controlling the nearby cycles of the family of functions,instead of merely controlling the vanishing cycles. As we show in Corollary 5.4, controlling thecharacteristic cycle of the vanishing cycles is sufficient for obtaining the af condition.

Our second fundamental idea is: the correct setting for all of our cohomological results is whereperverse sheaves are used as coefficients. While papers on intersection cohomology abound, andwhile perverse sheaves are occasionally used as a tool (e.g., [BMM, 4.2.1]), we are not aware of anyother work on general singularities in which arbitrary perverse sheaves of coefficients are used in anintegral fashion throughout. The importance of perverse sheaves in Part III begins with Theorem4.2, where we give a description of the critical locus of a function with respect to a perverse sheaf.

The third new feature of Part III is the recurrent use of the perverse cohomology of a complex ofsheaves. This device allows us to take our general results about perverse sheaves and translate theminto statements about the constant sheaf. The reason that we use perverse cohomology, insteadof intersection cohomology, is because perverse cohomology has such nice functorial properties: itcommutes with Verdier dualizing, and with taking nearby and vanishing cycles (shifted by [−1]).If we were only interested in proving results for local complete intersections (l.c.i.’s), we wouldnever need the perverse cohomology; however, we want to prove completely general results. Theperverse cohomology seems to be a hitherto unused tool for accomplishing this goal.

Part III is organized as follows:

In Chapter 1, we discuss seven different notions of the “critical locus” of a function. We giveexamples to show that, in general, all of these notions are different.

In Chapter 2, we discuss the relative polar curve of Le and Teissier. We need to relate theintrinsic definition to a conormal characterization in terms of gap sheaves.

Chapter 3 is devoted to proving an “index theorem”, Theorem 3.10, which provides the mainlink between the topological data of the Milnor fibre and the algebraic data obtained by blowing-up the image of df inside the appropriate space. This theorem is presented with coefficients in abounded, constructible complex of sheaves; this level of generality is absolutely necessary in orderto obtain the results in the remainder of Part III.

Chapter 4 uses the index theorem of Chapter 3 to show that ΣCf and the Betti numbers of theMilnor fibre really are fairly well-behaved. This is accomplished by applying Theorem 3.10 in thecase where the complex of sheaves is taken to be the perverse cohomology of the shifted constantsheaf. Perverse cohomology essentially gives us the “closest” perverse sheaf to the constant sheaf.Many of the results of Chapter 4 are stated for arbitrary perverse sheaves, for this seems to be themost natural setting.

Chapter 5 contains the necessary results from conormal geometry that we will need in order toconclude that topological data implies that Thom’s af condition holds. The primary result of thischapter is Corollary 5.4, which once again relies on the index theorem from Chapter 3.

Chapter 6 begins with a discussion of “continuous families of constructible complexes of sheaves”.We then prove in Theorem 6.7 that additivity of Milnor numbers occurs in continuous families ofperverse sheaves, and we use this to conclude additivity of the Betti numbers of the Milnor fibres, byonce again resorting to the perverse cohomology of the shifted constant sheaf. Finally, in Corollaries6.11 and 6.12, we prove that the constancy of the Milnor/Betti number(s) throughout a familyimplies that the af condition holds – we prove this first in the setting of arbitrary perverse sheaves,and then for perverse cohomology of the shifted constant sheaf. By translating our hypotheses fromthe language of the derived category back into more down-to-Earth terms, we obtain Corollary6.12, which leads to Theorem 0.2 above.

118 DAVID B. MASSEY

Chapter 1. CRITICAL AVATARS.

We continue with U , z, f , X, and f as in the introduction.

In this chapter, we will investigate seven possible notions of the “critical locus” of a function ona singular space, one of which is the C-critical locus already defined in 0.1.

Definition 1.1. The algebraic critical locus of f , Σalgf , is defined by

Σalgf := x ∈ X | f − f(x) ∈ m2X,x.

Remark 1.2. It is a trivial exercise to verify that

Σalgf = x ∈ X | there exists a local extension, f , of f to U such that dxf = 0.

Note that x being in Σalgf does not imply that every local extension of f has zero for itsderivative at x.

One might expect that Σalgf is always a closed set; in fact, it need not be. Consider the examplewhere X := V (xy) ⊆ C2, and f = y|X . We leave it as an exercise for the reader to verify thatΣalgf = V (y)− 0.

There are five more variants of the critical locus of f that we will consider. We let Xreg denotethe regular (or smooth) part of X and, if M is an analytic submanifold of U , we let T ∗

MU denote

the conormal space to M in U (that is, the elements (x, η) of the cotangent space to U such thatx ∈M and η annihilates the tangent space to M at x). We let N(X) denote the Nash modificationof X, so that the fibre Nx(X) at x consists of limits of tangent planes from the regular part of X.

We also remind the reader that complex analytic spaces possess canonical Whitney stratifications(see [Te6]).

Definition 1.3. We define the regular critical locus of f , Σregf , to be the critical locus of therestriction of f to Xreg, i.e., Σregf = Σ

(f|Xreg

).

We define the Nash critical locus of f , ΣNashf , to bex ∈ X | there exists a local extension, f , of f to U such that dxf(T ) ≡ 0, for all T ∈ Nx(X)

.

We define the conormal-regular critical locus of f , Σcnrf , to bex ∈ X | there exists a local extension, f , of f to U such that (x, dxf) ∈ T ∗

XregU

;

it is trivial to see that this set is equal tox ∈ X | there exists a local extension, f , of f to U such that dxf(T ) ≡ 0, for some T ∈ Nx(X)

.


Let S = Sα be a (complex analytic) Whitney stratification of X. We define the S-stratifiedcritical locus of f , ΣSf , to be

⋃α Σ(f|Sα

). If S is clear, we simply call ΣSf the stratified critical

locus.If S is, in fact, the canonical Whitney stratification of X, then we write Σcanf in place of ΣSf ,

and call it the canonical stratified critical locus.

We define the relative differential critical locus of f , Σrdff , to be the union of the singular setof X and Σregf .

If x ∈ X and h1, . . . , hj are equations whose zero-locus defines X near x, then x ∈ Σrdff if andonly if the rank of the Jacobian map of (f , h1, . . . , hj) at x is not maximal among all points ofX near x. By using this Jacobian, we could (but will not) endow Σrdff with a scheme structure(the critical space) which is independent of the choice of the extension f and the defining functionsh1, . . . , hn (see [Loo, 4.A]). The proof of the independence uses relative differentials; this is thereason for our terminology.

Remark 1.4. In terms of conormal geometry, ΣSf =

x ∈ X | (x, dxf) ∈⋃

α T∗SαU

or, using

Whitney’s condition a) again, ΣSf =

x ∈ X | (x, dxf) ∈⋃

α T∗SαU

.

Clearly, Σrdff is closed, and it is an easy exercise to show that Whitney’s condition a) impliesthat ΣSf is closed. On the other hand, Σregf is, in general, not closed and, in order to have anyinformation at singular points of X, we will normally look at its closure Σregf .

Looking at the definition of Σcnrf , one might expect that Σregf = Σcnrf . In fact, we shall seein Example 1.8 that this is false. That Σcnrf is, itself, closed is part of the following proposition.(Recall that f is our fixed extension of f to all of U .)

In the following proposition, we show that, in the definitions of the Nash and conormal-regularcritical loci, we could have used “for all” in place of “there exists” for the local extensions; inparticular, this implies that we can use the fixed extension f . Finally, we show that the conormal-regular critical locus is closed.

Proposition 1.5. The Nash critical locus of f is equal tox ∈ X | for all local extensions, f , of f to U , dxf(T ) ≡ 0, for all T ∈ Nx(X)

=

x ∈ X | dxf(T ) ≡ 0, for all T ∈ Nx(X)

.

The conormal-regular critical locus of f is equal tox ∈ X | for all local extensions, f , of f to U , (x, dxf) ∈ T ∗

XregU

=x ∈ X | (x, dxf) ∈ T ∗

XregU.

In addition, Σcnrf is closed.

120 DAVID B. MASSEY

Proof. Let Z :=x ∈ X | for all local extensions, f , of f to U , dxf(T ) ≡ 0, for all T ∈ Nx(X)

.

Clearly, we have Z ⊆ ΣNashf .Suppose now that x ∈ ΣNashf . Then, there exists a local extension, f , of f to U such that

dxf(T ) ≡ 0, for all T ∈ Nx(X). Let f be another local extension of f to U and let T∞ ∈ Nx(X);to show that x ∈ Z, what we must show is that dxf(T∞) ≡ 0.

Suppose not. Then, there exists v ∈ T∞ such that dxf(v) 6= 0, but dxf(v) = 0. Therefore,there exist xi ∈ Xreg and vi ∈ Txi

Xreg such that xi → x, TxiXreg → T∞, and vi → v.

Let V be an open neighborhood of x in U which in f and f are both defined. Let Φ : V∩TXreg →C be defined by Φ(p,w) = dp(f − f)(w). Then, Φ is continuous, and so Φ−1(0) is closed. As(f − f)|X∩V ≡ 0, (xi,vi) ∈ Φ−1(0), and thus (x,v) ∈ Φ−1(0) – a contradiction. Therefore,Z = ΣNashf .

It follows immediately that ΣNashf =x ∈ X | dxf(T ) ≡ 0, for all T ∈ Nx(X)

.

Now, let W :=x ∈ X | for all local extensions, f , of f to U , (x, dxf) ∈ T ∗

XregU

. Clearly, wehave W ⊆ Σcnrf .

Suppose now that x ∈ Σcnrf . Then, there exists a local extension, f , of f to U such that(x, dxf) ∈ T ∗

XregU . Let (xi, ηi) ∈ T ∗

XregU be such that (xi, ηi) → (x, dxf). Let f be another local

extension of f to U ; to show that x ∈W , what we must show is that (x, dxf) ∈ T ∗Xreg

U .

Since (f − f)|X∩V ≡ 0, for all q ∈ Xreg,(q, dq(f − f)

)∈ T ∗

XregU ; in particular,

(xi, dxi

(f − f))∈

T ∗Xreg

U . Thus,(xi, ηi + dxi

(f − f))∈ T ∗

XregU , and

(xi, ηi + dxi

(f − f))→ (x, dxf). Therefore,

(x, dxf) ∈ T ∗Xreg

U , and W = Σcnrf .

It follows immediately that Σcnrf =x ∈ X | (x, dxf) ∈ T ∗

XregU

.

Finally, we need to show that Σcnrf is closed. Let Ψ : X → T ∗U be given by Ψ(x) = (x, dxf).Then, Ψ is a continuous map and, by the above, Σcnrf = Ψ−1(T ∗

XregU).

Proposition 1.6. There are inclusions

Σregf ⊆ Σalgf ⊆ ΣNashf ⊆ Σcnrf ⊆ ΣCf ⊆ Σcanf ⊆ Σrdff.

In addition, if S is a Whitney stratification of X, then Σcanf ⊆ ΣSf .

Proof. Clearly, Σregf ⊆ Σalgf ⊆ ΣNashf ⊆ Σcnrf , and so the containments for their closuresfollows (recall, also, that Σcnrf is closed). It is also obvious that Σcanf ⊆ Σrdff and Σcanf ⊆ ΣSf .

That ΣZf ⊆ Σcanf follows from Stratified Morse Theory [Go-Mac1], and so, since Σcanf isclosed, ΣCf ⊆ Σcanf .

It remains for us to show that Σcnrf ⊆ ΣCf . Unfortunately, to reach this conclusion, we mustrefer ahead to Theorem 4.6, from which it follows immediately. (However, that Σalgf ⊆ ΣCffollows from A’Campo’s Theorem [A’C].)

Remark 1.7. For a fixed stratification S, for all x ∈ X, there exists a neighborhood W of x inX such that W ∩ ΣSf ⊆ f−1f(x). This is easy to show: the level hypersurfaces of f close to


V (f − f(x)) will be transverse to all of the strata of S near x. All of our other critical loci whichare contained in ΣSf (i.e., all of them except Σrdff) also satisfy this local isolated critical valueproperty.

Example 1.8. In this example, we wish to look at the containments given in Proposition 1.6, andinvestigate whether the containments are proper, and also investigate what would happen if wedid not take closures in the four cases where we do.

The same example that we used in Remark 1.2 shows that none of Σregf , Σalgf , ΣNashf , orΣCf are necessarily closed; if X := V (xy) ⊆ C2, and f = y|X , then all four critical sets areprecisely V (y)− 0. Additionally, since Σcnrf = V (y), this example also shows that, in general,Σcnrf 6⊆ ΣCf .

If we continue with X = V (xy) and let g := (x + y)2|X , then Σalgg = 0 and Σregg = ∅; thus,in general, Σregf 6= Σalgf .

While it is easy to produce examples where ΣNashf is not equal to Σalgf and examples whereΣNashf is not equal to Σcnrf , it is not quite so easy to come up with examples where all three ofthese sets are distinct. We give such an example here.

Let Z := V ((y − zx)(y2 − x3)) ⊆ C3 and L := y|Z . Then, one easily verifies that ΣalgL = ∅,ΣNashL = 0, and ΣcnrL = C× 0.

If X = V (xy) and h := (x+y)|X , then ΣCh = 0 and Σcnrh = ∅; thus, in general, Σcnrf 6= ΣCf .

Let W := V (z5 + ty6z + y7x+ x15) ⊆ C4; this is the example of Briancon and Speder [B-S] inwhich the topology along the t-axis is constant, despite the fact that the origin is a point-stratum inthe canonical Whitney stratification of W . Hence, if we let r denote the restriction of t to W , then,for values of r close to 0, 0 is the only point in Σcanr and 0 6∈ ΣCr. Therefore, 0 ∈ Σcanr − ΣCr,and so, in general, ΣCf 6= Σcanf .

Using the coordinates (x, y, z) on C3, consider the cross-product Y := V (y2 − x3) ⊆ C3. Thecanonical Whitney stratification of Y is given by Y − 0 × C, 0 × C. Let π := z|Y . Then,Σcanπ = ∅, while Σrdfπ = 0 × C. Thus, in general, Σcanf 6= Σrdff .

It is, of course, easy to throw extra, non-canonical, Whitney strata into almost any example inorder to see that, in general, Σcanf 6= ΣSf .

To summarize the contents of this example and Proposition 1.6: we have seven seeminglyreasonable definitions of “critical locus” for complex analytic functions on singular spaces (we arenot counting ΣSf , since it is not intrinsically defined). All of our critical locus avatars agreefor manifolds. The sets Σregf , Σalgf , ΣNashf , and ΣCf need not be closed. There is a chain ofcontainments among the closures of these critical loci, but – in general – none of the sets are equal.

However, we consider the sets Σregf , Σalgf , ΣNashf , and Σcnrf to be too small; these “criticalloci” do not detect the change in topology at the level hypersurface h = 0 in the simple exampleX = V (xy) and h = (x+ y)|X (from Example 1.8).

Despite the fact that the Stratified Morse Theory of [Go-Mac1] yields nice results and requiresone to consider the stratified critical locus, we also will not use Σcanf (or any other ΣSf) as ourprimary notion of critical locus; Σcanf is often too big. As we saw in the Briancon-Speder example

122 DAVID B. MASSEY

in Example 1.8, the stratified critical locus sometimes forces one to consider “critical points” whichdo not correspond to changes in topology.

Certainly, Σrdff is far too large, if we want critical points to have any relation to changes inthe topology of level hypersurfaces: if X has a singular set ΣX, then the critical space of theprojection π : X ×C → C would consist of ΣX ×C, despite the obvious triviality of the family oflevel hypersurfaces defined by π.

Therefore, we choose to concentrate our attention on the C-critical locus, and we will justifythis choice with the results in the remainder of Part III.

Note that we consider ΣCf , not its closure, to be the correct notion of critical locus; we thinkthat this is the more natural definition, and we consider the question of when ΣCf is closed to bean interesting one. It is true, however, that all of our results refer to ΣCf . We should mention herethat, while ΣCf need not be closed, the existence of Thom stratifications [Hi] implies that ΣCf isat least analytically constructible; hence, ΣCf is an analytic subset of X.

Before we leave this chapter, in which we have already looked at seven definitions of “criticallocus”, we need to look at one last variant. As we mentioned at the end of the introduction, eventhough we wish to investigate the Milnor fibre with coefficients in C, the fact that the shiftedconstant sheaf on a non-l.c.i. need not be perverse requires us to take the perverse cohomologyof the constant sheaf. This means that we need to consider the hypercohomology of Milnor fibreswith coefficients in an arbitrary bounded, constructible complex of sheaves of modules. As we wishto discuss Euler characteristics, we need for the rank of a finitely-generated module to be definedand additive over exact sequences; thus, we must choose our base ring to be a p.i.d. However, sincethe rank of a module over a p.i.d. equals the dimension of the associated vector space over thequotient field, we may as well restrict ourselves to the case where the base ring is, in fact, a field.

The C−critical locus is nicely described in terms of vanishing cycles (see [K-S] for generalproperties of vanishing cycles, but be aware that we use the more traditional shift):

ΣCf = x ∈ X | H∗(φf−f(x)C•X)x 6= 0.This definition generalizes easily to yield a definition of the critical loci of f with respect to arbitrarybounded, constructible complexes of sheaves on X.

Let R be a p.i.d. Let S := Sα be a Whitney stratification of X, and let F• be a boundedcomplex of sheaves of R-modules which is constructible with respect to S.

Definition 1.9. The F•-critical locus of f , ΣF• f , is defined by

ΣF• f := x ∈ X | H∗(φf−f(x)F•)x 6= 0.

Remark 1.10. Stratified Morse Theory (see [Go-Mac1]) implies that ΣF• f ⊆ ΣSf (alternatively,

this follows from 8.4.1 and 8.6.12 of [K-S], combined with the facts that complex analytic Whitneystratifications are w-stratifications, and w-stratifications are µ-stratifications.)

We could discuss three more notions of the critical locus of a function – two of which are obtainedby picking specific complexes for F• in Definition 1.9. However, we will defer the introduction ofthese new critical loci until Chapter 4; at that point, we will have developed the tools necessaryto say something interesting about these three new definitions.


Chapter 2. THE RELATIVE POLAR CURVE

In this chapter, we discuss the relative polar curve of Le and Teissier ([L-T2],[Te5], [Te6],[Te7], etc.). This object is now a standard part of singularity theory, and the reader is most likelyfamiliar with some of the results that appear here. However, we shall use our early results on gapvarieties and cycles to present the theory in the conormal form in which we will use it in the nextchapter. Our treatment of the higher-dimensional relative polar varieties, when the underlyingspace is singular, does not appear here; it is a major portion of Part IV.

We let U be an open subset of Cn+1, and let X be a reduced analytic subspace of U withanalytic components Xj. Let dj denote the dimension of Xj , let cj := n + 1 − dj denote itscodimension, and let d denote the global dimension of X. Let f : U → C be an analytic function,and let f := f|X .

For g1, . . . , gj ∈ OU , let Jac(g1, . . . , gj) denote the Jacobian matrix of (g1, . . . , gj), which hasthe partial derivatives of gi in its i-th row. For any matrix A of functions, we let Mini(A) denotethe sheaf of ideals generated by the determinants of the i× i minors of A. Let Ji(g1, . . . , gj) denoteMini(Jac(g1, . . . , gj)).

We use z := (z0, . . . , zn) as coordinates on U . We let η : T ∗U → U denote the cotangentbundle, and we identify the cotangent space T ∗U with U × Cn+1 by using dz0, . . . , dzn as a basis.We use w := (w0, . . . , wn) as coordinates for the cotangent vectors, i.e., a cotangent vector isw0dz0 + · · · + wndzn. If z is a linear change of coordinates applied to z, then we let w denotecotangent coordinates with respect to the new basis dz0, . . . , dzn, i.e., if A is in Gln+1(C) andz := Az, then w = Atw.

Definition 2.1. The relative polar curve of X with respect to f and z0, Γ1f,z0

(X), is defined as aset by

Γ1f,z0

:= Σ((f, z0)|Xreg−Σ(f|Xreg

)

).

Let V be an open subset of U , and suppose that h := (h1, . . . , hl) defines X in V. Then, Γ1f,z0

isdefined as a scheme on V by⋃

j

(Xj ∩ V

(Jcj+2(h, f , z0)

))¬ (ΣX ∪ Σ(f|Xreg

)).

(We remind the reader that the union is given a scheme structure by using the intersection of theunderlying ideal sheaves.)

These definitions are independent of the defining equations h and the choice of the extension,f , of f .

Le and Teissier prove:

Proposition 2.2. Let p ∈ X. Then, for a generic choice of the linear form, z0, in a neighborhoodof p, Γ1

f,z0is reduced, is purely 1-dimensional, and the restriction of either of the maps f or z0 to

this scheme is finite.

124 DAVID B. MASSEY

Moreover, if X is irreducible at p, and Y is a proper analytic subset of X, then for a genericchoice of z0, Γ1

f,z0has no component contained in Y at p.

Proof. The first paragraph is well-known and is proved in many places; see, for instance, [L-T2],4.2.1 and [Te6], 4.1.3.2. However, for lack of a convenient reference, we will prove the secondstatement.

Suppose that X is irreducible. If f is constant on X, then Γ1f,z0

is empty, and we are finished;so, assume that f is not constant. It clearly suffices to prove the result when Y is irreducible; so,we assume that it is. Also, assume that Y 6⊆ ΣX ∪ Σ(f|Xreg

), for otherwise there is nothing toprove. Finally, it suffices to prove that Γ1

f,z0has no component contained in Yreg at p; for if Γ1

f,z0

has a component contained in ΣY , we replace Y by ΣY and induct on the dimension of Y .

LetX := Xreg − Σ

(f|Xreg

), and consider the relative conormal variety

T ∗f| X

U := (x, η) ∈ T ∗U | x ∈X, η

(Tx

X ∩ ker dxf

)= 0.

The dimension of the fibre over a point x ∈X is precisely n + 2 − dimX, and (x, dxz0) ∈ T ∗f|

X

U

if and only if x ∈ Γ1f,z0

. It follows that η−1(Yreg) ∩ T ∗f| X

U is irreducible of dimension at most

n+ 2− dimX + dimY 6 n+ 1. Therefore, the fibre of this space over p is conic and of dimensionat most n, and so does not contain dpz0 for generic z0. The result follows.

In general, we do not care is z0 is chosen so that Γ1f,z0

is reduced, but we do want that therestrictions of f and z0 are finite.

We now wish to characterize the relative polar curve in terms of conormal spaces and gapsheaves.

We remind the reader that T ∗Xreg

U is purely (n+ 1)-dimensional. Also, if X = V (h) in an open

subset V of U , then it is trivial to see that, over V, the reduced space T ∗Xreg

U is given by

⋃j

((Xj × Cn+1) ∩Mincj+1

( wJac(h)

))¬(ΣX × Cn+1).

Note that a generic linear reorganization of z (and the corresponding reorganization of w) producesa generic linear reorganization of

(w0 − ∂f

∂z0, . . . , wn − ∂f

∂zn

).

Definition 2.3. The relative conormal polar curve of X with respect to f and z, Γ1f ,z

(T ∗Xreg

U), is

defined to be the 1-th gap variety of w − ∂f∂z :=

(w0 − ∂f

∂z0, . . . , wn − ∂f

∂zn

)restricted to T ∗

XregU ,

i.e.,

Γ1f ,z

(T ∗Xreg

U) :=

(T ∗

XregU ∩ V

(w1 −

∂f

∂z1, . . . , wn −

∂f

∂zn

))¬ V

(w − ∂f

∂z

).


Note that we are not claiming that the relative conormal polar curve is independent of theextension f .

We need to investigate the relation between the relative conormal polar curve and the ordinarypolar curve. The following lemma tells us that they agree over the regular part of X, regardless ofwhether the two varieties are even really curves.

Lemma 2.4. Let p ∈ X be a regular point and suppose that dpz0 6∈ (T ∗Xreg

U)p. Then, in a

neighborhood of p, the projection map η restricted to Γ1f ,z

(T ∗Xreg

U) induces an isomorphism ontoΓ1

f,z0.

Proof. Because X is smooth at p and dpz0 6∈ (T ∗Xreg

U)p, we may use an analytic change ofcoordinates at p to reduce ourselves to the case where X = V, where V is an open subset ofCd × 0 and T ∗

XregU = V × (0 × Cn+1−d).

Now, one sees easily that Γ1f,z0

=(V(

∂f∂z1

, . . . , ∂f∂zd−1

)¬V(

∂f∂z0

))× 0 and that

Γ1f ,z

(T ∗Xreg

U) =

((V × (0 × Cn+1−d)

)∩ V

(w1 −

∂f

∂z1, . . . , wn −

∂f

∂zn

))¬ V

(w0 −

∂f

∂z0

)=

((V × (0 × Cn+1−d)

)∩ V

( ∂f∂z1

, . . . ,∂f

∂zd−1, wd −

∂f

∂zd, . . . , wn −

∂f

∂zn

))¬ V

( ∂f∂z0

)=

(Γ1

f,z0× Cn+1

)∩ V

(wd −

∂f

∂zd, . . . , wn −

∂f

∂zn

).

Thus, η restricted to Γ1f ,z

(T ∗Xreg

U) (and to its image) has as its inverse the map τ : Γ1f,z0

→

Γ1f ,z

(T ∗Xreg

U) given by τ(x) =(x,0, ∂f

∂zd, . . . , ∂f

∂zn

).

We now prove the fundamental result for polar curves on general spaces.

Theorem 2.5. Let p ∈ X. For a generic linear reorganization, z,

0) dpz0 6∈(T ∗

XregU)p;

i) Γ1f,z0

is purely 1-dimensional at p;

ii) f|Γ1f,z0

is finite at p; and

iii) Γ1f ,z

(T ∗

XregU)

has no components contained in η−1(ΣX) at (p, dpf).

In addition, whenever 0)-iii) hold, then we also have

126 DAVID B. MASSEY

iv) z0|Γ1f,z0

is finite at p;

v) Γ1f ,z

(T ∗

XregU)

is purely 1-dimensional at (p, dpf);

vi) (f η)|Γ1

f,z(T∗

XregU)

is finite at (p, dpf);

vii) (z0 η)|Γ1

f,z(T∗

XregU)

is finite at (p, dpf); and

viii)(

Γ1f ,z

(T ∗

XregU)· V(w0 − ∂f

∂z0

))(p,dpf)

=(Γ1

f,z0· V (f − f(p))

)p−(Γ1

f,z0· V (z0 − z0(p))

)p.

Proof. For notational convenience, we shall assume that f(p) = z0(p) = 0.

That 0) holds generically is trivial. That i) and ii) hold generically follows from 2.2. Asη−1(ΣX)∩ T ∗

XregU has dimension at most n, I.2.11 implies that for a generic linear reorganization(

η−1(ΣX) ∩ T ∗Xreg

U ∩ V(w1 −

∂f

∂z1, . . . , wn −

∂f

∂zn

))¬ V

(w − ∂f

∂z

)is purely 0-dimensional at (p, dpf). As every component of Γ1

f ,z

(T ∗

XregU)

has dimension at least 1(by I.2.2), iii) holds for generic z.

Now suppose that 0)-iii) hold.

By the lemma, η yields a local isomorphism between the schemes Γ1f ,z

(T ∗

XregU)− η−1(ΣX) and

Γ1f,z0

−ΣX; combining this with iii), we conclude that i) implies v), ii) implies vi), and iv) impliesvii). It remains for us to show that iv) and viii) hold.

Let C be an irreducible component of Γ1f,z0

and suppose that γ(t) is an analytic parameterizationof C near p such that γ(0) = p. For t 6= 0, γ(t) ∈ Xreg, dγ(t)f 6∈ T ∗

XregU , and there must exist

complex numbers a(t) and b(t), not both zero, such that a(t)dγ(t)f − b(t)dγ(t)z0 ∈ T ∗XregU . If a(t)

were zero for an infinite number of t, then, since T ∗Xreg

U is conic, dγ(t)z0 would be in T ∗Xreg

U foran infinite number of t; this would contradict 0). Thus, for small t 6= 0, there exists c(t) such thatdγ(t)f − c(t)dγ(t)z0 ∈ T ∗

XregU . By evaluating this form on the tangent vector γ′(t), we conclude

that(f(γ(t))

)′ − c(t)(z0(γ(t))

)′ ≡ 0.If iv) were false, then z0(γ(t)) would be zero (for some component C), and, hence, we would

have that(f(γ(t))

)′ ≡ 0; but, this would imply that f(γ(t)) ≡ 0, in contradiction of ii). Therefore,we have shown iv).

Note that c(t) is uniquely determined by

c(t) =

(f(γ(t))

)′(z0(γ(t))

)′ ,where z0(γ(t)) 6≡ 0 by iv). If |c(t)| → ∞ as t→ 0, then since 1

c(t)dγ(t)f−dγ(t)z0 ∈ T ∗XregU , we would

once again conclude that dpz0 ∈(T ∗

XregU)p, in contradiction to 0). It follows that c(t) is analytic


at 0, and thus that c(t) approaches some finite value qC

as t → 0. Therefore, the component Ccorresponds to a component, C, of Γ1

f ,z

(T ∗

XregU)

through (p, dpf) if and only if qC

= 0; for C is

parameterized by γ(t) = (γ(t), dγ(t)f − c(t)dγ(t)z0).Now suppose that, at (p, dpf), the cycle Γ1

f ,z

(T ∗

XregU)

equals ΣnV

[V ]. As the restriction of η

to Γ1f ,z

(T ∗

XregU)

is a local isomorphism onto Γ1f,z0

over smooth points of X, we conclude that thecycle Γ1

f,z0can be written as

Γ1f,z0

=∑V

nV

[η(V )] +∑

qC6=0

nC

[C].

Therefore, to demonstrate viii), we need to show two things:

a) if qC6= 0, then (C · V (f))p − (C · V (z0))p = 0, and

b) if qC

= 0, then(C · V

(w0 − ∂f

∂z0

))(p,dpf)

= (C · V (f))p − (C · V (z0))p.

We show a) and b) by calculating intersection numbers via parameterizations (see Appendix A.9).

If qC6= 0, then

(f(γ(t))

)′ and(z0(γ(t))

)′ must have the same t-multiplicity. Thus, f(γ(t)) andz0(γ(t)) have the same t-multiplicity, and so

(C · V (f))p − (C · V (z0))p = multtf(γ(t))−multtz0(γ(t)) = 0.

If qC

= 0, then

(C · V

(w0 −

∂f

∂z0

))(p,dpf)

= multt

(w0(γ(t))− ∂f

∂z0∣∣

γ(t)

)=

multt

( ∂f∂z0

∣∣γ(t)

− c(t)− ∂f

∂z0∣∣

γ(t)

)= multt(c(t)) = multt

(f(γ(t))

)′ −multt

(z0(γ(t))

)′ =

multt

(f(γ(t))

)−multt

(z0(γ(t))

)= (C · V (f))p − (C · V (z0))p.

Remark 2.6. The point of 2.5.viii is that, for generic z, the quantity

(Γ1

f ,z

(T ∗

XregU)· V(w0 −

∂f

∂z0

))(p,dpf)

is, in fact, the multiplicity of the 0-th Vogel cycle of (w − ∂f∂z )∣∣

T∗Xreg

U

at (p, dpf) (recall I.2.14).

This enables us to apply results from Part I.Note, also, in the affine case where X = U , the equality of 2.5.viii reduces to the well-known

formula from Proposition II.1.20:(Γ1

f,z0· V( ∂f∂z0

))p

=(Γ1

f,z0· V (f − f(p))

)p−(Γ1

f,z0· V (z0 − z0(p))

)p.

128 DAVID B. MASSEY

Chapter 3. THE LINK BETWEEN THE ALGEBRAIC ANDTOPOLOGICAL POINTS OF VIEW.

We continue with our previous notation: X is a d-dimensional complex analytic space containedin some open subset U of some Cn+1, f : U → C is a complex analytic function, f = f|X , S = Sαis a Whitney stratification of X with connected strata, the base ring R is a p.i.d., and F• is abounded complex of sheaves of R-modules which is constructible with respect to S. In addition,Nα and Lα are, respectively, the normal slice and complex link of the dα-dimensional stratum Sα

(see [Go-Mac1]).In this chapter, we are going to prove a general result which describes the characteristic cycle

of φfF• in terms of blowing-up the image of df inside the conormal spaces to strata. We will haveto wait until the next chapter (on results for perverse sheaves) to actually show how this providesa relationship between Σ

F• f and ΣSf in the case where F• is perverse.Because d is the global dimension of X, and we are not assuming that X is pure-dimensional,

or that f is not constant on a d-dimensional component of X, if v ∈ C, then the dimension ofV (f−v) could be anything between 0 and d. Hence, we let dv := 1+dimV (f−v), and will usuallydenote d0 by simply d. Of course, if we work locally, or assume that X is pure-dimensional, andrequire f not to vanish on a component of X, then d will have attain its “expected” value of d.

Definition 3.1. Recall that the characteristic cycle, Ch(F•), of F• in T ∗U is the linear combina-tion

∑αmα(F•)

[T ∗

SαU], where the mα(F•) are integers given by

mα(F•) := (−1)dχ(φL|X[−1]F•)x = (−1)dχ(φL|Nα

[−1]F•|Nα[−dα])x =

(−1)d−dαχ(H∗(Nα,Lα; F•)

)for any point x in Sα, with normal slice Nα at x, and any L : (U , x) → (C, 0) such that dxL is anon-degenerate covector at x (with respect to our fixed stratification; see [Go-Mac1]) and L|Sα

has a Morse singularity at x. This cycle is independent of all the choices made (see, for instance,[K-S, Chapter IX]).

We need a number of preliminary results before we can prove the main theorem (Theorem 3.10)of this section.

Definition 3.2. Recall that, if M is an analytic submanifold of U and M ⊆ X, then the relativeconormal space (of M with respect to f in U), T ∗f|M U , is given by

T ∗f|MU := (x, η) ∈ T ∗U | x ∈M, η

(ker dx(f|M )

)= 0 =

(x, η) ∈ T ∗U | x ∈M, η(TxM ∩ ker dxf

)= 0.

We define the total relative conormal cycle, T ∗f,F•

U , by T ∗f,F•

U :=∑

Sα 6⊆f−1(0)

mα(F•)[T ∗f|Sα

U].


From this point, through Lemma 3.9, it will be convenient to assume that we haverefined our stratification S = Sα so that V (f) is a union of strata. By Remark 1.7,this implies that, in a neighborhood of V (f), if Sα 6⊆ V (f), then Σ(f|Sα

) = ∅.

We first stated Theorems 3.3 and 3.4 below in our earlier works [Mas2] and [Mas5]. In thosepapers, we were mainly concerned with local questions, and we also tacitly assumed that f was notconstant on any irreducible component of X. We did not correctly adjust the sign for degeneratecases. We correct this error in the statements below – the proofs remain the same.

We shall need the following important result from [BMM, 3.4.2].

Theorem 3.3. ([BMM]) The shifted characteristic cycle of the sheaf of nearby cycles of F•

along f , (−1)d−d

Ch(ψfF•

), is isomorphic to the intersection product T ∗

f,F•U ·(V (f) × Cn+1

)in

U × Cn+1.

We should note here that the context of [BMM] is that of D-modules and, hence, in that work,the complex of sheaves was a complex of C-vector spaces. However, Theorem 3.3 can easily berecovered by combining the first formula of Theorem 3.4 with Lemma 3.5 (see below), keeping inmind that Lemma 3.5 relies on the result of Theorem 3.3 with C-complexes only.

Let Γ1f,L(Sα) denote the relative polar curve of f|

Sαwith respect to a generic linear form L (see

Chapter 2 and [M1] and [M3]). It is important to note that the second part of 2.2 implies thatΓ1

f,L(Sα) has no components contained in any strata Sβ ⊆ Sα such that Sβ 6= Sα.It is convenient to have a specific point in X at which to work. Below, we concentrate our

attention at the origin; of course, if the origin is not in X (or, if the origin is not in V (f)), then weobtain zeroes for all the terms below. For any bounded, constructible complex A• on a subspaceof U , let m0(A•) equal the coefficient of

[T ∗0U

]in the characteristic cycle of A•.

We need to state one further result without proof – this result can be obtained from [BMM],but we give the result as stated in [Mas5, 4.6], with the added corrections of (−1)d−d in variousplaces.

Theorem 3.4. For generic linear forms L, we have the following formulas:

(−1)d−d

m0(ψfF•) =∑

Sα 6⊆V (f)

mα(F•)(Γ1

f,L(Sα) · V (f)

)0;

m0(F•) + (−1)d−d

m0(F•|V (f)) =

∑Sα 6⊆V (f)

mα(F•)(Γ1

f,L(Sα) · V (L)

)0; and

(−1)d−d

m0(φfF•) = m0(F•) +∑

Sα 6⊆V (f)

mα(F•)((

Γ1f,L

(Sα) · V (f))0−(Γ1

f,L(Sα) · V (L)

)0

).

130 DAVID B. MASSEY

Lemma 3.5. If Sα 6⊆ f−1(0), then the coefficient of[P(T ∗0U)

]in P

(T ∗

f|Sα

U)·(V (f) × Pn

)is

given by(Γ1

f,L(Sα) · V (f)

)0.

Proof. Take a complex of sheaves, F•, which has a characteristic cycle consisting only of[T ∗

SαU]

(see, for instance, [M1]). Now, apply the formula for m0(ψfF•) from Theorem 3.4 together withTheorem 3.3.

We need to establish some notation that we shall use throughout the remainder of this section.

Using the isomorphism, T ∗U ∼= U × Cn+1, we consider Ch(F•) as a cycle in X × Cn+1; we usez := (z0, . . . , zn) as coordinates on U and w := (w0, . . . , wn) as the cotangent coordinates.

Let I denote the sheaf of ideals on U given by the image of df , i.e., I =⟨w0− ∂f

∂z0, . . . , wn− ∂f

∂zn

⟩.

For all α, let Bα = Blim df T∗SαU denote the blow-up of T ∗

SαU along the image of I in T ∗

SαU , and let

Eα denote the corresponding exceptional divisor. For all α, we have Eα ⊆ Bα ⊆ X × Cn+1 × Pn.Let π : X×Cn+1×Pn → X×Pn denote the projection. Note that, if (x,w, [η]) ∈ Eα, then w = dxfand so, for all α, π induces an isomorphism from Eα to π(Eα). We refer to E :=

∑αmαEα as the

total exceptional divisor inside the total blow-up Blim df Ch(F•) :=∑

αmα Blim df

[T ∗

SαU].

Lemma 3.6. For all Sα, there is an inclusion π(

Blim df T∗SαU)⊆ P

(T ∗

f|Sα

U).

Proof. This is entirely straightforward. Suppose that

(x,w, [η]) ∈ Blim df T∗SαU = Blim df T

∗SαU .

Then, we have a sequence (xi,wi, [ηi]) ∈ Blim df T∗SαU such that (xi,wi, [ηi]) → (x,w, [η]).

By definition of the blow-up, for each (xi,wi, [ηi]), there exists a sequence (xji ,w

ji ) ∈ T ∗

SαU −

im df such that (xji ,w

ji , [w

ji−dxj

if ]) → (xi,wi, [ηi]). Now, (xj

i , [wji−dxj

if ]) is clearly in P

(T ∗

f|Sα

U),

and so each (xi, [ηi]) is in P(T ∗

f|Sα

U). Therefore, (x, [η]) ∈ P

(T ∗

f|Sα

U).

Lemma 3.7. If Sα 6⊆ f−1(0), then the coefficient of[P(T ∗0U)]

= 0 × Pn in π∗(Eα) equals(Γ1

f,L(Sα) · V (f)

)0−(Γ1

f,L(Sα) · V (L)

)0.

Proof. By I.2.23, the multiplicity of 0×Pn in π∗(Eα) equals(∆0(

w− ∂f∂z

)|T∗

SαU

)(0,d0f)

, for a generic

choice of z. By 2.5.viii and 2.6, this is equal to(Γ1

f,L(Sα) ·V (f)

)0−(Γ1

f,L(Sα) ·V (L)

)0

for a genericlinear choice of L.


Lemma 3.8. For all α such that Sα ⊆ V (f), there is an inclusion of the exceptional divisor

Eα∼= π(Eα) ⊆ P

(T ∗

f|Sα

U)∩(V (f)× Pn

).

Proof. That π is an isomorphism when restricted to the exceptional divisor is trivial: (x,w, [η]) ∈Eα implies that w = dxf . From Lemma 3.6, π(Eα) ⊆ π

(Blim df T

∗SαU)⊆ P

(T ∗

f|Sα

U). The result

follows.

Lemma 3.9. If Sα ⊆ f−1(0), then Eα∼= π(Eα) = P(T ∗

SαU).

Proof. If Sα ⊆ f−1(0), then P(T ∗

f|Sα

U)

= P(T ∗

SαU), and so, by 3.8, π(Eα) ⊆ P(T ∗

SαU). We will

demonstrate the reverse inclusion.Suppose that we have (x, [η]) ∈ P(T ∗

SαU). Then, there exists a sequence (xi, ηi) ∈ T ∗

SαU such

that (xi, ηi) → (x, η). Hence,(xi,

1i ηi + dxi f

)∈ T ∗

SαU − im df and(

xi,1iηi + dxi

f ,[(1iηi + dxi

f)− dxi

f])

→ (x, dxf , [η]) ∈ Eα.

We come now to the main theorem of this section. This theorem relates the topological dataprovided by the vanishing cycles of a function f to the algebraic data given by blowing-up theimage of the differential of an extension of f .

Theorem 3.10. The projection π induces an isomorphism between the total exceptional divisorE ⊆ Blim df Ch(F•) and the sum over all v ∈ C of the projectivized characteristic cycles of thesheaves of vanishing cycles of F• along f − v, i.e.,

E ∼= π∗(E) =∑v∈C

(−1)d−dv P(Ch(φf−vF•)).

Proof. Remarks 1.7 and 1.10 imply that, locally, suppφf−vF• ⊆ f−1(v). As the P(Ch(φf−vF•))are disjoint for different values of v, we may immediately reduce ourselves to the case where weare working near 0 ∈ X and where f(0) = 0. We refine our stratification so that, for all α,Σ(f|Sα

) = ∅ unless Sα ⊆ V (f). As any newly introduced stratum will appear with a coefficientof zero in the characteristic cycle, the total exceptional divisor will not change. We need to showthat E ∼= π(E) = P(Ch(φfF•)).

Now, we will first show that π(E) is Lagrangian.

If Sα ⊆ f−1(0), then π(Eα) = P(T ∗SαU) by 3.9. If Sα 6⊆ f−1(0), then, by Theorem 3.3,

P(T ∗

f|Sα

U)∩(V (f) × Pn

)is Lagrangian and, in particular, is purely n-dimensional. By Lemma

132 DAVID B. MASSEY

3.8, π(Eα) is a purely n-dimensional analytic set contained in P(T ∗

f|Sα

U)∩(V (f)× Pn

). We need

to show that π(Eα) is closed.Suppose we have a sequence (xi, [ηi]) ∈ π(Eα) and (xi, [ηi]) → (x, [η]) in U × Pn. Then, there

exists a sequence wi so that (xi,wi, [ηi]) ∈ Eα; by definition of the exceptional divisor, this implieswi = dxi

f . Therefore, (xi,wi, [ηi]) → (x, dxf , [η]), which is contained in Eα since Eα is closed inU × Cn+1 × Pn. Thus, (x, [η]) ∈ π(Eα), and so π(Eα) is closed and, hence, Lagrangian.

Now, π(E) and P(Ch(φfF•)) are both supported over ΣSf and, by taking normal slices tostrata, we are reduced to the point-stratum case. Thus, what we need to show is: the coefficientof[P(T ∗0U)]

in E equals the coefficient of[P(T ∗0U)]

in (−1)d−dP(Ch(φfF•)). Using 3.4, this is

equivalent to showing that the coefficient of[P(T ∗0U)]

in E equals

m0(F•) +∑

Sα 6⊆V (f)

mα

((Γ1

f,L(Sα) · V (f)

)0−(Γ1

f,L(Sα) · V (L)

)0

)

for a generic linear form L.But, by 3.9,

E =∑α

mαEα =∑

Sα⊆V (f)

mα

[P(T ∗

SαU)]

+∑

Sα 6⊆V (f)

mαEα

and the coefficient of[P(T ∗0U)]

in∑

Sα⊆V (f)

mα

[P(T ∗

SαU)]

is precisely m0(F•).

Therefore, we will be finished if we can show that the coefficient of[P(T ∗0U)]

in Eα equals(Γ1

f,L(Sα) · V (f)

)0−(Γ1

f,L(Sα) · V (L)

)0

if Sα 6⊆ V (f). However, this is exactly the content ofLemma 3.7.

Remark 3.11. In special cases, Theorem 3.10 was already known.

Consider the case where X = U and F• is the constant sheaf. Then, Ch(F•) = U ×0, and theimage of df in U×0 is simply defined by the Jacobian ideal of f . Hence, our result reduces to theresult obtained from the work of Kashiwara in [K] and Le-Mebkhout in [L-M] – namely, that theprojectivized characteristic cycle of the sheaf of vanishing cycles is isomorphic to the exceptionaldivisor of the blow-up of the Jacobian ideal in affine space.

As a second special case, suppose that X and F• are completely general, but that x is an isolatedpoint in the image of Ch(φfF•) in X (for instance, x might be an isolated point in suppφfF•).Then, for every stratum for which mα 6= 0, (x, dxf) is an isolated point of im df ∩ T ∗

SαU or is not

contained in the intersection at all.Therefore, the last part of I.2.23 implies that the exceptional divisor of the blow-up of im df in

T ∗SαU has one component over (x, dxf) and that that component occurs with multiplicity precisely

equal to the intersection multiplicity(

im df · T ∗SαU)

(x,dxf)in T ∗U . Thus, we recover the results of

three independent works appearing in [Gi], [Le3], and [Sab2] – that the coefficient of x×Cn+1


in (−1)d−d

Ch(φfF•) is given by(

im df · Ch(F•))

(x,dxf). This result is usually stated in terms of

the Euler characteristic: if x is an isolated point in suppφfF•, then

χ(φf [−1]F•)x = (−1)d(

im df · Ch(F•))

(x,dxf).

In addition to generalizing the above results, Theorem 3.10 fits in well with Theorem 3.4.2 of[BMM]; that theorem contains a nice description of the characteristic cycles of the nearby cyclesand of the restriction of a complex to a hypersurface. However, [BMM] does not contain a nicedescription of the vanishing cycles, nor does our Theorem 3.10 seem to follow easily from the resultsof [BMM]; in fact, Example 3.4.3 of [BMM] makes it clear that the general result contained inour Theorem 3.10 was unknown – for Briancon, Maisonobe, and Merle only derive the vanishingcycle result from their nearby cycle result in the easy, known case where the vanishing cycles aresupported on an isolated point and, even then, they must make half a page of argument.

Corollary 3.12. For each extension f of f , let Ef denote the exceptional divisor in Blim df T∗Xreg

U .

Then, π(Ef

)is independent of f .

Proof. We apply Theorem 3.10 to a complex of sheaves F• such that mα = 1 for each smoothcomponent of Xreg and mα = 0 for every other stratum in some Whitney stratification of X (it iseasy to produce such an F• – see, for instance, Lemma 3.1 of [M1]). The corollary follows fromthe fact that P(Ch(φfF•)) does not depend on the extension.

134 DAVID B. MASSEY

Chapter 4. THE SPECIAL CASE OF PERVERSE SHEAVES.

We continue with our previous notation, except that in this chapter we must assumethat our base ring is a field.

For the purposes of Part III, perverse sheaves are important because the vanishing cycles functor(shifted by −1) applied to a perverse sheaf once again yields a perverse sheaf and because of thefollowing lemma.

Lemma 4.1. If P• is a perverse sheaf on X, then Ch(P•) =∑

αmα

[T ∗

SαU], where

mα = (−1)d dimH0(Nα,Lα; P•|Nα

[−dα]);

in particular, (−1)d Ch(P•) is a non-negative cycle.If P• is perverse on X (or, even, perverse up to a shift), then supp P• equals the image in X

of the characteristic cycle of P•.

Proof. The first statement follows from the definition of the characteristic cycle, together with thefact that a perverse sheaf supported on a point has non-zero cohomology only in degree zero.

The second statement follows at once from the fact that if P• is perverse up to a shift, then sois the restriction of P• to its support. Hence, by the support condition on perverse sheaves, thereis an open dense set of the support, Ω, such that, for all x ∈ Ω, H∗(P•)x is non-zero in a singledegree. The conclusion follows.

The fact that the above lemma refers to the support of P•, which is the closure of the set ofpoints with non-zero stalk cohomology, means that we can use it to conclude something about theclosure of the P•-critical locus (recall Definition 1.9).

Theorem 4.2. Let P• be a perverse sheaf on X, and suppose that the characteristic cycle of P•

in U is given by Ch(P•) =∑

αmα

[T ∗

SαU].

Then, the closure of the P•-critical locus of f is given by

ΣP• f =

x ∈ X

∣∣ (x, dxf) ∈ |Ch(P•)|

=⋃

mα 6=0

Σcnr

(f|

Sα

).

Proof. Let q ∈ X, and let v = f(q). Let W be an open neighborhood of q in X such that W ∩Σ

P• f ⊆ V (f−v) (see the end of Remark 1.7). Then, W∩ΣP• f = W∩suppφf−vP•. As φf−vP•[−1]

is perverse, Lemma 4.1 tells us that suppφf−vP• equals the image in X of Ch(φf−vP•). Now,Theorem 3.10 tells us that this image is precisely⋃

mα 6=0

x ∈ Sα | (x, dxf) ∈ T ∗

SαU,

since there can be no cancellation as all the non-zero mα have the same sign.Therefore, we have the desired equality of sets in an open neighborhood of every point; the

theorem follows.


We will use the perverse cohomology of the shifted constant sheaf, C•X [k], in order to deal withnon-l.c.i.’s; this perverse cohomology is denoted by µH0(C•X [k]) = µHk(C•X) (see [BBD], [K-S],or Appendix B). Like the intersection cohomology complex, this sheaf has the property that it isthe shifted constant sheaf on the smooth part of any component of X with dimension equal tod = dimX.

We now list some properties of the perverse cohomology and of vanishing cycles that we willneed later. For further properties, see Appendix B.

The perverse cohomology functor on X, µH0, is a functor from the derived category of bounded,constructible complexes on X to the Abelian category of perverse sheaves on X.

If F• is constructible with respect to S, then µH0(F•) is also constructible with respect to S,and

(µH0(F•)

)|Nα

[−dα] is naturally isomorphic to µH0(F•|Nα[−dα]).

The functor µH0, applied to a perverse sheaf P• is canonically isomorphic to P•. In addition, abounded, constructible complex of sheaves F• is perverse if and only µH0(F•[k]) = 0 for all k 6= 0.In particular, if X is an l.c.i., then µH0(C•X [d]) ∼= C•X [d] and µH0(C•X [k]) = 0 if k 6= d.

The functor µH0 commutes with vanishing cycles with a shift of −1, nearby cycles with a shiftof −1, and Verdier dualizing. That is, there are natural isomorphisms

µH0 φf [−1] ∼= φf [−1] µH0, µH0 ψf [−1] ∼= ψf [−1] µH0, and D µH0 ∼= µH0 D.

Let F• be a bounded complex of sheaves on X which is constructible with respect to a connectedWhitney stratification Sα of X. Let Smax be a maximal stratum contained in the support ofF•, and let m = dimSmax. Then,

(µH0(F•)

)|Smax

is isomorphic (in the derived category) to thecomplex which has (H−m(F•))|Smax

in degree −m and zero in all other degrees.In particular, supp F• =

⋃i supp µH0(F•[i]), and if F• is supported on an isolated point, q,

then H0(µH0(F•))q ∼= H0(F•)q.

Throughout the remainder of Part III, we let kP• denote the perverse sheaf µH0(C•X [k + 1]); itwill be useful later to have a nice characterization of the characteristic cycle of kP•.

Proposition 4.3. The complex kP• is a perverse sheaf on X which is constructible with respectto S and the characteristic cycle Ch(kP•) is equal to

(−1)d∑α

bk+1−dα(Nα,Lα)[T ∗

SαU],

where bj denotes the j-th (relative) Betti number.

In particular, H∗(Lα; C) ∼= H∗(point; C) if and only if mα

(kP•) = 0 for all k.

Proof. The constructibility claim follows from the fact that the constant sheaf itself is clearlyconstructible with respect to any Whitney stratification. The remainder follows trivially from thedefinition of the characteristic cycle, combined with two properties of µH0; namely, µH0 commuteswith φf [−1], and µH0 applied to a complex which is supported at a point simply gives ordinarycohomology in degree zero and zeroes in all other degrees. See [K-S, 10.3].

136 DAVID B. MASSEY

Remark 4.4. As Nα is contractible, it is possible to give a characterization of bk+1−dα(Nα,Lα)without referring to Nα; the statement gets a little complicated, however, since we have to worryabout what happens near degree zero and because the link of a maximal stratum is empty. However,if we slightly modify the usual definitions of reduced cohomology and the corresponding reducedBetti numbers, then the statement becomes quite easy.

What we want is for the “reduced” cohomology Hk(A; C) to be the relative cohomology vectorspace Hk+1(B,A; C), where B is a contractible set containing A, and we want b∗() to be the Bettinumbers of this “reduced” cohomology. Therefore, letting bk() denote the usual k-th Betti number,we define b∗() by

bk(A) =

bk(A), if k 6= 0 and A 6= ∅b0(A)− 1, if k = 0 and A 6= ∅0, if k 6= −1 and A = ∅1, if k = −1 and A = ∅.

Thus, bk(A) is the k-th Betti number of the reduced cohomology, provided that A is not the emptyset.

The special definition of bk() for the empty set implies that if Sα is maximal, so that Nα = pointand Lα = ∅, then

bk+1−dα(Nα,Lα) =

0, if k + 1 6= dα

1, if k + 1 = dα

= bk−dα(Lα).

Thus, with this new notation,

Ch(kP•) = (−1)d∑α

bk−dα(Lα)

[T ∗

SαU].

By combining 4.2 with 4.3 and 4.4, we can now give a result about ΣCf . First, though, it willbe useful to adopt the following terminology.

Definition 4.5. We say that the stratum Sα is visible (or, C-visible) if H∗(Lα; C) 6∼= H∗(point; C)(or, equivalently, if H∗(Nα,Lα; C) 6= 0). Otherwise, the stratum is invisible.

The final line of Proposition 4.3 tells us that a stratum is visible if and only if there exists aninteger k such that

[T ∗

SαU]

appears with a non-zero coefficient in Ch(kP•).

Note that if Sα has an empty complex link (i.e., the stratum is maximal), then Sα is visible.

Theorem 4.6. Then,

ΣCf =d−1⋃

k=−1

ΣkP•

f =⋃

visible Sα

x ∈ Sα | (x, dxf) ∈ T ∗

SαU

=⋃

visible Sα

Σcnr

(f|

Sα

).

In particular, since all maximal strata are visible, Σcnrf ⊆ ΣCf (as stated in Proposition 1.6).Moreover, if x is an isolated point of ΣCf ,then, for all Whitney stratifications, Rβ, of X, theonly possibly visible stratum which can be contained in f−1f(x) is x.


Proof. Recall that, for any complex F•, supp F• =⋃

k supp µH0(F•[k]). In addition, we claim thatkP• = 0 unless −1 6 k 6 d− 1. By Lemma 4.1, kP• = 0 is equivalent to Ch(kP•) = 0; if k is notbetween −1 and d − 1, then, using Proposition 4.3, Ch(kP•) = 0 follows from the fact that thecomplex link of a stratum has the homotopy-type of a finite CW complex of dimension no morethan the complex dimension of the link (see [Go-Mac1]).

Now, in an open neighborhood of any point q with v := f(q), we have

ΣCf = suppφf−vC• =⋃k

supp µH0(φf−vC•X [k]) =

⋃k

suppφf−v[−1](µH0(C•X [k + 1])

)=⋃k

ΣkP•

f.

Now, applying Theorem 4.2, we have

ΣCf =⋃k

⋃mα(kP•) 6=0

x ∈ Sα | (x, dxf) ∈ T ∗

SαU.


Remark 4.7. Those familiar with stratified Morse theory should find the result of Theorem 4.6 veryun-surprising – it looks like it results from some break-down of the C-critical locus into normaland tangential data, and naturally one gets no contributions from strata with trivial normal data.This is the approach that we took in Theorem 3.2 of [Ma1]. There is a slightly subtle, technicalpoint which prevents us from taking this approach in our current setting: by taking normal slicesat points in an open, dense subset of suppφf−vC•X , we could reduce ourselves to the case whereΣCf consists of a single point, but we would not know that the point was a stratified isolatedcritical point. In particular, the case where suppφf−vC•X consists of a single point, but wheref has a non-isolated (stratified) critical locus coming from an invisible stratum causes difficultieswith the obvious Morse Theory approach.

Remark 4.8. At this point, we wish to add to our hierarchy of critical loci from Proposition 1.6.Theorem 4.6 tells us that Σ

kP•f ⊆ ΣCf for all k. If X is purely (m + 1)-dimensional, then 4.2

implies that Σcnrf ⊆ ΣmP• f .Now, suppose that X is irreducible of dimension m+1. Let IC• be the intersection cohomology

sheaf (with constant coefficients) on X (see [Go-Mac2]); IC• is a simple object in the category ofperverse sheaves. As the category of perverse sheaves on X is (locally) Artinian, and since mP• isa perverse sheaf which is the shifted constant sheaf on the smooth part of X, it follows that IC•

appears as a simple subquotient in any composition series for mP•. Consequently, |Ch(IC•)| ⊆|Ch(mP•)|, and so 4.2 implies that Σ

IC• f ⊆ ΣmP• f . Moreover, 4.2 also implies that Σcnrf ⊆Σ

IC• f . Therefore, we can extend our sequence of inclusions from Proposition 1.6 to:

Σregf ⊆ Σalgf ⊆ ΣNashf ⊆ Σcnrf ⊆ ΣIC• f ⊆ ΣmP• f ⊆ ΣCf ⊆ Σcanf ⊆ Σrdff.

Why not use one of these new critical loci as our most fundamental notion of the critical locusof f? Both Σ

IC• f and ΣmP• f are topological in nature, and easy examples show that they canbe distinct from ΣCf . However, 4.6 tells us that ΣmP• f is merely one piece that goes into makingup ΣCf – we should include the other shifted perverse cohomologies. On the other hand, given

138 DAVID B. MASSEY

the importance of intersection cohomology throughout mathematics, one should wonder why wedo not use Σ

IC• f as our most basic notion.Consider the node X := V (y2 − x3 − x2) ⊆ C2 and the function f := y|X . The node has a

small resolution of singularities (see [Go-Mac2]) given by simply pulling the branches apart. Asa result, the intersection cohomology sheaf on X is the constant sheaf shifted by one on X − 0,and the stalk cohomology at 0 is a copy of C2 concentrated in degree −1. Therefore, one can easilyshow that 0 6∈ Σ

IC• f .As Σ

IC• f fails to detect the simple change in topology of the level hypersurfaces of f as theygo from being two points to being a single point, we do not wish to use Σ

IC• f as our basic type ofcritical locus. That is not to say that Σ

IC• f is not interesting in its own right; it is integrally tiedto resolutions of singularities. For instance, it is easy to show (using the Decomposition Theorem[BBD]) that if X π−→ X is a resolution of singularities, then Σ

IC• f ⊆ π(Σ(f π)).

Now that we can “calculate” ΣCf using Theorem 4.6, we are ready to generalize the Milnornumber of a function with an isolated critical point.

Definition 4.9. If P• is a perverse sheaf on X, and x is an isolated point in ΣP• f (or, if x 6∈ Σ

P• f),then we call dimC H

0(φf−f(x)[−1]P•)x the Milnor number of f at x with coefficients in P• andwe denote it by µx(f ; P•).

This definition is reasonable for, in this case, φf−f(x)[−1]P• is a perverse sheaf supported atthe isolated point x. Hence, the stalk cohomology of φf−f(x)[−1]P• at x is possibly non-zero onlyin degree zero. Normally, we summarize that x is an isolated point in Σ

P• f or that x 6∈ ΣP• f by

writing dimxΣP• f 6 0 (we consider the dimension of the empty set to be −∞).

Before we state the next proposition, note that it is always the case that(im df · T ∗0U

)(0,d0f)

= 1.

Proposition 4.10. For notational convenience, we assume that 0 ∈ X and that f(0) = 0.Then, dim0ΣCf 6 0 if and only if, for all k, dim0ΣkP•f 6 0. Moreover, if dim0ΣCf 6 0, then,

i) for all visible strata, Sα, such that dimSα > 1, the intersection of im df and T ∗SαU is, at

most, 0-dimensional at (0, d0f),and (

im df · T ∗SαU)

(0,d0f)=(Γ1

f,L(Sα) · V (f)

)0−(Γ1

f,L(Sα) · V (L)

)0,

where L is a generic linear form, and

ii) for all k,

µ0(f ; kP•) = bk(Ff,0) = (−1)dim X(

im df · Ch(kP•))(0,d0f)

=


∑visible Sα

bk−dα(Lα)

(im df · T ∗

SαU)

(0,d0f)=

∑visible Sα

Sα not maximal

bk−dα(Lα)

(im df · T ∗

SαU)

(0,d0f)+

∑Sα maximaldim Sα=k+1

(im df · T ∗

SαU)

(0,d0f).

Proof. It follows immediately from 4.6 that dim0ΣCf 6 0 if and only if, for all k, dim0ΣkP•f 6 0.

i) follows immediately from Lemma 3.7 (combined with Remark 3.11).

It remains for us to prove ii). As in the proof of 4.6, we have

µH0(φfC•X [k]) = φf [−1](µH0(C•X [k + 1])

)= φf [−1]kP•.

It follows that

µ0(f ; kP•) = dimC H0(φf [−1]kP•)0 = dimC H

0(µH0(φfC•X [k])

)0

= dimC H0(φfC•X [k]

)0,

where the last equality is a result of the fact that 0 is an isolated point in the support of φfC•X [k].Therefore,

µ0(f ; kP•) = dimC H0(φfC•X [k]

)0

= dimC Hk(φfC•X

)0

= dim Hk(Ff,0; C).

That we also have the equality

µ0(f ; kP•) = (−1)dim X(

im df · Ch(kP•))(0,d0f)

is precisely the content of Theorem 3.10, interpreted as in the last paragraph of Remark 3.11.The remaining equalities in ii) follow from the description of Ch(kP•) given in Proposition 4.3

and Remark 4.4.

Remark 4.11. The formulas from 4.10 provide a topological/algebraic method for “calculating”the Betti numbers of the Milnor fibre for isolated critical points on arbitrary spaces. It should notbe surprising that the data that one needs is not just the algebraic data – coming from the polarcurves and intersection numbers – but also includes topological data about the underlying space:one has to know the Betti numbers of the complex links of strata.

Example 4.12. The most trivial, non-trivial case where one can apply 4.10 is the case where X isan irreducible local, complete intersection with an isolated singularity (that is, X is an irreduciblei.c.i.s). Let us assume that 0 ∈ X is the only singular point of X and that f has an isolatedC-critical point at 0. Let d denote the dimension of X.

Let us write LX,0 for the complex link of X at 0. By [Le1], LX,0 has the homotopy-type of afinite bouquet of (d − 1)-spheres. Applying 4.10.ii, we see, then, that the reduced cohomology ofFf,0 is concentrated in degree (d− 1), and the (d− 1)-th Betti number of Ff,0 is equal to

bd−1(LX,0)(

im df · T ∗0U)

(0,d0f)+(

im df · T ∗XregU)

(0,d0f)=

140 DAVID B. MASSEY

bd−1(LX,0) +(Γ1

f,L(Xreg) · V (f)

)0−(Γ1

f,L(Xreg) · V (L)

)0,

for generic linear L.Now, the polar curve and the intersection numbers are quite calculable in practice; see Remark

1.8 and Example 1.9 of [Ma1]. However, there remains the question of how one can computebd−1(LX,0). Corollary 5.6 and Example 5.4 of [Ma1] provide an inductive method for computingthe Euler characteristic of LX,0 (the induction is on the codimension of X in U) and, since weknow that LX,0 has the homotopy-type of a bouquet of spheres, knowing the Euler characteristicis equivalent to knowing bd−1(LX,0).

The obstruction to using 4.10 to calculate Betti numbers in the general case is that, if X is notan l.c.i., then a formula for the Euler characteristic of the link of a stratum does not tell us theBetti numbers of the link.

Example 4.13. In this example, X will be a hypersurface with a non-isolated singularity. Use(x, y, z) as coordinates for U := C3, and let X := V (xy). Let f := xα + yβ + zγ , where α, β, γ > 2.

The strata are S0 := V (x, y), S1 := V (x) − V (y), and S2 := V (y) − V (x), with correspondinglinks L0 = two points, L1 = ∅, and L2 = ∅. As Ch(kP•) = (−1)d

∑α bk−dα(Lα)

[T ∗

SαU], we see

that Ch(kP•) = 0 unless k = 1, and

Ch(1P•) =[T ∗

S0U]

+[T ∗

S1U]

+[T ∗

S2U]

= [V (x, y, w2)] + [V (x,w1, w2)] + [V (y, w0, w2)] .

Now, we have thatim df = V (w0 − αxα−1, w1 − βyβ−1, w2 − γzγ−1)

and im df ∩ |Ch(1P•)| = (0,0). Therefore, dim0ΣCf = 0, the only non-zero reduced Betti numberof the Milnor fibre of f at 0 is

b1(Ff,0) =(

im df · Ch(1P•))(0,0)

=

(c− 1) + (b− 1)(c− 1) + (a− 1)(c− 1) = (c− 1)(a+ b− 1).

One can actually verify this computation. The Milnor fibre Ff,0 is easily seen to be the unionof the Milnor fibre, F1, of yβ + zγ restricted to V (x) and the Milnor fibre, F2, of xα + zγ restrictedto V (y); these two fibres intersect in c distinct points. The classical calculation of the Milnornumbers tells us that F1 is homotopy-equivalent to a bouquet of (β − 1)(γ − 1) 1-spheres, whileF2 is homotopy-equivalent to a bouquet of (α− 1)(γ − 1) 1-spheres. Applying the Mayer-Vietorisexact sequence, we recover the equality above.

Example 4.14. In this example, X will be the simplest non-l.c.i. Use (u, x, y, z) as coordinates forU := C4, and let X := V (u, x) ∪ V (y, z). Let f := uα + xβ + yγ + zδ, where α, β, γ, δ > 2.

The strata are S0 := 0, S1 := V (u, x) − 0, and S2 := V (y, z) − 0, with correspondinglinks L0 = two complex disks (sets of complex dimension one), L1 = ∅, and L2 = ∅.

We see that Ch(kP•) = 0 unless k = 0 or 1, and

Ch(1P•) =[T ∗

S1U]

+[T ∗

S2U]

= [V (u, x, w2, w3)] + [V (y, z, w0, w1)] ,


whileCh(0P•) =

[T ∗

S0U]

= [V (u, x, y, z)] .

Now, we findim df = V (w0 − αuα−1, w1 − βxβ−1, w2 − γyγ−1, w3 − δzδ−1),

im df ∩ |Ch(1P•)| = 0, and im df ∩ |Ch(0P•)| = 0.Therefore, dim0ΣCf = 0, the only non-zero reduced Betti numbers of the Milnor fibre of f at 0

are b1 and b0, and

b1(Ff,0) =(

im df · Ch(1P•))(0,0)

= (γ − 1)(δ − 1) + (α− 1)(β − 1)

andb0(Ff,0) =

(im df · Ch(0P•)

)(0,0)

= 1.

Again, one can actually verify this computation. The Milnor fibre Ff,0 is easily seen to be thedisjoint union of the Milnor fibre, F1, of yγ + zδ restricted to V (u, x) and the Milnor fibre, F2, ofuα + xβ restricted to V (y, z). The classical calculation of the Milnor numbers tells us that F1 ishomotopy-equivalent to a bouquet of (γ − 1)(δ − 1) 1-spheres, while F2 is homotopy-equivalent toa bouquet of (α− 1)(β − 1) 1-spheres. Thus, we recover the equalities above.

142 DAVID B. MASSEY

Chapter 5. THOM’S af CONDITION.

We continue with the notation from Chapter 3.

In this section, we explain the fundamental relationship between Thom’s af condition and thevanishing cycles of f .

Definition 5.1. Let M and N be analytic submanifolds of X such that f has constant rank onN . Then, the pair (M,N) satisfies Thom’s af condition at a point x ∈ N if and only if we have

the containment(T ∗f|M

U)x⊆(T ∗f|N

U)x

of fibres over x.

In particular, if f is, in fact, constant on N , then the pair (M,N) satisfies Thom’s af condition

at a point x ∈ N if and only if we have the containment(T ∗f|M

U)x⊆(T ∗

NU)x

of fibres over x.

We have been slightly more general in the above definition than is sometimes the case; wehave not required that the rank of f be constant on M . Thus, if X is an analytic space, wemay write that (Xreg, N) satisfies the af condition, instead of writing the much more cumbersome(Xreg −Σ

(f|Xreg

), N) satisfies the af condition. If f is not constant on any irreducible component

of X, it is easy to see that these statements are equivalent:

LetX := Xreg − Σ

(f|Xreg

), which is dense in Xreg (as f is not constant on any irreducible

components of X). We claim that T ∗f| X

U = T ∗f|XregU ; clearly, this is equivalent to showing that

T ∗f|XregU ⊆ T ∗f|

X

U . This is simple, for if x ∈ Σ(f|Xreg

), then (x, η) ∈ T ∗f|Xreg

U if and only if

(x, η) ∈ T ∗Xreg

U , and T ∗Xreg

U ⊆ T ∗XU ⊆ T ∗f|

X

U .

The link between Theorem 3.10 and the af condition is provided by the following theorem, whichdescribes the fibre in the relative conormal in terms of the exceptional divisor in the blow-up ofim df . Originally, we needed to assume Whitney’s condition a) as an extra hypothesis; however,T. Gaffney showed us how to remove this assumption by using a re-parameterization trick.

Theorem 5.2. Let π : U × Cn+1 × Pn → U × Pn denote the projection.Suppose that f is not constant on any irreducible component of X. Let E denote the exceptional

divisor in Blim df T∗Xreg

U ⊆ U × Cn+1 × Pn.

Then, for all x ∈ X, there is an inclusion of fibres over x given by(π(E)

)x⊆(P(T ∗f|Xreg

U))

x.

Moreover, if x ∈ ΣNashf , then this inclusion is actually an equality.

Proof. By 3.12, it does not matter what extension of f we use.

That(π(E)

)x⊆(P(T ∗f|Xreg

U))

xis easy. Suppose that (x, [η]) ∈ π(E), that is (x, dxf , [η]) ∈ E.

Then, there exists a sequence (xi, ωi) ∈ T ∗XregU−im df such that (xi, ωi, [ωi−dxi

f ]) → (x, dxf , [η]).

Hence, there exist scalars ai such that ai(ωi − dxif) → η, and these ai(ωi − dxi

f) are relativeconormal covectors whose projective class approaches that of η. Thus,

(π(E)

)x⊆(P(T ∗f|Xreg

U))

x.


We must now show that(P(T ∗f|Xreg

U))

x⊆(π(E)

)x, provided that x ∈ ΣNashf .

LetX := Xreg − Σ

(f|Xreg

). Suppose that (x, [η]) ∈ P

(T ∗f|

X

U). Then, there exists a complex

analytic path α(t) = (x(t), ηt) ∈ T ∗f| X

U such that α(0) = (x, η) and α(t) ∈ T ∗f| X

U for t 6= 0.

As f has no critical points onX, each ηt can be written uniquely as ηt = ωt + λ(x(t))dx(t)f ,

where ωt(Tx(t)

X) = 0 and λ(x(t)) is a scalar. By evaluating each side on x′(t), we find that

λ(x(t)) = ηt(x′(t))

ddt f(x(t))

.Thus, as λ(x(t)) is a quotient of two analytic functions, there are only two possibilities for what

happens to λ(x(t)) as t→ 0.

Case 1: |λ(x(t))| → ∞ as t→ 0.

In this case, since ηt → η, it follows thatηt

λ(x(t))→ 0 and, hence, − ωt

λ(x(t))→ dxf . Therefore,

(x(t),− ωt

λ(x(t)),

[− ωt

λ(x(t))− dx(t)f

])=(

x(t),− ωt

λ(x(t)), [ηt(x(t))]

)→ (x, dxf , [η]),

and so (x, [η]) ∈ π(E).

Case 2: λ(x(t)) → λ0 as t→ 0.

In this case, ωt must possess a limit as t → 0. For t small and unequal to zero, let projtdenote the complex orthogonal projection from the fibre

(T ∗f|

X

U)x(t)

to the fibre(T ∗

XU)x(t)

. Let

γt := projt(ηt) = ωt + λ(x(t)) projt(dx(t)f). Since x ∈ ΣNashf , we have that projt(dx(t)f) → dxfand, thus, γt → η.

As η is not zero (since it represents a projective class), we may define the (real, non-negative)scalar

at :=

√||projt(dx(t)f)− dx(t)f ||

||γt||.

One now verifies easily that

(x(t), atγt + projt(dx(t)f), [atγt + projt(dx(t)f)− dx(t)f ]) −→ (x, dxf , [η]),

and, hence, that (x, [η]) ∈ π(E).

Remark 5.3. In a number of results throughout the remainder of Part III, the reader will find thehypotheses that x ∈ ΣNashf or that x ∈ Σalgf . While Theorem 5.2 explains why the hypothesisx ∈ ΣNashf is important, it may not be so clear why the hypothesis x ∈ Σalgf is of interest.

If Y is an analytic subset of X, then one shows easily that Y ∩ Σalgf ⊆ Σalg(f|Y ). The Nashcritical locus does not possess such an inheritance property. Thus, the easiest hypothesis to make

144 DAVID B. MASSEY

in order to guarantee that a point, x, is in the Nash critical locus of any analytic subset containingx is the hypothesis that x ∈ Σalgf , for then if x ∈ Y , we conclude that x ∈ Σalg(f|Y ) ⊆ ΣNash(f|Y ).

A further remark is that the fibre(π(E)

)x

being non-empty is trivially seen to be equiva-

lent to x ∈ Σcnrf . As the fibre(P(T ∗f|Xreg

U))

xis always non-empty, the equality

(π(E)

)x

=(P(T ∗f|Xreg

U))

ximplies that x ∈ Σcnrf . This is slightly short of being a converse to the statement

in the theorem, unless we are in a situation where we know that Σcnrf = ΣNashf .

We come now to the result which tells one how the topological information provided by thesheaf of vanishing cycles controls the af condition.

Corollary 5.4. Let N be a submanifold of X such that N ⊆ V (f), and let x ∈ NLet Ch(F•) =

∑αmα

[T ∗

MαU], where Mα is a collection of connected analytic submanifolds of

X such that either mα > 0 for all α, or mα 6 0 for all α. Let Ch(φfF•) =∑

β kβ

[T ∗

RβU], where

Rβ is a collection of connected analytic submanifolds.

Finally, suppose that, for all β, there is an inclusion of fibres over x given by(T ∗

RβU)x

⊆(T ∗

NU)x.

Then, the pair((Mα

)reg, N)

satisfies Thom’s af condition at x for every Mα for which f|Mα6≡

0, mα 6= 0, and x ∈ ΣNash(f|Mα

).

Proof. LetSγ

be a Whitney stratification for X such that each Mα is a union of strata and

such that Σ(f|Sγ

)= ∅ unless Sγ ⊆ V (f). Hence, for each α, there exists a unique Sγ such that

Mα = Sγ ; denote this stratum by Sα. It follows at once that Ch(F•) =∑

αmα

[T ∗

SαU].

From Theorem 3.10, E =∑

αmαEα∼= P

(Ch(φfF•)

). Thus, since all non-zero mα have the

same sign, if mα is not zero, then Eα appears with a non-zero coefficient in P(

Ch(φfF•)).

The result now follows immediately by applying Theorem 5.2 to each Mα in place of X.

Theorem 5.2 also allows us to prove an interesting relationship between the characteristic vari-eties of the vanishing and nearby cycles – provided that the complex of sheaves under considerationis perverse.

Corollary 5.5. Let P• be a perverse sheaf on X. If x ∈ Σalgf and (x, η) ∈ |Ch(ψfP•)|, then(x, η) ∈ |Ch(φfP•)|.

Proof. Let S := Sα be a Whitney stratification with connected strata such that P• is con-structible with respect to S and such that V (f) is a union of strata. For the remainder of theproof, we will work in a neighborhood of V (f) – a neighborhood in which, if Sα 6⊆ V (f), thenΣ(f|Sα

) = ∅.


Let Ch(P•) =∑mα

[T ∗

SαU]. As P• is perverse, all non-zero mα have the same sign. Thus,

3.10 tells us – using the notation from 3.10 – that

(†) |P(Ch(φfP•))| =⋃

mα 6=0

π(Eα),

where Eα denotes the exceptional divisor in the blow-up of T ∗SαU along im df (in a neighborhood

of V (f)). In addition, 3.3 tells us that

|Ch(ψfP•)| =(V (f)× Cn+1

)∩

⋃mα 6=0

Sα 6⊆V (f)

T ∗f|Sα

U .

Assume (x, η) ∈ |Ch(ψfP•)|. Then, there exists Sα 6⊆ V (f) such that mα 6= 0 and (x, η) ∈T ∗f|Sα

U . Clearly, then, (x, η) ∈ T ∗f|(Sα)reg

U . Now, if x ∈ Σalgf and η 6= 0, then x ∈ Σalg(f|Sα

) and

so Theorem 5.2 implies that (x, [η]) ∈ π(Eα), where [η] denotes the projective class of η and Eα

denotes the exceptional divisor of the blow-up of T ∗(Sα)reg

U = T ∗SαU along im df . Thus, by (†),

(x, η) ∈ |Ch(φfP•)|.We are left with the trivial case of when (x, 0) ∈ |Ch(ψfP•)|. Note that, if (x, 0) ∈ |Ch(ψfP•)|,

then there must exist some non-zero η such that (x, η) ∈ |Ch(ψfP•)|. For, otherwise, the stratum(in some Whitney stratification) of suppψfP• containing x must be all of U . However, ψfP• issupported on V (f), and so f would have to be zero on all of U ; but, this implies that |Ch(ψfP•)| =∅. Now, if we have some non-zero η such that (x, η) ∈ |Ch(ψfP•)|, then by the above argument,(x, η) ∈ |Ch(φfP•)| and, thus, certainly (x, 0) ∈ |Ch(φfP•)|.

The following result helps to illuminate the connection between the Le-Iomdine (-Vogel) cyclesand Thom’s af condition (see II.6 and IV.2). The result tells us that adding a large power of asecond function, g, to f reduces the critical locus, but expands the fibre of the relative conormal.For a generic choice of g, we can obtain effective lower bounds on the power to which g must beraised (see II.4.3.iii and IV.2.1.ii); however, the g below is completely general.

Corollary 5.6 (Thom reduction). Suppose that x ∈ ΣNashf . Assume that f(x) = 0, and thatwe have a second function g : X → C such that g(x) = 0. Suppose that [η] ∈

(P(T ∗f|Xreg

U))

x,

Then, for all j 2, [η] ∈(P(T ∗(f+gj)|Xreg

U))

x, and there exists a neighborhood W of x in X

such that, in W, Σreg(f + gj) ⊆ V (g) ∩ Σregf .

Proof. Let g denote a local extension of g to U . Let E denote the exceptional divisor inBlim df T

∗Xreg

U ⊆ U×Cn+1×Pn, and let Ej denote the exceptional divisor in Blim d(f+gj) T∗Xreg

U ⊆U × Cn+1 × Pn. Let π : U × Cn+1 × Pn → U × Pn denote the projection.

It is trivial to show that if [η] ∈(π(E)

)x, then, for all j 2, [η] ∈

(π(Ej)

)x. For [η] ∈

(π(E)

)x

if and only if (x, dxf , [η]) ∈ E, which means that there is an analytic path α(t) = (p(t), ω(t)) inT ∗

XregU such that α(0) = (x, dxf), α(t) ∈ T ∗

XregU−im df for t 6= 0, and [ω(t)−dp(t)f ] → [η]. Clearly,

since g(x) = 0, we may now choose j large enough so that [ω(t)−dp(t)f− jgj−1(p(t))dp(t)g] → [η].

146 DAVID B. MASSEY

Moreover, if α(t) ∈ im d(f + gj) for two different j’s, then g(p(t)) ≡ 0 and we are finished withthe proof of the first statement; otherwise, α(t) 6∈ im d(f + gj) for large j, and we are once againfinished.

Therefore, if [η] ∈(π(E)

)x, then, for all j 2, [η] ∈

(π(Ej)

)x. Now, one shows easily that

x ∈ ΣNashf implies that x ∈ ΣNash(f + gj) for all j > 2. One now applies Theorem 5.2 twice toconclude the first part of the corollary.

It is somewhat lengthier to prove that there exists a neighborhood W of x in X such that, inW, Σreg(f + gj) ⊆ V (g) ∩ Σregf , but the idea is simple: we prove it first when X is smooth at x(using an inequality of Lojasiewicz), and then we resolve the singularity in the general case.

So, assume that X is smooth at x. Perform an analytic change of coordinates to place ourselvesin an open subset of affine space. By an inequality of Lojasiewicz ([ Loj], p. 238), there exists aneighborhood W of x and a real θ, with 0 < θ < 1, such that, for p ∈ W, |f(p)|θ 6 | grad f(p)|.We will show how to pick j large depending on the size of θ.

Suppose that Σ(f+gj) 6⊆ V (g)∩Σf ; we wish to derive a contradiction. Then, there would existan analytic path α(t) ∈ X such that α(0) = x and, for t 6= 0, α(t) ∈ Σ(f + gj) − V (g) ∩ Σf . ByRemark 1.7, we know that, near x, Σ(f+gj) ⊆ V (f+gj). Thus, along α(t), grad f = −jgj−1 grad gand f = −gj . Hence, along α(t), |g|jθ 6 j|g|j−1| grad g| and so, as g(α(t)) 6≡ 0, we conclude that|g|jθ−j+1 6 j| grad g|. As g(α(0)) = 0, we would have a contradiction if jθ− j + 1 < 0. Therefore,if j > 1/(1 − θ), we obtain the desired conclusion. Actually, in the smooth case, we have shownthe stronger result that there exists a single neighborhood W which can be used for all large j.

Now, allow X to be singular at x. Let X π−→ X be a local analytic resolution of the singularitiesof X, i.e., a proper map from the smooth space X such that π is an isomorphism over Xreg. As πis proper, π−1(x) is compact. Applying the smooth case to f π and g π at each point of π−1(x),and using compactness, we conclude that there is a neighborhood, W, of π−1(x) such that, in W,for all j 2, Σ((f + gj) π) ⊆ V (g π); fix a j this large.

As in the smooth case, suppose that Σ(f + gj) 6⊆ V (g)∩Σf ; we wish to derive a contradiction.Then, there would exist an analytic path α(t) ∈ X such that α(0) = x and, for t 6= 0, α(t) ∈Σreg(f + gj)−V (g)∩Σregf . Let Γ denote the image of α. The proper transform, Γ, of Γ is a curvewhich intersects π−1(x) in a unique point; the existence of such a curve contradicts the choice ofj and W.


Chapter 6. CONTINUOUS FAMILIES OF CONSTRUCTIBLE COMPLEXES.

We wish to prove statements of the form: the constancy of certain data in a family implies thatsome nice geometric facts hold. As the reader should have gathered from the last section, it is veryadvantageous to use complexes of sheaves for cohomology coefficients; in particular, being able touse perverse coefficients is very desirable. The question arises: what should a family of complexesmean?

Let X be a d-dimensional analytic space, let t : X → C be an analytic function, and let F• bea bounded, constructible complex of C-vector spaces. We could say that F• and t form a “nice”family of complexes, since, for all a ∈ C, we can consider the complex F•|t−1(a)

on the space X|t−1(a).

However, this does yield a satisfactory theory, because there may be absolutely no relation betweenF•|t−1(0)

and F•|t−1(a)for a close to 0. What we need is a notion of continuous families of complexes

– we want F•|t−1(0)to equal the “limit” of F•|t−1(a)

as a approaches 0. Fortunately, such a notionalready exists; it just is not normally thought of as continuity.

Definition 6.1. Let X, t, and F• be as above. We define the limit of F•a := F•|t−1(a)[−1] as a

approaches b, lima→b

F•a, to be the nearby cycles ψt−bF•[−1].

We say that the family F•a is continuous at the value b if the comparison map from F•b toψt−bF•[−1] is an isomorphism, i.e., if the vanishing cycles φt−bF•[−1] = 0. We say that the familyF•a is continuous if it is continuous for all values b.

We say that the family F•a is continuous at the point x ∈ X if there is an open neighborhoodW of x such that the family defined by restricting F• to W is continuous at the value t(x).

If P• is a perverse sheaf on X and P•a := P•

|t−1(a)[−1] is a continuous family of complexes, then

we say that P•a is a continuous family of perverse sheaves.

Remark 6.2. The reason for the shifts by −1 in the families is so that if P• is perverse, and P•a

is a continuous family, then each P•a is, in fact, a perverse sheaf (since P•

a∼= ψt−aP•[−1]).

It is not difficult to show that: if the family F•a is continuous at the value b, and, for all a 6= b,each F•a is perverse, then, near the value b, the family F•a is a continuous family of perverse sheaves.

For the remainder of this section, we will be using the following additional notation. Let t bean analytic function on U , and let t denote its restriction to X. Let P• be a perverse sheaf on X.Consider the families of spaces, functions, and sheaves given by Xa := X ∩ V (t − a), fa := f|Xa

,and P•

a := P•|Xa

[−1] (normally, if we are not looking at a specific value for t, we write Xt, ft,and P•

t for these families). Note that, if we have as an hypothesis that P•t is continuous, then the

family P•t is actually a family of perverse sheaves.

We will now prove three fundamental lemmas; all of them have trivial proofs, but they arenonetheless extremely useful.

The first lemma uses Theorem 4.2 to characterize continuity at a point for families of perversesheaves.

148 DAVID B. MASSEY

Lemma 6.3. Let x ∈ X. The following are equivalent:

i) The family P•t is continuous at x;

ii) x 6∈ ΣP• t;

iii) (x, dxt) 6∈ |Ch(P•)| for some local extension, t, of t to U in a neighborhood of x; and

iv) (x, dxt) 6∈ |Ch(P•)| for every local extension, t, of t to U in a neighborhood of x.

Proof. The equivalence of i) and ii) follows from their definitions, together with Remark 1.7. Theequivalence between ii), iii), and iv) follows immediately from Theorem 4.2.

The next lemma is a necessary step in several proofs.

Lemma 6.4. Suppose that the family P•t is continuous at t = b, and that the characteristic cycle

of P• is given by∑

αmα

[T ∗

SαU]. Then, Sα 6⊆ V (t− b) if mα 6= 0.

Proof. This follows immediately from 6.3.

The last of our three lemmas is the stability of continuity result.

Lemma 6.5 (Stability of Continuity). Suppose that the family P•t is continuous at x ∈ X.

Then, P•t is continuous at all points near x. In addition, if

D is an open disk around the origin

in C, h :D×X → C is an analytic function, hc(z) := h(c, z), and h0 = t, then the family P•

hcis

continuous at x for all c sufficiently close to 0.

Proof. Let t be an extension of t to a neighborhood of x in U , and let Π1 : T ∗U → U be the cotangentbundle. As T ∗U is isomorphic to U × Cn+1, there is a second projection Π2 : T ∗U → Cn+1.

Now, Π−11 (x)∩|Ch(P•)| and Π−1

2 (dxt)∩|Ch(P•)| are closed sets. Therefore, the lemma followsimmediately from 6.3.

The following lemma allows us to use intersection-theoretic arguments for families of generalizedisolated critical points.

Lemma 6.6. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x). Let Sα be a

Whitney stratification of X with connected strata with respect to which P• is constructible. Supposethat Ch(P•) is given by

∑αmα

[T ∗

SαU]. If dimx Σ

P•b

fb 6 0, then there exists an open neighborhoodW of x in U such that:


i) im df properly intersects∑α

mα

[T ∗t|Sα

U]

in W;

ii) for all y ∈ X ∩W, V (t− t(y)) properly intersects

im df ·∑α

mα

[T ∗t|Sα

U]

at (y, dyf) in (at most) an isolated point; and

iii) for all y ∈ X ∩W, if a := t(y), then dimy ΣP•afa 6 0 and

µy(fa; P•a) = (−1)d

[(im df ·

∑α

mα

[T ∗t|Sα

U])

· V (t− a)](y,dy f)

.

Proof. First, note that we may assume that X = supp P•; for, otherwise, we would immediatelyreplace X by supp P•. We may refine our stratification so that V (t− b) is a union of strata; for byLemma 6.4, if Sα ⊆ V (t− b), then mα = 0. This also explains why we may index over all strata inthe formulas. Finally, 6.4 implies that V (t − b) does not contain an entire irreducible componentof X; thus, dimX0 = d− 1.

We use f as a common extension of ft to U , for all t. Proposition 4.10 tells us that µx(fb; P•b) =

(−1)d−1(

im df · Ch(P•b))(b,dbf)

. Then, continuity, implies that Ch(P•b) = Ch(ψt−b[−1]P•), and

(∗) Ch(ψt−b[−1]P•) = −Ch(ψt−bP•) = −(V (t− b)× Cn+1

)·

∑Sα 6⊆V (t−b)

mα

[T ∗t|Sα

U],

by Theorem 3.3.Therefore,

(†) µx(fb; P•b) = (−1)d

(im df ·

(V (t− b)× Cn+1

)·∑α

mα

[T ∗t|Sα

U])

(x,dxf)=

(−1)d((

im df ·∑α

mα

[T ∗t|Sα

U])

·(V (t− b)× Cn+1

))(x,dxf)

.

Thus,C := (−1)d

(im df ·

∑α

mα

[T ∗t|Sα

U])

is a non-negative cycle such that (x, dxf) is an isolated point in (or, is not in) C · V (t − b).Statements i) and ii) of the lemma follow immediately.

Now, Lemma 6.5 tells us that the family P•t is continuous at all points near x; therefore, if y is

close to x and a := t(y), then, by repeating the argument for (∗), we find that

Ch(P•a) = −Ch(ψt−aP•) = −

(V (t− a)× Cn+1

)·∑α

mα

[T ∗t|Sα

U]

150 DAVID B. MASSEY

and we know that the intersection of this cycle with im df is (at most) zero-dimensional at (y, dyf)(since C ∩ V (t − b) is (at most) zero-dimensional at x). By considering f an extension of fa andapplying Theorem 4.2, we conclude that dimy Σ

P•afa 6 0.

Finally, now that we know that P•t is continuous at y and that dimy Σ

P•afa 6 0, we may argue

as we did at x to conclude that (†) holds with x replaced by y and b replaced by a. This provesiii).

We can now prove an additivity/upper-semicontinuity result. We prove this result for amore general type of family of perverse sheaves; instead of parametrizing by the values of a function,we parametrize implicitly. We will need this more general perspective in Theorem 6.10.

Theorem 6.7. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x), and suppose

that dimx ΣP•

b

fb 6 0.

LetD be an open disk around the origin in C, let h :

D ×X → C be an analytic function, for

all c ∈D, let hc(z) := h(c, z), let cP• := P•

|V (hc−b)[−1] and cf := f|V (hc−b)

. Suppose that h0 = t.Then, there exists an open neighborhood W of x in U such that, for all small c, for all y ∈

V (hc − b) ∩W, dimy ΣcP• cf 6 0.

Moreover, for fixed c close to 0, there are a finite number of points y ∈ V (hc− b)∩W such thatµy(cf ; cP•) 6= 0 and

µx(bf ; bP•) =∑

y∈V (hc−b)∩W

µy(cf ; cP•).

In particular, for all small c, for all y ∈ V (hc − b) ∩W, µy(cf ; cP•) 6 µx(bf ; bP•).

Proof. We continue to let P•c = P•

|V (t−c)[−1] and fc = f|V (t−c)

. Note that, if we let h(w, z) :=t(z) − w, then the statement of the theorem would reduce to a statement about the ordinaryfamilies P•

c and fc. Moreover, this statement about the families P•c and fc follows immediately

from Lemma 6.6. We wish to see that this apparently weak form of the theorem actually impliesthe stronger form.

ShrinkingD and U if necessary, let h :

D×U → C denote a local extension of h to

D×U . We use

w as our coordinate onD. Note that replacing h(w, z) by h(w2, z) does not change the statement

of the theorem. Therefore, we can, and will, assume that d(0,x)h vanishes on C× 0.

Let p :D × U → U denote the projection, and let p := p|

D×X

. Let Q• := p∗P•[1]; as P• is

perverse, so is Q•. Let Y := (D ×X) ∩ V (h − b), and let w : Y →

D denote the projection. Let

R• := Q•|Y [−1]. Let f : Y → C be given by f(w, z) := f(z). As we already know that the theorem

is true for ordinary families of functions, we wish to apply it to the family of functions fw and thefamily of sheaves R•

w; this would clearly prove the desired result.

Thus, we need to prove two things: that R• is perverse near (0,x), and that the family R•w is

continuous at (0,x).


Let Sα be a Whitney stratification, with connected strata, of X with respect to whichP• is constructible. Refining the stratification if necessary, assume that V (t − b) is a union ofstrata. Let Ch(P•) =

∑mα

[T ∗

SαU]. Clearly, Q• is constructible with respect to the Whitney

stratification D × Sα, and the characteristic cycle of Q• in T ∗(

D × U) is given by Ch(Q•) =

−∑mα

[T ∗

D×Sα

(D× U)

].

Note that, for all (z, η) ∈ T ∗U , (z, η) ∈ T ∗SαU if and only if (0, z, η d(0,z)p) ∈ T ∗

D×Sα

(D× U).

As we are assuming that d(0,x)h vanishes on C × 0 and that h0 = t, we know that d(0,x)h =

dxt d(0,z)p. Thus, (x, dxt) ∈ T ∗SαU if and only if (0,x, d(0,x)h) ∈ T ∗

D×Sα

(D× U). Therefore,

(x, dxt) ∈ |Ch(P•)| if and only if (0,x, d(0,x)h) ∈ |Ch(Q•)|. As we are assuming that the familyP•

t is continuous at x, we may apply Lemma 6.3 to conclude that (x, dxt) 6∈ |Ch(P•)| and, hence,(0,x, d(0,x)h) 6∈ |Ch(Q•)|. It follows that, for all (w, z) near (0,x), (w, z, d(w,z)h) 6∈ |Ch(Q•)| andthat the family Q•

h is continuous at (0,x); that is, there exists an open neighborhood, Ω×W, of

(0,x) inD × U , in which φh−b[−1]Q• = 0 and such that, if (w, z) ∈ Ω × W and mα 6= 0, then

(w, z, d(w,z)h) 6∈ T ∗D×Sα

( D× U

). For the remainder of the proof, we assume that

D and U have

been rechosen to be small enough to use for Ω and W.As φh−b[−1]Q• = 0, R• ∼= ψh−b[−1]Q• is a perverse sheaf on Y . It remains for us to show that

the family R•w is continuous at (0,x).

Of course, we appeal to Lemma 6.3 again – we need to show that (0,x, d(0,x)w) 6∈ |Ch(R•)|.Now, |Ch(R•)| = |Ch(ψh−b[−1]Q•)|, and we wish to use Theorem 3.3 to describe this character-

istic variety. If (w, z) ∈ Ω×W and mα 6= 0, then (w, z, d(w,z)h) 6∈ T ∗D×Sα

( D× U

); thus, if mα 6= 0,

then h has no critical points when restricted toD× Sα, and, using the notation of 3.2 and 3.3,

T ∗h−b,Q•

( D× U

)=∑α

mα

[T ∗h|

D×Sα

( D× U

)].

Now, using Theorem 3.3, we find that

|Ch(R•)| =(V (h− b)× Cn+2

)∩⋃

mα 6=0

T ∗h|D×Sα

( D× U

).

We will be finished if we can show that, if mα 6= 0, then (0,x, d(0,x)w) 6∈ T ∗h|D×Sα

( D× U

).

Fix an Sα for which mα 6= 0. Suppose that (0,x, η) ∈ T ∗h|D×Sα

( D× U

). Then, there exists a

sequence (wi, zi, ηi) ∈ T ∗h|D×Sα

( D × U

)such that (wi, zi, ηi) → (0,x, η). Thus, ηi

((C × Tzi

Sα) ∩

ker d(wi,zi)h)

= 0. By taking a subsequence, if necessary, we may assume that TziSα converges

to some T in the appropriate Grassmanian. Now, we know that ker d(wi,zi)h → ker d(0,x)h =C×ker dxt. As (x, dxt) 6∈ T ∗Sα

U , C×ker dxt transversely intersects C×T . Therefore, (C×TziSα)∩

ker d(wi,zi)h→ (C×T )∩(C×ker dxt) , and so C×0 ⊆ ker η. However, ker d(0,x)w = 0×Cn+1,and we are finished.

152 DAVID B. MASSEY

We would like to translate Theorem 6.7 into a statement about Milnor fibres and the constantsheaf. First, though, it will be convenient to prove a lemma.

Lemma 6.8. Let x ∈ X, and let b := t(x). Suppose that dimx

(V (t− b)∩ΣCt

)6 0. Fix an integer

k. If Hk(Ft,x; C) = 0, then the family kP•t is continuous at x. In addition, if Hk(Ft,x; C) = 0

and Hk−1(Ft,x; C) = 0, then kP•b∼= µH0(C•Xb

[k]) near x.

Proof. By Remark 1.7, the assumption that dimx

(V (t−b)∩ΣCt

)6 0 is equivalent to dimxΣCt 6 0

and, by Theorem 4.6, this is equivalent to dimxΣjP•t 6 0 for all j. Thus, suppφt−b[−1]kP• ⊆

x near x. We claim that the added assumption that Hk(Ft,x; C) = 0 implies that, in fact,φt−b[−1]kP• = 0 near x.

For, near x, suppφt−b[−1]C•X [k + 1] ⊆ x, and so

φt−b[−1]kP• = φt−b[−1]µH0(C•X [k + 1]) ∼= µH0(φt−b[−1]C•X [k + 1]) ∼= H0(φt−b[−1]C•X [k + 1]).

Near x, φt−b[−1]C•X [k + 1] is supported at, at most, the point x and, hence, φt−b[−1]kP• = 0provided that H0(φt−b[−1]C•X [k + 1])x = 0, i.e., provided that Hk(Ft,x; C) = 0. This proves thefirst claim in the lemma.

Now, if the family kP•t is continuous at x, then, near x,

kP•b = kP•

|V (t−b)[−1] ∼= ψt−b[−1]µH0(C•X [k + 1]) ∼= µH0(ψt−b[−1]C•X [k + 1]),

and we claim that, if Hk(Ft,x; C) = 0 and Hk−1(Ft,x; C) = 0, then there is an isomorphism (inthe derived category) µH0(ψt−b[−1]C•X [k + 1]) ∼= µH0(C•Xb

[k]).To see this, consider the canonical distinguished triangle

C•Xb[k] → ψt−b[−1]C•X [k + 1] → φt−b[−1]C•X [k + 1]

[1]−→ C•Xb[k].

A portion of the long exact sequence (in the category of perverse sheaves) resulting from applyingperverse cohomology is given by

pH−1(φt−b[−1]C•X [k + 1]) → pH0(C•Xb[k]) → µH0(ψt−b[−1]C•X [k + 1]) → µH0(φt−b[−1]C•X [k + 1]).

We would be finished if we knew that the terms on both ends of the above were zero. However,since φt−b[−1]C•X [k+1] has no support other than x (near x), we proceed as we did above to showthat pH−1(φt−b[−1]C•X [k+ 1]) and pH0(φt−b[−1]C•X [k+ 1]) are zero precisely when Hk−1(Ft,x; C)and Hk(Ft,x; C) are zero.

Theorem 6.9. Let x ∈ X and let b := t(x). Suppose that x 6∈ ΣCt, and that dimx ΣC(fb) 6 0.Then, there exists a neighborhood, W, of x in X such that, for all a near b, there are a finite

number of points y ∈ W ∩ V (t − a) for which H∗(Ffa,y; C) 6= 0; moreover, for all integers, k,bk−1(Ffa,y) = µy(fa; kP•

a), and

bk−1(Ffb,x) =∑

y∈W∩V (t−a)

bk−1(Ffa,y),


where H∗() and b∗() are as in Remark 4.4.

Proof. Let v := fb(x). Fix an integer k.By the lemma, the family kP•

t is continuous at x and kP•b∼= µH0(C•Xb

[k]) near x. Thus,

φfb−v[−1]kP•b∼= φfb−v[−1]µH0(C•Xb

[k]) ∼= µH0(φfb−v[−1]C•Xb[k]).

We are assuming that dimx ΣC(fb) 6 0; this is equivalent to: suppφfb−v[−1]C•Xb[k] ⊆ x near x,

it follows from the above line and Theorem 4.6 that dimx ΣkP•

b

fb 6 0 and that

(‡) µx(fb; kP•b) = dimH0(φfb−v[−1]C•Xb

[k])x = bk−1(Ffb,x).

Applying Theorem 6.7, we find that there exists an open neighborhood W ′ of x in U such that,for all y ∈ W ′, if a := t(y), then (∗) dimy Σ

kP•afa 6 0, and, for fixed a close to b, there are a finite

number of points y ∈ W ′ ∩ V (t− a) such that µy(fa; kP•a) 6= 0 and

(†) µx(fb; kP•b) =

∑y∈W∩V (t−a)

µy(fa; kP•a).

Now, using the above argument for all k with 0 6 k 6 d − 1 and intersecting the resultingW ′-neighborhoods, we obtain an open neighborhood W of x such that (∗) and (†) hold for allsuch k. We claim that, if a is close to b, then W ∩ΣCfa consists of isolated points, i.e., the pointsy ∈ W ∩ V (t− a) for which H∗(Ffa,y; C) 6= 0 are isolated.

If a = b, then there is nothing to show. So, assume that a 6= b, and assume that we are workingin W throughout. By Remark 1.7, t satisfies the hypotheses of Lemma 6.8 at t = a; hence, for allk, not only is kP•

t continuous at t = a, but we also know that kP•a∼= µH0(C•Xa

[k]). By Theorem4.6, ΣCfa =

⋃ΣkP•

afa, where the union is over k where 0 6 k 6 dimXa. As dimXa 6 d− 1, the

claim follows from (∗) and the definition of W.

Now that we know that kP•t is continuous at t = a and that W ∩ ΣCfa consists of isolated

points, we may use the argument that produced (‡) to conclude that µy(fa; kP•a) = bk−1(Ffa,y).

The theorem follows from this, (‡), (∗), and (†).

We want to prove a result which generalizes that of Le and Saito [L-S]. We need to make theassumption that the Milnor number is constant along a curve that is embedded in X. Hence, itwill be convenient to use a local section of t : X → C at a point x ∈ X; that is, an analytic functionr from an open neighborhood, V, of t(x) in C into X such that r(t(x)) = x and t r equals theinclusion morphism of V into C. Note that existence of such a local section implies that x 6∈ Σalgt;in particular, V (t− t(x)) is smooth at x.

Theorem 6.10. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x), and let

v := fb(x). Let r : V → X be a local section of t at x, and let C := im r. Assume that C ⊆ V (f−v),that dimx Σ

P•b

fb 6 0, and that, for all a close to b, the Milnor number µr(a)(fa; P•a) is non-zero

and is independent of a; denote this common value by µ.

Then, C is smooth at x, V (t − b) transversely intersects C in U at x , and there exists aneighborhood, W, of x in X such that W ∩ ΣP•f ⊆ C and

(φf−v[−1]P•)

|W∩C

∼=(CµW∩C [1]

)•. In

154 DAVID B. MASSEY

particular, if we let t denote the restriction of t to V (f − v), then the family(φf−v[−1]P•)

tis

continuous at x.

If, in addition to the other hypotheses, we assume that x ∈ Σalgf , then the two families(ψf−v[−1]P•)

tand

(P•|V (f−v)

[−1])t

are continuous at x. (Though P•|V (f−v)

[−1] need not be per-verse.)

Proof. Let us first prove that the last statement of the theorem follows easily from the first portionof the theorem. So, assume that φt−b[−1]φf−v[−1]P• = 0 near x. Therefore, working near x,we have that φt−b[−1]

(P•|V (f−v)

[−1]) ∼= φt−b[−1]ψf−v[−1]P•, and we need to show that this is

the zero-sheaf. By Lemma 6.3, what we need to show is that (x, dxt) 6∈ |Ch(ψf−v[−1]P•)| =|Ch(ψf−vP•)|. As we are assuming that x ∈ Σalgf , we may apply Corollary 5.5 to find that itsuffices to show that (x, dxt) 6∈ |Ch(φf−vP•)| = |Ch(φf−v[−1]P•)|. By 6.3, this is equivalent toφt−b[−1]φf−v[−1]P• = 0 near x, which we already know to be true. This proves the last statementof the theorem.

Before proceeding with the remainder of the proof, we wish to make some simplifying assump-tions. As x 6∈ Σalgt, we may certainly perform an analytic change of coordinates in U to reduceourselves to the case where t is simply the restriction to X of a linear form t. Moreover, it isnotational convenient to assume, without loss of generality, that x = 0 and that b and v are bothzero.

Let Sα be a Whitney stratification of X with connected strata with respect to which P• isconstructible and such that V (t) and V (f) are each unions of strata. Suppose that Ch(P•) is givenby∑

αmα

[T ∗

SαU].

Let C := (r(a), dr(a)f) | a ∈ V; the projection, ρ, onto the first component induces anisomorphism from C to C. By Lemma 6.6, the assumption that the Milnor number, µr(a)(fa; kP•

a),is independent of a is equivalent to:

(†) there exists an open neighborhood W of (0, d0f) in T ∗U in which C equals

im df ∩⋃

mα 6=0

T ∗t|Sα

U

and C is a smooth curve at (0, d0f) such that (0, d0f) 6∈ Σ(t ρ|C

).

It follows immediately that C is smooth at 0 and 0 6∈ Σ(t|C ). We need to show that (†) impliesthat W ∩ ΣP•f ⊆ C and

(φf [−1]P•)

|W∩C

∼=(CµW∩C [1]

)•, where W := ρ(W).

As T ∗SαU ⊆ T ∗t|Sα

U , we have that |Ch(P•)| ⊆⋃

mα 6=0 T∗t|Sα

U and, thus, im df ∩ |Ch(P•)| ⊆ C

inside W. It follows from Theorem 4.2 that W ∩ ΣP•f ⊆ C.

It remains for us to show that(φf [−1]P•)

|W∩C

∼=(CµW∩C [1]

)•. As φf [−1]P• is perverse andwe have just shown that the support of φf [−1]P•, near 0, is a smooth curve, it follows from thework of MacPherson and Vilonen in [M-V] that what we need to show is that, for a generic linearform L, Q• := φL[−1]φf [−1]P• = 0 near 0. By definition of the characteristic cycle (and since 0is an isolated point in the support of Q•), this is the same as showing that the coefficient of T ∗0U


in Ch(φf [−1]P•) equals zero. To show this, we will appeal to Theorem 3.4 and use the notationfrom there.

We need to show that m0(φf [−1]P•) = 0. By 3.4, if suffices to show that m0(P•) = 0 andΓ1

f,L(Sα) = ∅ near 0, for all Sα which are not contained in V (f) and for which mα 6= 0 (where Lstill denotes a generic linear form). As P•

t is continuous at 0, Lemma 6.3 tells us that m0(P•) = 0.Now, near 0, if y ∈ Γ1

f,L(Sα) − 0, then (y, dyf) ∈ T ∗L|Sα

U . If we knew that, near (0, d0f), C

equals im df ∩⋃

mα 6=0 T∗L|Sα

U , then we would be finished – for C is contained in V (f) while Sα

is not; hence, Γ1f,L(Sα) would have to be empty near 0.

Looking back at (†), we see that what we still need to show is that if C equals im df ∩⋃mα 6=0 T

∗t|Sα

U near (0, d0f), then the same statement holds with t replaced by a generic linearform L. We accomplish this by perturbing t until it is generic, and by then showing that thisperturbed t satisfies the hypotheses of the theorem.

As C is smooth and transversely intersected by V (t) at 0, by performing an analytic change ofcoordinates, we may assume that t = z0, that C is the z0-axis, and that r(a) = (a,0). Since the

set of linear forms for which 3.4 holds is generic, there exists an open disk,D, around the origin

in C and an analytic family h : (D × U ,

D × 0) → (C, 0) such that h0(z) := h(0, z) = t(z) and

such that, for all small non-zero c, hc(z) := h(c, z) is a linear form for which Theorem 3.4 holds.Let h := h|

D×X

.

As the family P•t is continuous at 0, Lemma 6.5 tells us that P•

hcis continuous at 0 for all

small c. As we are now considering these two different families with the same underlying sheaf,the expression P•

a for a fixed value of a is ambiguous, and we need to adopt some new notation.We continue to let P•

a := P•|V (t−a)

[−1] and fa := f|V (t−a), and let cP•

a := P•|V (hc−a)

[−1] and

cfa := f|V (hc−a).

Since V (h0) = V (z0) transversely intersects C at 0 in U , for all small c, V (hc) transverselyintersects C at 0 in U . Hence, for all small c, there exists a local section rc(a) for hc at 0 suchthat im rc ⊆ C.

We claim that, for all small c:

i) dim0 ΣcP•

0(cf0) 6 0 and µ0(cf0; cP•

0) 6 µ0(0f0; 0P•0) = µ0(f0; P•

0);

ii) for all small a, dimrc(a) ΣcP•

a(cfa) 6 0 and µrc(a)(cfa; cP•

a) 6 µ0(0f0; 0P•0); and

iii) for all small a 6= 0, µrc(a)(cfa; cP•a) = µrc(a)(fz0(rc(a)); P•

z0(rc(a))).

Note that proving i), ii), and iii) would complete the proof of the theorem, for they imply thatthe hypotheses of the theorem hold with t replaced by hc for all small c. To be precise, we wouldknow that P•

hcis continuous at 0, dim0 Σ

cP•0(cf0) 6 0, and, for all small a, µrc(a)(cfa; cP•

a) =µ0(cf0; cP•

0); this last equality follows from i), ii), and iii), since, for all small a 6= 0, we wouldhave

µ = µrc(a)(fz0(rc(a)); P•z0(rc(a))) = µrc(a)(cfa; cP•

a) 6 µ0(cf0; cP•0) 6 µ0(f0; P•

0) = µ.

However, i), ii) and iii) are easy to prove. i) and ii) follow immediately from Theorem 6.7, andiii) follows simply from the fact that, for all small a 6= 0, V (z0−z0(rc(a))) and V (hc−hc(rc(a))) aresmooth and transversely intersect all strata of any analytic stratification of X in a neighborhoodof (0,0). This concludes the proof.

156 DAVID B. MASSEY

Corollary 6.11. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x), and let

v := fb(x). Let r : V → X be a local section of t at x, and let C := im r. Assume that C ⊆ V (f−v),that dimx Σ

P•b

fb 6 0, and that, for all a close to b, the Milnor number µr(a)(fa; P•a) is non-zero

and is independent of a. Let Ch(P•) =∑

αmα

[T ∗

SαU], where Sα is a collection of connected

analytic submanifolds of U .

Then, C is smooth at x, and there exists a neighborhood, W, of x in X such that, for all Sα forwhich Sα 6⊆ V (f − v) and mα 6= 0:

W ∩ Σ(f|(Sα)reg

)⊆ C and, if x ∈ ΣNash(f|

Sα), then the pair

((Sα

)reg, C

)satisfies Thom’s af

condition at x.

Proof. One applies Theorem 6.10. The fact that W∩Σ(f|(Sα)reg

)⊆ C, for all Sα for which mα 6= 0

follows from Theorem 4.2, since W∩ΣP•f ⊆ C. The remainder of the corollary follows by applyingCorollary 5.4, where one uses C for the submanifold N .

Just as we used perverse cohomology to translate Theorem 6.7 into a statement about theconstant sheaf in Theorem 6.9, we can use perverse cohomology to translate Corollary 6.11. Wewill use the notation and results from Proposition 4.3 and Remark 4.4.

Corollary 6.12. Let b := t(x), and let v := fb(x). Suppose that x 6∈ ΣCt. Suppose, further, that,dimx ΣC(fb) 6 0.

Let r : V → X be a local section of t at x, and let C := im r. Assume that C ⊆ V (f − v).

Let Sα be a visible stratum of X of dimension dα, not contained in V (f − v), and let j be aninteger such that bj−1(Lα) 6= 0. Let Y := Sα and let k := dα + j− 1. In particular, Y could be anyirreducible component of X, j could be zero, and k would be (dimY )− 1.

Suppose that the reduced Betti number bk−1(Ffa,r(a)) is independent of a for all small a, andthat either

a) x ∈ ΣNash(f|Y ); or that

b) x 6∈ Σcnr(f|Y ), C is smooth at x, and (Yreg, C) satisfies Whitney’s condition a) at x.

Then, C is smooth at x, and the pair (Yreg, C) satisfies the af condition at x.

Moreover, in case a), bk−1(Ffa,r(a)) 6= 0, C is transversely intersected by V (t − b) at x, andΣ(f|Yreg

) ⊆ C near x.

In addition, if x ∈ Σalgf and, for all small a and for all i, bi(Ffa,r(a)) is independent of a, thenx 6∈ ΣC(t|V (f−v)

).

Proof. We will dispose of case b) first. Suppose that x 6∈ Σcnr(f|Y ), C is smooth at x, and (Yreg, C)

satisfies Whitney’s condition a) at x. LetY := Yreg − Σ(f|Yreg

).


Suppose that we have an analytic path (x(t), ηt) ∈ T ∗f| Y

U , where (x(0), η0) = (x, η) and, for

t 6= 0, (x(t), ηt) ∈ T ∗f| Y

U . We wish to show that (x, η) ∈ T ∗CU .

For t 6= 0, x(t) ∈Y , and thus ηt can be written uniquely as ηt = ωt + λtdxt f , where ωt ∈ T ∗

YU

and λt ∈ C. As we saw in Theorem 5.2, this implies that either |λt| → ∞ or that λt → λ0, forsome λ0 ∈ C. If |λt| → ∞, then ηt

λt→ 0 and, therefore, −ωt

λt→ dxf ; however, this implies that

x ∈ Σcnr(f|Y ), contrary to our assumption. Thus, we must have that λt → λ0.It follows at once that ωt converges to some ω0. By Whitney’s condition a), (x, ω0) ∈ T ∗

CU . As

C ⊆ V (f − v), (x, dxf) ∈ T ∗CU . Hence, (x, η) ∈ T ∗

CU and we have finished with case b).

We must now prove the results in case a). The main step is to prove that bk−1(Ffb,x) 6= 0.

We may refine our stratification, if necessary, so that V (t− b) is a union of strata. By the firstpart of Theorem 6.9, bk−1(Ffb,x) = µx(fb; kP•

b). Hence, by Lemma 6.6.iii, bk−1(Ffb,x) would beunequal to zero if we knew, for some Sβ for which mβ

(kP•) 6= 0, that (x, dxf) ∈ T ∗t|Sβ

U . However,

our fixed Sα is such a stratum, for bk+1−dα(Nα,Lα) 6= 0 and, since x ∈ ΣNash(f|Y ), x ∈ Σcnr(f|Y )

and so (x, dxf) ∈ T ∗SαU ⊆ T ∗t|Sα

U .

Now, applying the first part of 6.9 again, we have that µr(a)(fa; kP•a) = bk−1(Ffa,r(a)) for all

small a. The conclusions in case a) follow from Corollary 6.11.

We must still demonstrate the last statement of corollary.

Suppose that if bi(Ffa,r(a)) is independent of a for all small a and for all i. Let t denote therestriction of t to V (f − v). We will work in a small neighborhood of x. Applying the last twosentences of Theorem 6.10, we find that φt−b[−1]φf−v[−1]iP• = 0 and φt−b[−1]ψf−v[−1]iP• = 0for all i. Commuting nearby and vanishing cycles with perverse cohomology, we find that

µH0(φt−b[−1]φf−v[−1]C•X [i+ 1]

)= 0 and µH0

(φt−b[−1]ψf−v[−1]C•X [i+ 1]

)= 0,

for all i. Therefore, φt−b[−1]φf−v[−1]C•X = 0 and φt−b[−1]ψf−v[−1]C•X = 0. It follows fromthe existence of the distinguished triangle (relating nearby cycles, vanishing cycles, and restric-tion to the hypersurface) that φt−b[−1]C•V (f−v)[−1] = 0. This proves the last statement of thecorollary.

Remark 6.13. If X is a connected l.c.i., then each Lα has (possibly) non-zero cohomology concen-trated in middle degree. Hence, for each visible Sα, bj−1(Lα) 6= 0 only when j = codim

XSα; this

corresponds to k = d − 1. Therefore, the degree d − 2 reduced Betti number of Ffa,r(a) controlsthe af condition between all visible strata and C.

Corollary 6.14. Let W be an analytic subset of an open subset of Cn. Let Z be a d-dimensional

irreducible component of W . Let X :=D×W be the product of an open disk about the origin with

W , and let Y :=D× Z. Let f : (X,

D× 0) → (C, 0) be an analytic function such that f|Y 6≡ 0,

and let ft(z) := f(t, z).

158 DAVID B. MASSEY

Suppose that 0 is an isolated point of ΣC(f0), and that the reduced Betti number bd−1(Ffa,(a,0))is independent of a for all small a.

If either a) 0 ∈ ΣNash(f|Y ) or b) 0 6∈ Σcnr(f|Y ), then the pair (Yreg,D×0) satisfies Thom’s

af condition at 0.

Moreover, in case a), bd−1(Ffa,(a,0)) 6= 0 and, near 0, Σ(f|Yreg) ⊆

D× 0.

Remark 6.15. A question naturally arises: how effective is the criterion appearing in Corollary6.14 that bd−1(Ffa,(a,0)) is independent of a?

By Proposition 4.10, if Rβ is a Whitney stratification of W , then (using the notation from4.10)

bd−1(Ffa,(a,0)) =

bd−1(L0) +∑

Rβ visibledim Rβ>1

bd−1−dβ(Lβ)

((Γ1

fa,L(Rβ) · V (fa)

)0−(Γ1

fa,L(Rβ) · V (L)

)0

),

where L0 denotes the complex link of the origin. As the Betti numbers do not vary with a,bd−1(Ffa,(a,0)) will be independent of a provided that

(Γ1

fa,L(Rβ) · V (fa)

)0−(Γ1

fa,L(Rβ) · V (L)

)0

is independent of a for all visible strata, Rβ , of dimension at least one.This condition is certainly very manageable to check if the dimension of the singular set of X

at the origin is zero or one.

The final statement of Corollary 6.12 has as its conclusion that the constant sheaf onX∩V (f−v),parametrized by the restriction of t, is continuous at x; this is useful for inductive arguments, sincethe hypothesis on the ambient space in Corollary 6.12 is that the constant sheaf, parametrized byt, should be continuous at x. For instance, we can prove the following corollary.

Corollary 6.16. Suppose that f1, . . . , fk are analytic functions from U into C which define asequence of local complete intersections at the origin, i.e., are such that, for all i with 1 6 i 6 k, thespace Xn+1−i := V (f1, . . . , f i) is a local complete intersection of dimension n+ 1− i at the origin.If, for all i, Xn+1−i

t has an isolated singularity at the origin and the restrictions f i+1t : Xn+1−i

t → Care such that dim0 Σcanf

i+1t 6 0 and have Milnor numbers (in the sense of [Loo]) which are

independent of t, then Σ(f|

Xn+1−(k−1)reg

)⊆ C×0 and the pair

(X

n+1−(k−1)reg ,C×0

)satisfies the

afk condition at the origin.

Proof. Recall that C•X [d] is a perverse sheaf if X is a local complete intersection. The “ordinary”Milnor number of f i+1

t at the origin is equal to µ0(f i+1t ; C•

Xn+1−it

[n− i]). Hence, using Proposition4.10.ii, this Milnor number is equal to the degree n − i − 1 (the “middle” degree) reduced Bettinumber of the Milnor fibre of f i+1

t at the origin – the only possible non-zero reduced Betti number.Now, use Corollary 6.12 and induct; the inductive requirement on the Milnor fibre of z0 followsfrom the last statement of the corollary.

Remark 6.17. In [G-K], Gaffney and Kleiman deal with families of local complete intersections asabove. In this setting, they obtain the result of Corollary 6.16 using multiplicities of modules.

Part IV. NON-ISOLATED CRITICAL POINTS OF FUNCTIONSON SINGULAR SPACES


In Part II, we generalized many results from the study of isolated critical points to the case ofnon-isolated critical points. We accomplished this by developing the Le cycles and Le numbersof a non-isolated critical point; the Le numbers are a generalization of the Milnor number of anisolated critical point. However, throughout Part II, the domains of our analytic functions wererequired to be open subsets of affine space.

In Part III, we investigated what an “isolated critical point” of a function on an arbitrarilysingular space should mean, and we developed a theory of Milnor numbers.

In Part IV, we wish to use our construction of the Le cycles and numbers as a guide in order todecide how to generalize our work in Part III to the non-isolated case. We will produce Le-Vogelcycles and numbers, and use them to generalize many previous results.

159

160 DAVID B. MASSEY

Chapter 1. LE-VOGEL CYCLES

We will adopt some notation that we will use throughout Part IV; much of this notation wasused in Part III.

We let U be an open subset of Cn+1, and let X be a (not necessarily purely) d-dimensionalanalytic subset of U . Let f : U → C be an analytic function, and let f := f|X . Let Sα denote aWhitney stratification, with connected strata, of X. We let dα := dimSα.

As in Part III, Chapter 3, we let dv := 1 + dimV (f − v), and will usually denote d0 by simplyd. If we work locally, or assume that X is pure-dimensional, and require f not to vanish on acomponent of X, then d will have attain its “expected” value of d.

We use z0, . . . , zn as coordinates on U . We let η : T ∗U → U denote the cotangent bundle, andwe identify the cotangent space T ∗U with U × Cn+1 by using dz0, . . . , dzn as a basis. We usew0, . . . , wn as coordinates for the cotangent vectors, i.e., a cotangent vector is w0dz0 + · · ·+wndzn.If z is a linear change of coordinates applied to z, then we let w denote cotangent coordinates withrespect to the new basis dz0, . . . , dzn, i.e., if A is in Gln+1(C) and z := Az, then w = Atw.

We shall be blowing-up subspaces of T ∗U ∼= U × Cn+1 along (n + 1)-tuples. This blow-up willlie in U × Cn+1 × Pn; we let π : U × Cn+1 × Pn → U × Pn, τ : U × Cn+1 × Pn → U × Cn+1, andν : U × Pn → U denote the projections. Note that η τ = ν π.

We remind the reader that we slightly modified the definition of the reduced Betti number bj()in III.4.4, so that the empty set has a non-zero reduced Betti number precisely in degree −1.

Recall, from Part III, that we defined kP• := µH0(C•X [k + 1]). In Part IV, a different shift will

be of more use to us. Thus, we define kQ• := d−k−1P• = µH0(C•X [dimX − k]).

We wish to produce Le-Vogel (LeVo) cycles in much the same way that we produce the Lecycles: by taking the Vogel cycles of the Jacobian tuple. We immediately run into the problem ofwhat ideal we should use. Theorem III.3.10 provides us with a clue: the vanishing cycles of theconstant sheaf along f are integrally related to blowing-up im df in T ∗

SαU for various strata Sα.

Hence, we make the following definition.

Definition 1.1. The conormal Jacobian tuple of f with respect to z (and the corresponding choiceof w) is given by

J∗z (f) :=

(w0 −

∂f

∂z0, . . . , wn −

∂f

∂zn

)∈ (O

T∗U )n+1.

Thus, im df is the zero-locus of J∗z (f).

We shall normally be blowing-up T ∗SαU along the restriction of J∗z (f) to (O

T∗Sα

U)n+1; we will

follow the standard practice of simply writing Blim df

T ∗SαU .

In Part I, we defined the gap cycles and Vogel cycles with respect to given cycle M ; we developedthe theory in this generality precisely so that we could now make the appropriate choice(s) for M .Our choice of M is guided by our work in Part III; in particular, we use III.4.3.

PART IV. NON-ISOLATED CRITICAL POINTS 161

Definition 1.2. Let kM be the cycle in T ∗U given by

kM := (−1)d Ch(kQ•) =

∑α

bd−k−1−dα

(Lα)[T ∗

SαU].

Note that kM will be zero unless 0 6 k 6 d. Note also that if X is purely d-dimensional, then thefinal expression for kM above can be written more simply as∑

α

b(dim Lα)−k(Lα)

[T ∗

SαU],

where we mean that dim Lα = −1 if Lα = ∅.

We define kmα := bd−k−1−dα

(Lα), and so kM =∑

αkmα

[T ∗

SαU].

It is also convenient to define kX, the image in X of kM , i.e.,

kX := |η∗(kM)| =⋃

bd−k−1−dα

(Lα) 6=0

Sα.

We can now define polar and Le-Vogel cycles in the cotangent space by using the theory of gapand Vogel cycles developed in Part I. We can then push-forward these “conormal” Le-Vogel cyclesto arrive at the Le-Vogel cycles in X. We will get one set of Le-Vogel cycles for each kM ; note thateach kM > 0 and that all the components of kM have dimension n+1.

Definition 1.3. The i-th k-shifted conormal polar cycle of f with respect to z in T ∗U , kΓif ,z

, is

defined to be ΠiJ∗z (f)

(kM), the i-th inductive gap cycle of J∗z (f) with respect to kM .

The i-th k-shifted conormal Le-Vogel (LeVo) cycle of f with respect to z in T ∗U , kΛif ,z

, is defined

to be ∆iJ∗z (f)

(kM), the i-th Vogel cycle of J∗z (f) with respect to kM , provided that these Vogelcycles exist (see I.2.14).

If the k-shifted LeVo cycles of f with respect to z exist, then each |kΛif ,z| is contained in im df .

Therefore, the proper push-forward η∗ induces an isomorphism between kΛif ,z

and its image in U ; we

define the i-th k-shifted Le-Vogel (LeVo) cycle of f with respect to z, kΛif ,z

, by kΛif ,z

:= η∗(kΛi

f ,z

).

We have

Proposition 1.4. Suppose that the k-shifted Le-Vogel cycles of f with respect to z exist. Then,

0) kΓif ,z

, kΛif ,z

, and kΛif ,z

are non-negative and purely i-dimensional;

i)⋃i

∣∣kΛif ,z

∣∣ = |kM | ∩ im df ;

162 DAVID B. MASSEY

ii)⋃i

∣∣kΛif ,z

∣∣ = ΣkQ• f ; and

iii)⋃i,k

∣∣kΛif ,z

∣∣ = ΣCf .

Proof. As kM is non-negative, all of the cycles defined in 1.3 are non-negative. I.2.2.i implies thatkΓi

f ,zis purely i-dimensional, and I.2.15 implies that kΛi

f ,zand kΛi

f ,zare purely i-dimensional.

i) follows from I.2.4 (and I.2.15). By applying η to each side of i), and using that each kΛif ,z

is

purely i-dimensional, we obtain that⋃i,k

∣∣kΛif ,z

∣∣ =x ∈ X | (x, dxf) ∈ |kM | =

∣∣Ch(kQ•)∣∣. ii)

now follows immediately from III.4.2. Finally, iii) follows from ii) by Theorem III.4.6.

Example 1.5. Consider the case where dimx ΣCf 6 0. By III.4.6, this is equivalent to requiringthat, for all k, dim(x,dxf)

(|kM | ∩ im df

)6 0.

Now, fix k. If (x, dxf) 6∈ |kM |, then, for all i, kΛif ,z

and kΛif ,z

are defined and are zero near

(x, dxf) and x, respectively. Suppose, then, that (x, dxf) is an isolated point of |kM | ∩ im df .Then, as we saw in I.2.8 and I.2.16, near (x, dxf), the conormal LeVo cycles, kΛi

f ,z, are defined

for all i, kΛif ,z

= 0 for i > 1, and, at (x, dxf),

kΛ0f ,z

= kM · V(wn −

∂f

∂zn

)· . . . · V

(w0 −

∂f

∂z0

)=

kM · V(w0 −

∂f

∂z0, . . . , wn −

∂f

∂zn

)= kM · im df = bd−1−k(Ff,x)

[(x, dxf)

],

where the second equality follows from the fact that w0 − ∂fz0, . . . , wn − ∂f

znmust determine a

regular sequence in OU at points in |kM |, and the last equality follows from III.4.10.Thus, the LeVo cycles are all defined, kΛi

f ,z= 0 for i > 1, and, at x, kΛ0

f ,z= bd−1−k(Ff,x)

[x].

Remark 1.6. In Remark 2.16, we discussed how one actually calculates Vogel cycles in practice;we wish to do this again in our present setting, in order to describe how one calculates the LeVocycles.

By definition,

kΛif ,z

= ∆iJ∗z (f)

(kM) =∑α

bd−k−1−dα

(Lα)[∆i

J∗z (f)(T ∗

SαU)].

So, how does one calculate ∆iα := ∆i

J∗z (f)(T ∗

SαU)? One begins with

Πn+1α := Πn+1

J∗z (f)(T ∗

SαU) = T ∗

SαU ¬ V

(wn −

∂f

∂zn

);


thus, Πn+1

J∗z (f)(T ∗

SαU) is either 0 or T ∗

SαU . Next, one calculates the intersection Πn+1

α ·V(wn − ∂f

∂zn

).

This intersection cycle has components contained in W := V(wn − ∂f

∂zn, . . . , wn − ∂f

∂zn

)and com-

ponents which are not contained in W . By I.2.12, the sum of the components which are notcontained in W is precisely Πn

α and the sum of the components which are contained in W is∆n

α := ∆nJ∗z (f)

(T ∗SαU). Having calculated Πn+1

α ·V(wn − ∂f

∂zn

)= Πn

α +∆nα, we use our newly found

Πnα in the next step: the calculation of Πn

α · V(wn−1 − ∂f

∂zn−1

)= Πn−1

α + ∆n−1α . One proceeds

downward inductively.As we pointed out in I.2.15, if one is working near a point of T ∗

SαU ∩ im df , the slightly subtle

point here is that – to know that this method of calculation is valid– one has only to verify thateach ∆i

α is purely i-dimensional as one performs the calculations.We demonstrate such a calculation in Example 1.14, after we have discussed when the LeVo

cycles are independent of the extension of f .

If X is a pure-dimensional l.c.i. (e.g., a connected l.c.i.), then, by [Le9], the complex links ofstrata of X have the homotopy-types of bouquets of spheres of middle dimension. Consequently,for such a space, the only kM which can be non-zero is 0M and, therefore, it is only 0Γi

f ,z, 0Λi

f ,z,

and 0Λif ,z

which are of interest.

Note that, at this point, there is no claim that the LeVo cycles of f are independent of theextension of f . However, we will now use the Segre-Vogel Relation of I.2.22 to find a manageablecriterion guaranteeing the existence of the LeVo cycles, to relate the LeVo cycles to our work inPart III, and to see that, under reasonable hypotheses, the LeVo cycles are independent of theextension f . In fact, we shall prove an analog of Theorem II.1.26, and so we need to define analogsof good stratifications and prepolar coordinates in our current, more general, setting.

Definition 1.7 Let R := Rβ be a Whitney stratification of X with connected strata.

Then, R is a Le-Vogel stratification for f provided that, for all visible Rβ ∈ R,

i) Σcnr(f|Rβ

) is a union of strata; and

ii) for all Rγ ⊆ Σcnr(f|Rβ

), the pair (Rβ , Rγ) satisfies Thom’s af condition.

Since, near a point x, Σcnr(f|Rβ

) ⊆ f−1f(x) (see III.1.7), the condition that the pair (Rβ , Rγ)

satisfies Thom’s af condition in ii) is equivalent to:(T ∗

f|Rβ

U)|Rγ

⊆ T ∗RγU .

We call the strata comprising Σcnr(f|Rβ

) the good strata of R associated to Rβ (with respect to

f). As ΣCf =⋃

visible Rβ

Σcnr

(f|

Rβ

), we refer to any stratum contained in ΣCf as a good stratum

(of R with respect to f).

Let x ∈ X, and let Rβ be a Le-Vogel stratification for f in an open neighborhood of x. Thetuple (z0, . . . , zk) is a Le-Vogel tuple for f at x with respect to Rβ provided that, for all i with0 6 i 6 k, if Rβ ⊆ ΣCf and dimRβ > i+1, then V (z0−z0(x), . . . , zi−zi(x)) transversely intersects

164 DAVID B. MASSEY

Rβ near x; as we saw in II.1.24, this is equivalent to there existing an open neighborhood Ω of xsuch that

P(T ∗Rβ

Ω) ∩(V (z0 − z0(x), . . . , zi − zi(x))× Pi × 0

)= ∅.

Naturally, we define a Le-Vogel tuple for f at x to be a tuple (z0, . . . , zk) such that there existsa Le-Vogel stratification for f near x with respect to which (z0, . . . , zk) is a Le-Vogel tuple at x.

Proposition 1.8. Le-Vogel stratifications always exist and, for all x ∈ X, for a generic linearreorganization, z, of z, z is a Le-Vogel tuple for f at x.

In particular, if dimx

(ΣCf ∩ V (z0 − z0(x))

)6 0, then z is a Le-Vogel tuple for f at x.

Proof. By [Hi], Thom stratifications always exist (since f has codomain C); if we now refine a Thomstratification, Rβ, so that, for all visible Rβ , Σcnr(f|

Rβ

) is a union of strata, then we will have aLe-Vogel stratification. Now that we know that we can always produce a Le-Vogel stratification,it is completely trivial, and standard, to conclude that a generic linear reorganization will be aLe-Vogel tuple. We leave it as an exercise for the reader.

The last sentence follows from the fact that if dimx

(ΣCf∩V (z0−z0(x))

)6 0, then dimx ΣCf 6 1

and V (z0 − z0(x)) does not contain a 1-dimensional component of ΣCf through x. If Ci are theirreducible components of ΣCf of dimension 1 through x, then, in a neighborhood of x, we mayrefine any Le-Vogel stratification so that Ci − x and x are the good strata. The conclusionfollows.

Recalling the notation that we used in I.2.22, we let

Blim df (kM) := (−1)d Blim df

(Ch(kQ•)) :=

∑α

kmα Blim df

(T ∗

SαU)⊆ U × Cn+1 × Pn

andEim df (kM) := (−1)dEim df

(Ch(kQ•)) :=

∑α

kmαEim df

(T ∗

SαU),

where Bl denotes the blow-up and E denotes the corresponding exceptional divisor.

The following theorem is analogous to the first part of Theorem II.1.2. It tells us that z beinga Le-Vogel tuple implies the correct hypotheses hold for us to apply the Segre-Vogel Relation ofPart I.

Theorem 1.9. Let x ∈ X, and suppose that z is a Le-Vogel tuple for f at x. Then, there exists anopen neighborhood, Ω, of x such that, for all i, |π∗

(Eim df (kM)

)| properly intersects Ω× (Pi×0).

Proof. Our goal is to reduce the proof to the point where it precisely follows the proofs of LemmaII.1.25 and the first part of Theorem II.1.26.


As kM is independent of the Whitney stratification, we may assume that Sα is a Le-Vogelstratification for f at x, and that z is a Le-Vogel tuple for f at x with respect to Sα.

Fix an Sα such that T ∗SαU appears in kM ; such an Sα is necessarily visible. Let Eα :=

Eim df (T ∗SαU). We need to show that π(Eα) properly intersects Ω× (Pi × 0), for all i.

By III.5.2, π(Eα) ⊆ P(T ∗f|Sα

U). Since we are assuming that we have a Le-Vogel stratification, if

Sβ ⊆ Σcnr(f|Sα

), then(T ∗

f|Sα

U)|Sβ

⊆ T ∗SβU . Therefore, if Sβ ⊆ Σcnr(f|

Sα), then ν−1(Sβ)∩π(Eα) ⊆

P(T ∗SβU). As ν(π(Eα)) = Σcnr(f|

Sα), it follows that

π(Eα) ⊆⋃

Sβ⊆Σcnr(f|Sα

)

P(T ∗SβU).

The proof is now exactly the arguments of Lemma II.1.25 and the first part of TheoremII.1.26.

Theorem 1.10. The analytic set |Eim df (kM)| properly intersects U × Cn+1 × (Pi × 0) for alli if and only if |π∗

(Eim df (kM)

)| = |

∑v P(Ch(φf−v(kQ•)))| properly intersects U × (Pi × 0) for

all i, and whenever these equivalent conditions hold:

i) the k-shifted LeVo and conormal LeVo cycles of f with respect to z exist;

ii) for all i, kΛif ,z

= τ∗(Eim df (kM) ·

(U × Cn+1 × (Pi × 0)

));

iii) for all i,kΛi

f ,z= η∗τ∗

(Eim df (kM) ·

(U × Cn+1 × (Pi × 0)

))ν∗

(π∗(Eim df (kM)

)· (U × (Pi × 0))

)=

ν∗

(∑v

(−1)dv P(Ch(φf−v(kQ•))) · (U × (Pi × 0)));

and there exists a neighborhood Ω of |kM | ∩ im df such that

iv) the k-shifted conormal polar cycles of f|Ω with respect to z exist inside Ω;

v) for all i, |Blim df (kM)| properly intersects Ω× (Pi × 0) in Ω× Pn;

vi) inside Ω, for all i, kΓi+1

f ,z= τ∗

(Blim df (kM) ·

(U × Cn+1 × (Pi × 0)

)).

Note that iii) implies that the k-th shifted LeVo cycles are independent of the extension f .

Proof. That π∗(Eim df (kM)

)=∑

v(−1)dv P(Ch(φf−v(kQ•))) follows from Theorem III.3.10. Theequivalence of the two intersection conditions is a trivial consequence of the fact that the pointsin Eim df (kM) lie above the graph im df .

166 DAVID B. MASSEY

Now, i), ii), iv), v), and vi) follow immediately from the Segre-Vogel Relation (I.2.20). iii) followsby applying η∗ to each side of ii), using that ν π = η τ , and using again that π∗

(Eim df (kM)

)=∑

v(−1)dv P(Ch(φf−v(kQ•))).

Remark 1.11. Statement iii) of 1.10 tells us that – under the hypotheses of the theorem – kΛif ,z

only depends on f and the choice of w0, . . . , wi. As w0, . . . , wi are determined by z0, . . . , zi, onemight be tempted to reference only z0, . . . , zi in the notation for kΛi

f ,z, e.g., kΛ0

f ,z0. Note, however,

the hypotheses of the theorem put conditions on all the z’s. This should not be surprising – theVogel cycles are defined in terms of the inductive gap cycles, which are defined by downwardinduction. Thus, the higher-dimensional data needs to behave well before we can work with thelower-dimensional data.

Of course, we could use 1.10.iii to define kΛif ,z

, and thereby avoid needing to impose conditions

on all the coordinates and also avoid referring to f at all. However, we prefer the algorithmic,Vogel cycle definition, because it is the most useful for calculation. Moreover, in general, we willnot be interested in working in isolation with individual LeVo cycles, but, rather, will want torequire that all of them are well-behaved.

On the other hand, it is desirable to have the LeVo cycles be independent of the extension off . Therefore, we make the following definition.

Definition 1.12. If |Eim df (kM)| properly intersects U × Cn+1 × (Pi × 0) for all i, then we saythat the k-shifted Le-Vogel (LeVo) cycles of f with respect to z exist; we write kΛi

f,z in place ofkΛi

f ,zand refer to it as the i-th k-shifted Le-Vogel (LeVo) cycle of f with respect to z (that is, we

eliminate the reference to the extension f).If x ∈ X and the k-shifted Le-Vogel cycles of f with respect to z exist in a neighborhood of x,

then we say that i-th k-shifted Le-Vogel (LeVo) number of f at x with respect to z exists providedthat |kΛi

f,z| properly intersects V (z0 − x0, . . . , zi−1 − xi−1) at x, and then we define this Le-Vogelnumber to be kλi

f,z(x) :=(kΛi

f,z · V (z0 − x0, . . . , zi−1 − xi−1))x. When i = 0, we mean that

kλ0f,z(x) = (kΛ0

f,z)x.

Note that the LeVo cycles and numbers are only (possibly) non-zero for 0 6 k 6 d and 0 6 i 6 d,and they exist near a point x provided that z is a Le-Vogel tuple for f at x.

Example 1.13. As in Example 1.5, consider the case where dimx ΣCf 6 0. Thus, for all k,dim(x,dxf)

(|kM | ∩ im df

)6 0. It follows that, in a neighborhood of x, |π∗

(Eim df (kM)

)| =

P(T ∗xU) = x × Pn, which certainly properly intersects U × (Pi × 0) for all i.

Therefore, the equivalent hypotheses of Theorem 1.10 hold, and so the LeVo cycles of f (not f)exist, and they equal the LeVo cycles of f as given in 1.5.

Example 1.14. We return to the underlying space of Example III.4.14, the simplest non-l.c.i., butuse a function with non-isolated critical points.


Use (u, x, y, z) as coordinates for U := C4, and let X := V (u, x) ∪ V (y, z). Let

f := (uα + xβ)τ + yγ + zδ,

where α, β, γ, δ, τ > 2.Since d = 2, kQ• := 1−kP• and our calculation in III.4.14 tells us that Ch(kQ•) = 0 unless

k = 0 or 1, andCh(0Q•) = [V (u, x, w2, w3)] + [V (y, z, w0, w1)] ,

whileCh(1Q•) = [V (u, x, y, z)] .

One easily shows that

im df = V (w0 − τ(uα + xβ)τ−1αuα−1, w1 − τ(uα + xβ)τ−1βxβ−1, w2 − γyγ−1, w3 − δzδ−1),

im df ∩ V (u, x, w2, w3) = 0,

im df ∩ V (y, z, w0, w1) = V (uα + xβ , y, z, w0, w1, w2, w3),

im df ∩ |Ch(0Q•)| = V (uα + xβ , y, z, w0, w1, w2, w3)

andim df ∩ |Ch(1Q•)| = im df ∩ V (u, x, y, z) = 0.

Thus, ΣCf is the 1-dimensional set V (uα + xβ , y, z), and we calculate in the manner discussed inRemark 1.6.

Let C0 := V (u, x, y, z), C1 := V (u, x, w2, w3), and C2 := V (y, z, w0, w1). We need to calculateΠi

J∗z (f)and ∆i

J∗z (f)for each Cj ; let us denote the corresponding inductive gap cycles and Vogel

cycles by simply Πij and ∆i

j .As im df intersects C0 and C1 in the isolated point 0, ∆i

0 and ∆i1 are easy to calculate – they

are both 0 unless i = 0 and, then,

∆00 = (im df · V (u, x, y, z))0[0] = [0],

and∆0

1 = (im df · V (u, x, w2, w3))0[0] = (γ − 1)(δ − 1)[0].

Now,Π4

2 = [V (y, z, w0, w1)],

Π42 · V (w3 − δzδ−1) = [V (y, z, w0, w1, w3)] = Π3

2,

Π32 · V (w2 − γyγ−1) = [V (y, z, w0, w1, w2, w3)] = Π2

2,

Π22 · V (w1 − τ(uα + xβ)τ−1βxβ−1) =

(β − 1)[V (x, y, z, w0, w1, w2, w3)] + (τ − 1)[V (uα + xβ , y, z, w0, w1, w2, w3)] =

168 DAVID B. MASSEY

Π12 + ∆1

2,

Π12 · V (w0 − τ(uα + xβ)τ−1αuα−1) =

(β − 1)[α(τ − 1) + (α− 1)][0] = (β − 1)(ατ − 1)[0] = ∆02.

Therefore, the only non-zero LeVo cycles are

0Λ1f,z = (τ − 1)[V (uα + xβ , y, z)],

0Λ0f,z =

((γ − 1)(δ − 1) + (β − 1)(ατ − 1)

)[0],

and1Λ0

f,z = [0].

The non-zero LeVo numbers at the origin are

0λ1f,z(0) = β(τ − 1),

0λ0f,z(0) = (γ − 1)(δ − 1) + (β − 1)(ατ − 1),

and1λ0

f,z(0) = 1.

We have the following analog of Theorem II.7.2.

Theorem 1.15. Let x ∈ X, let Ω be an open neighborhood of x, and let Rβ be a Le-Vogelstratification for f|Ω . Suppose that z is a Le-Vogel tuple for f with respect to Rβ at all points ofΩ. Then, inside Ω, the k-shifted LeVo cycles of f , kΛi

f,z, exist and

|kΛif,z| ⊆

⋃Rβ⊆ΣCfdim Rβ6i

Rβ .

Proof. The existence of kΛif,z follows from 1.7 and 1.8. As we saw in the proof of 1.7, if Rβ is

visible,π(Eim df (T ∗Rβ

U))⊆

⋃Rγ⊆Σcnr(f|

Rβ

)

P(T ∗RγU).

The assumption on the coordinates z is that, if Rβ is visible, Rγ ⊆ Σcnr(f|Rβ

), and dimRγ > i+1,then

P(T ∗RγU) ∩

(Ω× (Pi × 0)

)= ∅.

The result now follows at once from 1.8.iii.


Chapter 2. LE-IOMDINE FORMULAS AND THOM’S CONDITION

We developed the Le-Iomdine-Vogel formulas in extreme generality in I.3.4 specifically so thatwe would be able to apply them to the LeVo cycles at this point. As we saw in Part II, Chapter 4,such formulas allow us to reduce questions concerning non-isolated singularities to questions aboutthe isolated case. Hence, we will be able to use III.6.12 in order give conditions which imply thatThom’s af condition holds.

We continue with our notation from the previous chapter.

For simplicity, we assume in the following two results that x ∈ V (z0); clearly, this causes no lossof generality.

Theorem 2.1. Let x ∈ V (z0)∩ |η∗(kM)| = V (z0)∩ kX. Let a be a non-zero complex number, andlet j > 1 be an integer. Let z denote the “rotated” coordinates (z1, . . . , zn, z0).

Suppose that the k-shifted LeVo cycles of f , kΛif,z, exist in a neighborhood of x and that, for all

i > 1, V (z0) properly intersects each kΛif,z at x.

Then, kλ0f,z(x) and kλ1

f,z(x) exist and, if j > 1 + kλ0f,z(x), then, in a neighborhood of x,

i) ΣkQ• (f + azj+1

0 ) = V (z0) ∩ ΣkQ• f ;

ii) dimxΣkQ• (f + azj+1

0 ) =(dimxΣ

kQ• f)− 1, provided that dimxΣ

kQ• f > 1;

iii) the k-shifted LeVo cycles of f + azj+10 with respect to z exist; and

iv) kλ0f+azj+1

0 ,z(x) = kλ0

f,z(x)+j(kλ1f,z(x)) and, for 1 6 i 6 n−1, kΛi

f+azj+10 ,z

= j(kΛi+1

f,z ·V (z0)).

Proof. This is simply a translation I.3.4 in our current situation. We have also used that (p, dpf) ∈|kM | ∩ V

(w0 − ∂f

∂z0, . . . , wn − ∂f

∂zn

)if and only if x ∈ Σ

kQ• f .

We immediately conclude

Corollary 2.2 (Le-Iomdine Formulas). Let x ∈ V (z0) ∩ |η∗(kM)| = V (z0) ∩ kX. Let a be anon-zero complex number, and let j > 1 be an integer. Let z denote the “rotated” coordinates(z1, . . . , zn, z0).

Suppose that the k-shifted LeVo numbers at x of f , kλif,z(x), exist and that j > 1 + kλ0

f,z(x).

Then, in a neighborhood of x,

i) ΣkQ• (f + azj+1

0 ) = V (z0) ∩ ΣkQ• f ;

170 DAVID B. MASSEY

ii) dimxΣkQ• (f + azj+1

0 ) =(dimxΣ

kQ• f)− 1, provided that dimxΣ

kQ• f > 1;

iii) the k-shifted LeVo numbers of f + azj+10 with respect to z exist; and

iv) kλ0f+azj+1

0 ,z(x) = kλ0

f,z(x)+j(kλ1f,z(x)) and, for 1 6 i 6 n−1, kλi

f+azj+10 ,z

(x) = j(kλi+1

f,z (x)).

Just as our generalized Le-Saito Theorem of II.6.5 followed immediately by applying the Le-Iomdine formulas to the actual result of Le and Saito on families of isolated affine hypersurfacesingularities, so too does a “super” general Le-Saito result follow by applying 2.2 above to CorollaryIII.6.12.

Since we wish to apply 2.2 to families, we first need to introduce some new notation. LetD

be an open disc about the origin in C, let Ω :=D × U , let t : Ω →

D denote the projection, let

g : Ω → C be an analytic function, let X be a (d+1)-dimensional analytic subset of Ω, let t denote

the restriction of t to X , and let g denote the restriction of g to X . We suppose thatD×0 ⊆ X

and that g(D × 0) = 0. For a ∈

D, use z as coordinates on each t−1(a) ∼= U , let Xa := t−1(a),

and let ga : Xa → C be given by ga := g|Xa.

Theorem 2.3 (General Le-Saito Theorem). Suppose that 0 6∈ ΣCt.Let Sα be a visible stratum of X of dimension dα such that g|Sα

6= 0, and let j be an integer suchthat bj−1(Lα) 6= 0. Let Y := Sα and let k := d− dα − j. In particular, Y could be any irreduciblecomponent of X , j could be zero, and k would be 0.

Suppose that, for all i, kλiga,z(0) is independent of a, for all small a, and that either

a) 0 ∈ ΣNash(g|Y ); or that

b) 0 6∈ Σcnr(g|Y ), and (Yreg,D× 0) satisfies Whitney’s condition a) at 0.

Then, the pair (Yreg,D × 0) satisfies the ag condition at 0. Moreover, in case a), there exists

an i such that kλiga,z(0) 6= 0.

Proof. The argument in case b) is exactly that of III.6.12; so, assume that we are in case a).

Let s := dim0 ΣC(g0). Let 0 j0 · · · js, and let h := g + zj00 + · · · + zjs

s . Certainly,h|Sα

6≡ 0, since g|Sα6≡ 0.

Consider the family ha := ga + zj00 + · · · + zjs

s . By 2.2.ii, dim0 ΣC(h0) = 0 and so, by III.6.7,dim0 ΣC(ha) 6 0 for all small a. By 1.5 and 1.13, kλ0

ha,z(0) = bdα+j−2(Fha,0). By an inductiveapplication of the Le-Iomdine formulas, kλ0

ha,z(0) is a function of only

kλiga,z(0)

i(and the fixed

j’s); thus, bdα+j−2(Fha,0) is independent of a for small a.It is trivial to show that, since 0 ∈ ΣNash(g|Y ), 0 ∈ ΣNash(h|Y ). Therefore, we may apply III.6.2

to conclude that (Yreg,D× 0) satisfies the ah condition at 0 and bdα+j−2(Fha,0) 6= 0.


As kλ0ha,z(0) = bdα+j−2(Fha,0) 6= 0, the Le-Iomdine formulas imply that there exists an i such

that kλiga,z(0) 6= 0.

As in II.6.5, by inducting, we would be finished if we could show that:

0 ∈ ΣNash(g|Y ) and [η] ∈ P(T ∗g|Yreg

Ω)0

implies that, for all j sufficiently large,

[η] ∈ P(T ∗

(g+zj0)|Yreg

Ω)0.

By III.5.2, what we need to show is that [η] ∈ p(Eg)0 implies that, for all j sufficiently large,[η] ∈ P

(T ∗

(g+zj0)|Yreg

Ω)0, where Eg denotes the exceptional divisors of Blim dg

T ∗Yreg

Ω, and p denotes

the projection Ω× Cn+2 × Pn+1 → Ω× Pn+1.Now, [η] ∈ p(Eg)0 if and only if there exists an analytic path γ(u) = (x(u), ω(u)) in T ∗

YregΩ

such that γ(0) = (0, d0g), γ(u) ∈ T ∗Yreg

Ω − im dg, and [ω(u) − dx(u)g] → [η] as u → 0. That[ω(u)− dx(u)g] → [η] is equivalent to

ξ ·ω(u)− dx(u)g

|ω(u)− dx(u)g|→ η

|η|,

for some root of unity ξ. One concludes easily that

|η|ξ|ω(u)− dx(u)(g + zj

0)|(ω(u)− dx(u)(g + zj

0))→ η,

for all large j. However, the terms on the left side of the above expression are clearly elements of(T ∗

(g+zj0)|Yreg

Ω)x(u)

whose projective class approaches that of η.

172 DAVID B. MASSEY

Chapter 3. LE-VOGEL CYCLES AND THE EULER CHARACTERISTIC

In this final chapter, we will relate the Le-Vogel cycles to the Euler characteristic of the Milnorfibre, in a way that generalizes our result in II.10.3. In order to accomplish this, we must recall adefinition and a result from [Mas11].

We continue with the notation from the previous two chapters.

Proposition/Definition 3.1. Let p ∈ X and let F• be a bounded, constructible complex onX. Then, for a generic choice of the coordinates z, there exists an open neighborhood, W, ofp and cycles Λi

F•,zin W such that each Λi

F•,zis purely i-dimensional, Λi

F•,zproperly intersects

V (z0 − z0(x), . . . , zi−1 − zi−1(x)) and, for all x ∈ W,

χ(F•)x = (−1)d∑

i

(−1)i(Λi

F•,z· V (z0 − z0(x), . . . , zi−1 − zi−1(x))

)x,

(here, when i = 0, we mean that the intersection number is simply(Λ0

F•,z

)x).

Moreover, whenever such cycles exist, they are unique.

In the case where dimp suppF• = 1, such cycles exist if dimp

(V (z0 − z0(p)) ∩ suppF•) 6 0.

We call ΛiF•,z

the i-dimensional characteristic polar cycle of F• with respect to z, and refer toλi

F•,z(x) :=

(Λi

F•,z· V (z0 − z0(x), . . . , zi−1 − zi−1(x))

)x

as the i-th characteristic polar number ofF• with respect to z at x.

Proof. The statement about the case where dimp suppF• 6 1 is trivial. The remaining statementsare a combination of Propositions 2.4 and 3.1 from [Mas11].

Below, we refer to the absolute polar varieties of Le and Teissier ([L-T2], [Te4], [Te5]); however,we need to explain our notation. We let Γi

z(Sα) denote the i-dimensional polar variety of Sα withrespect to the flag 0 ⊆ V (z0, z1, . . . , zn−1) ⊆ . . . V (z0, z1) ⊆ V (z0) ⊆ Cn+1. Thus, as a set,Γi

z(Sα) = crit((z0, . . . zi)|(Sα)reg

). See also our treatment in Section 7 of [Mas11].

Theorem 3.2. Let p ∈ X and let F• be a bounded complex on X, which is constructible withrespect to Sα. Suppose that Ch(F•) =

∑αmα

[T ∗

SαU].

Then, for a generic choice of the coordinates z, there exists an open neighborhood, W, of p inwhich all of the Λi

F•,zexist, such that, for all i, P(Ch(F•)) and W × (Pi × 0) intersect properly

in P(T ∗W), and the restriction of ν to W × Pn yields the equalities

ΛiF•,z

=∑α

mαΓiz(Sα) = ν∗

(P(Ch(F•)) · (W × (Pi × 0))

).

In the case where Y := suppF• is 1-dimensional at p, the conclusions hold if z0 is finite at p,i.e., if dimp(V (z0 − z0(p)) ∩ Y

)p

= 0.


Proof. Aside from the statement about the case where dimp suppF• = 1, this follows immediatelyfrom Theorem 7.5 of [Mas11]. However, when dimp suppF• = 1, the hypotheses of 7.5 of [Mas11]are strictly stronger than saying that z0 is finite at p, so we must provide a proof of this portion.

Assume that Y := suppF• is 1-dimensional at p and that z0 is finite at p. Let C denote acomponent of Y through p, and let x

Cdenote a point in C−p. By shrinking our neighborhood,

we may assume that C −p is smooth, and that χ(F•)xC

is independent of the choice of xC. As

z0 is not constant along C, we may also assume that (x, dxz0) 6∈ T ∗Creg

U for x 6= p. It follows atonce that P(Ch(F•)) and W× (Pi ×0) intersect properly for all i. Also, since the hypotheses of7.5 of [Mas11] are satisfied at all x 6= p, 7.5 of [Mas11] implies that

ν∗(P(Ch(F•)) · (W × (P1 × 0))

)=∑C

χ(F•)xC

[C].

Moreover, by applying Theorem III.3.10 to the function z0, we find that

ν∗(P(Ch(F•)) · (W × (P0 × 0))

)= (Ch(F•) · im dz0)p [p] = χ(φz0−z0(p)F•)p[p].

As

χ(φz0−z0(p)F•)p = χ(ψz0−z0(p)F•)p − χ(F•)p = −χ(F•)p +∑C

(C · V (z0 − z0(p)))p χ(F•)xC,

the conclusion follows.

We relate the LeVo cycles and numbers to the Euler characteristic by the following theorem.

Theorem 3.3. Let p ∈ X and assume that f(p) = 0. Fix a k and let F• := φf (kQ•). Then, fora generic choice of the coordinates z, there exists an open neighborhood, W, of p in which all ofthe Λi

F•,zexist, all of the k-shifted LeVo numbers, kλi

f,z, exist, and such that kΛif,z = (−1)dΛi

F•,z

and kλif,z(x) = (−1)dλi

F•,z(x) for all x ∈ W, i.e.,

χ(φf [−1](kQ•)

)x

=∑

i

(−1)i kλif,z(x).

In the case where Y := suppF• is 1-dimensional at p, the conclusions hold if

dimp(V (z0 − z0(p)) ∩ Y)p

= 0.

Proof. Throughout the proof, we will work in an arbitrarily small neighborhood of p.From 1.10.iii, we have

kΛif,z = (−1)d ν∗

(P(Ch(φf (kQ•))) · (W × (Pi × 0))

).

174 DAVID B. MASSEY

Hence, by 3.2, kΛif,z = (−1)d Λi

F•,zand kλi

f,z(x) = (−1)d λiF•,z

(x).

By definition, χ(F•)x = (−1)d−1∑

i(−1)iλiF•,z

(x). Therefore,

χ(F•)x = −∑

i

(−1)i kλif,z(x),

or, equivalently,χ(F•[−1])x =

∑i

(−1)i kλif,z(x).

Before we can connect Theorem 3.3 to the ordinary cohomology of the Milnor fibre with constantcoeeficients, we need to prove a lemma. While we suspect that this lemma is well-known, we canfind no reference.

Lemma 3.4. Let F• be a bounded, constructible complex on X. Then, for all x ∈ X,

χ(F•)x =∑

k

(−1)kχ(µH0(F•[k])

)x.

Proof. For convenience, assume that x = 0. The proof is by induction on the dimension of X at 0.

If dim0X = 0, then Hi(µH0(F•[k]))0 = 0 unless i = 0, and then

H0(µH0(F•[k]))0 ∼= H0(F•[k])0 = Hk(F•)0.

Thus, the lemma holds if dim0X = 0.

Now, assume the lemma for spaces of dimension j, and suppose that dim0X = j + 1. Let L bea generic linear form. Consider the distinguished triangle(

µH0(F•[k]))|V (L)

[−1] −→ ψL[−1]µH0(F•[k]) −→ φL[−1]µH0(F•[k])[1]−→;

it yields the equality

χ(ψL[−1]µH0(F•[k])

)0

= χ((

µH0(F•[k]))|V (L)

[−1])0

+ χ(φL[−1]µH0(F•[k])

)0.

Thus,χ(µH0(ψL[−1]F•[k])

)0

= −χ(µH0(F•[k])

)0

+ χ(µH0(φL[−1]F•[k])

)0,

and so∑k

(−1)kχ(µH0(ψL[−1]F•[k])

)0

= −∑

k

(−1)kχ(µH0(F•[k])

)0

+∑

k

(−1)kχ(µH0(φL[−1]F•[k])

)0.

Applying our inductive hypothesis twice, we obtain


∑k

(−1)kχ(µH0(F•[k])

)0

= χ(φL[−1]F•)

0−χ(ψL[−1]F•)

0= −χ

(φLF•)

0+χ(ψLF•)

0= χ(F•)0,

and we are finished.

The relationship between the characteristic polar cycles of φfC•X and the k-shifted LeVo cycles

is given in

Theorem 3.5. Let p ∈ X and suppose that f(p) = 0. Then, for a generic choice of the coordinatesz, for all k, there exists an open neighborhood, W, of p in which all of the Λi

φf C•X

,zexist, all of the

k-shifted LeVo numbers, kλif,z, exist, and such that Λi

φf C•X

,z= (−1)d−d

∑k(−1)

k kΛif,z and, for all

x ∈ W ∩ V (f),χ(Ff,x

)=∑

k

(−1)k∑

i

(−1)d−i−1 kλi

f,z(x),

where Ff,x denotes the Milnor fibre of f at x.In the case where Y := ΣCf has dimension 1 at p, the conclusions hold for all k provided that

dimp

(V (z0 − z0(p)) ∩ Y

)p

= 0.

Proof. Using 3.3 and 3.4,

χ(Ff,x

)= χ(φfC•

X)x =∑

k

(−1)kχ(µH0(φfC•

X [k]))x

=∑

k

(−1)kχ(φf [−1]µH0(C•

X [k + 1]))x

=

∑k

(−1)kχ(φf [−1] d−k−1Q•)

x=∑

k

(−1)k∑

i

(−1)i d−k−1λif,z(x) =

∑k

(−1)k∑

i

(−1)d−i−1 kλif,z(x).

That Λiφf C•

X,z

= (−1)d−d∑

k(−1)k kΛi

f,z follows immediately.

Example 3.6. Recall Example 1.14 where X = V (u, x) ∪ V (y, z) ⊆ C4, and

f := (uα + xβ)τ + yγ + zδ,

where α, β, γ, δ, τ > 2.Applying Theorem 3.5, we obtain

χ(Ff,0

)= −0λ0

f,z(0) + 0λ1f,z(0) + 1λ0

f,z(0) =

−(γ − 1)(δ − 1)− (β − 1)(ατ − 1) + β(τ − 1) + 1 = −(γ − 1)(δ − 1) + τ(−αβ + α+ β).

176 DAVID B. MASSEY

We can verify this calculation. One easily sees that Ff,0 is the disjoint union of F1, the Milnorfibre of (uα + xβ)τ restricted to V (x, y), and F2, the Milnor fibre of yγ + zδ restricted to V (u, x).Thus, F1 is homotopy-equivalent to the disjoint union of τ copies of a bouquet of (α − 1)(β − 1)circles, and F2 is homotopy-equivalent to a bouquet of (γ − 1)(δ − 1) circles. Therefore,

χ(Ff,0

)= −τ(α− 1)(β − 1)− (γ − 1)(δ − 1) + τ + 1− 1,

where the last −1 is due to the fact that we use the reduced cohomology. One sees, then, thatthe calculations agree.

Appendix A: Analytic Cycles and Intersections

We wish to consider schemes, cycles, and sets. Frequently, we will be in the algebraic settingand, hence, we may use algebraic schemes, cycles, and sets. However, as we wish to treat the moregeneral analytic case, we should clarify what we mean by the terms scheme and cycle.

In the analytic setting, by scheme, we actually mean a (not necessarily reduced) complex analyticspace, (X,OX), in the sense of [G-R1] and [G-R2]. By the irreducible components of X, we meansimply the irreducible components of the underlying analytic set X. If we concentrate our attentionon the germ of X at some point p, then we may discuss embedded subvarieties and (non-embedded,or isolated) components of the germ of X at p – these correspond to non-minimal and minimalprimes, respectively, in the set of associated primes of the Noetherian local ring O

X,p .If X is a complex space and α is a coherent sheaf of ideals in OX , then we write V (α) for the

possibly non-reduced analytic subspace defined by the vanishing of α.By the intersection of a collection of closed subschemes, we mean the scheme defined by the

sum of the underlying ideal sheaves. By the union of a finite collection of closed subschemes, wemean the scheme defined by the intersection (not the product) of the underlying ideal sheaves. Wesay that two subschemes, V and W , are equal up to embedded subvariety provided that, in eachstalk, the isolated components of the defining ideals (those corresponding to minimal primes) areequal. Our main concern with this last notion is that it implies that the cycles [V] and [W] areequal (see below).

A.1 Given an analytic space X (with its reduced structure), an analytic cycle in X is a formalsum

∑m

V[V ], where the V ’s are irreducible analytic subsets of X, the m

V’s are integers, and the

collection V is a locally finite collection of subsets of X. As a cycle is a locally finite sum, andas we will normally be concentrating on the germ of an analytic space at a point, usually we cansafely assume that a cycle is actually a finite formal sum.

Throughout this book, whenever we write a cycle∑

mV[V ], we shall assume that the V ’s are

distinct and that none of the mV’s are zero. This is the same as saying that the presentation is

minimal, in the sense that no further cancellations are possible.

We say that a cycle∑

mV[V ] is positive if m

V> 0 for all V ; a cycle is non-negative if it is the

zero-cycle or is positive.

A.2 Given an analytic space, (X,OX

), we wish to define the (positive) cycle associated to(X,O

X). In the algebraic context, this is given by Fulton in [Fu, 1.5] as

[X] :=∑

mV[V ],

where the V ’s run over all the irreducible components of X, and mV

equals the length of the ringO

X,V, the local ring of X along V . In the analytic context, we wish to use the same definition, but

we must be more careful in defining the mV.

Define mV

as follows. Take a point p in V . The germ of V at p breaks up into irreducible germcomponents (Vp)i. Take any one of the (Vp)i and let m

Vequal the length of the ring (O

X,p)(Vp)i

(that is, the local ring of X at p localized at the prime corresponding to (Vp)i). This number isindependent of the point p in V and the choice of (Vp)i.

177

178 DAVID B. MASSEY

Note that any embedded subvarieties of a scheme do not contribute to the associated cycle.

One can easily show that, if f, g ∈ OX

, then [V (fg)] = [V (f)]+ [V (g)]; in particular, [V (fm)] =m[V (f)].

If Y is an analytic subset of X and C is a cycle in Y , then we may naturally consider C as acycle in X.

We shall be dealing with analytic schemes, cycles, and analytic sets. For clarification of whatstructure we are considering, we shall at times enclose cycles in square brackets, [ ], and analyticsets in a pair of vertical lines,||. Occasionally, when the notation becomes cumbersome, we shallsimply state explicitly whether we are considering V as a scheme, a cycle, or a set.

We say that two cycles are equal at a point, p, provided that the portions of each cycle whichpass through p are equal. When we say that a space, X, is purely k-dimensional at a point, p, wemean to allow for the vacuous case where X has no components through p.

We wish to describe some aspects of intersection theory. Of course, [Fu] is the definitive referencefor this subject. However, we deal only with cycles, not cycle classes, and we deal only with properintersections inside complex manifolds; this makes much of the theory fairly trivial to describe.

A.3 If V and W are irreducible subschemes of a connected complex manifold, M , and Z is anirreducible component of V ∩W such that codimM Z = codimM V + codimM W , then we say thatV and W intersect properly along Z, or that Z is a proper component of V ∩W . Two irreduciblesubschemes V and W in a connected complex manifold, M , are said to intersect properly in Mprovided that they intersect properly along each component of V ∩W ; when this is the case, theintersection product, ([V ] · [W ];M), of [V ] and [W ] in M is characterized axiomatically by fourproperties listed below: openness, transversality, projection, and continuity (see [Fu], Example11.4.4).

A.4 If α is a coherent sheaf of ideals in OM and f ∈ OM is such that V (f) contains no embeddedsubvarieties or irreducible components of V (α), then V (α) and V (f) intersect properly in M and[V (α)] · [V (f)] = [V (α + 〈f〉)] (see [Fu], 7.1.b). This statement immediately implies one which,a priori, seems stronger: if α and f are as before and V (f) contains no irreducible componentof V (α) and contains no embedded subvariety which is of codimension one inside someirreducible component of V (α), then V (α) and V (f) intersect properly in M and [V (α)] ·[V (f)] = [V (α + 〈f〉)]

More generally, if W := V (α) is a subscheme of M and f1, . . . , fk ∈ OM determine regularsequences in the stalks OW,p and OM,p at all points p ∈ W ∩ V (f1, . . . , fk), then

[V (α)] · [V (f1, . . . , fk)] = [V (α + 〈f1, . . . , fk〉)] .

The two paragraphs above allow one to define the intersection with a hypersurface (or, moregenerally, a Cartier divisor) without having to refer to an ambient manifold. Suppose that V (α) is asubscheme of an analytic space X, that X is contained in an analytic manifold M , and that f ∈ OX .Then, locally, OX

∼= OM/γ for some coherent sheaf of ideals γ ⊆ OM . Let α ⊆ OM be a coherentsheaf of ideals such that γ ⊆ α and such that α/γ corresponds to α, i.e., α is such that V (α) = V (α).Let f be an extension of f to M . If V (f) contains no embedded subvarieties or isolated componentsof V (α), then V (f) contains no embedded subvarieties or isolated components of V (α) and so, bythe previous paragraph,

([V (α)] · [V (f)]; M

)= [V (α+ < f >)], which defines the same cycle in X

APPENDIX A: ANALYTIC CYCLES AND INTERSECTIONS 179

as does [V (α+ < f >)]. Therefore, we may unambiguously define ([V (α)] · [V (f)] ; X) by settingit equal to [V (α+ < f >)].

A.5 Two cycles∑

mi[Vi] and∑

nj [Wj ] are said to intersect properly if Vi and Wj intersectproperly for all i and j; when this is the case, the intersection product is extended bilinearly bydefining ∑

mi[Vi] ·∑

nj [Wj ] =∑

minj ([Vi] · [Wj ]) .

Occasionally it is useful to include the ambient manifold in the notation; in these cases we write(C1 · C2; M) for the proper intersection of cycles C1 and C2 in M .

If two cycles C1 and C2 intersect properly and C1 ·C2 =∑

nk[Zk], where the Zk are irreducible,then the intersection number of C1 and C2 at Zk, (C1 · C2)Zk

, is defined to be nk; that is, thenumber of times Zk occurs in the intersection, counted with multiplicity. Note that, when C1 andC2 have complementary codimensions, all the Zk are merely points.

If V is irreducible at p, then the multiplicity of V at p, multpV , is the minimum value of([V ] · [W ])p, where W ranges over all analytic subsets which are irreducible at p and which havep as a component of the proper intersection of V and W ; in fact, when working in affine space,W may be chosen to be a generic affine linear subspace through p of dimension complementary tothat of V .

Suppose that p is an isolated point in the proper intersection of V and W , where V and Ware irreducible. Then, ([V ] · [W ])p >

(multpV

)(multpW

)with equality holding if and only if the

projectivized tangent cones P(TpV ) and P(TpW ) are disjoint.

A.6 It is fundamental that (C1 · C2)Zkcan be calculated locally; that is, if U is an open subset

of M such that Zk ∩ U 6= ∅, then

(openness) (C1 ∩ U · C2 ∩ U ; U)Zk∩U = (C1 · C2; M)Zk

(see [Fu], 11.4.4).

A.7 If V and W are two irreducible subvarieties of M and P is an irreducible component ofV ∩W , we say that V and W are generically transverse along P in M provided that V and W arereduced and, at generic points of P , V and W are smooth and intersect transversely in M ; naturally,we say that V and W are generically transverse in M provided they are generically transverse alongevery component of the intersection. Another fundamental property of intersection numbers is thetransversality characterization:

if V and W are irreducible subschemes of M which intersect properly along an irreducible compo-nent P , then (V ·W )P = 1 if and only if V and W are generically transverse along P in M ([Fu],8.2.c and 11.4.4).

A.8 If C1, C2, and C3 are positive cycles such that C1 and C2 intersect properly, and C3 properlyintersects C1 · C2, then C2 and C3 intersect properly, C1 properly intersects C2 · C3, and

(associativity) (C1 · C2) · C3 = C1 · (C2 · C3) .

We wish to introduce a slight generalization of proper intersections of cycles. If V and W areirreducible subschemes of a connected complex manifold, M , and Z is an irreducible component of

180 DAVID B. MASSEY

V ∩W along which V and W intersect properly, then, for every open neighborhood U ⊆ M suchthat V ∩ W ∩ U = Z ∩ U , the value of (V ∩ U · W ∩ U ; U)Z∩U is independent of U ; we define(V · W )Z to be this common value. If Z is a proper component of V ∩ W , then, by [Fu], 8.2.a,(V ·W )Z 6 [V ∩W ]Z .

We define [V ] ·p [W ] :=∑

Z(V ·W )Z , where the sum is over all Z along which V and W intersect

properly. We extend bilinearly∑mi[Vi] ·p

∑nj [Wj ] =

∑minj ([Vi] ·p [Wj ]) .

We refer to this as the proper intersection of the two cycles.One easily verifies that, if C1, C2, and C3 are positive cycles, then

(C1 ·p C2) ·p C3 = C1 ·p (C2 ·p C3) .

A.9 Given a point p ∈ M , a curve W = V (α) in M which is reduced and irreducible at p, anda hypersurface V (f) ⊆ M which intersects W properly at p, there is a very useful way to calculatethe intersection number ([W ] · [V (f)])p. One takes a local parameterization φ(t) of W which takes0 to p, and then ([W ] · [V (f)])p = multtf(φ(t)), the degree of the lowest non-zero term. This iseasy to see, for composition with φ induces an isomorphism

OM,p

α+ < f >

φ−−−→ Ctf(φ(t))

.

Of course, if c is small and unequal to zero, multtf(φ(t)) is precisely the number of roots off(φ(t))− c which occur near zero.

More generally, given a point p ∈ M , a curve W = V (α) in M (which need not be reduced orirreducible at p), and a hypersurface V (f) ⊆ M which intersects W properly at p, consider themap given by multiplication by f

OM,p

α

·f−−→ OM,p

α;

the intersection number ([W ] · [V (f)])p = dimC(coker(·f))− dimC(ker(·f)).

A.10 Combining this with transversality, we obtain the following dynamic intersection property:

(V (α) · V (f))p =∑

q∈

Bε∩V (α)∩V (f−c)

(V (α) · V (f − c)

)q,

where ε > 0 is sufficiently small,

Bε is an open ball of radius ε centered at p, and |c| ε.This formula may seem ridiculously complex, since all the

(V (α) · V (f − c)

)q

equal 1; however,it is the form which generalizes nicely: if C is a purely one-dimensional cycle and V (f) properlyintersects |C| at p, then

(C · V (f))p =∑

q∈

Bε∩|C|∩V (f−c)

(C · V (f − c)

)q.

This is a special case of conservation of number, which we shall discuss more generally below.


A.11 The projection formula ([Fu], 11.4.4.iii) allows us to calculate intersections inside normalslices. Let C =

∑ni[Vi] be a cycle in M and let N be a closed submanifold of M such that N

generically transversely intersects each Vi in M (this is equivalent to: for each component Z ofVi ∩N , (Vi ·N)Z = 1).

We may consider (C ·N ; M) as a cycle in N ; denote this cycle by C. Let B be a cycle in N ;we may also consider B as a cycle in M . Then, the projection formula states:

(projection formula) (C ·p B; N) = (C ·p B; M).

The projection formula lets us reduce the problem of calculating intersection numbers to the casewhere the intersection consists of isolated points. To see this, suppose that C1 and C2 are two cyclesin M which intersect properly and let C1 ·C2 =

∑ni[Vi]. To calculate ni0 , first let p be a smooth

point of Vi0 which is not contained in any other Vi of dimension less than or equal to that of Vi0 .Now, take a normal slice, N , to Vi0 at a smooth point, p, of Vi0 ; that is, in an open neighborhood,U , of p in M , N is a closed submanifold of U of complementary codimension to Vi0 such that Ntransversely intersects Vi0 inside U in the single point p and such that N is generically transverse toall other Vi and to all components of C1 and C2 in U . By locality ni0 = (C1∩U · C2∩U ; U)Vi0∩U .As Vi0 ∩ U is the only component of (C1 ∩ U · C2 ∩ U ; U) whose intersection with N gives p,the transversality characterization yields that ni0 = (C1 ∩ U · C2 ∩ U ·N ; U)p. But, we wish tocalculate this intersection inside of the normal slice N – this is what we get from the projectionformula.

Replace the M , N , C, and B in the projection formula as stated above by letting M = U ,N = N , C = C1 ∩U , and B = (C2 ∩U) ·N (consider B as a cycle in N). Then, the formula yieldsthat (

(C1 ∩ U) ·N) · ((C2 ∩ U) ·N); N)

= (C1 ∩ U · C2 ∩ U ·N ; U).

Thus, we see that taking normal slices reduces calculating proper intersections of cycles to the casewhere the dimension of the intersection is zero.

A.12 There is one last property of intersections of cycles that we need – continuity ([Fu],11.4.4.iii). This property is what makes intersections dynamic; one can move the intersections ina family.

Let M be a analytic manifold and letD be an open disc centered at the origin in C. Then,

the projection M ×D π−→

D determines a one-parameter family of spaces, and any subscheme

W ⊆ M ×D determines a one-parameter family of schemes Wt := W ∩ (M × t). Hence, any

cycle C :=∑

ni[Vi] in M ×D determines a family of cycles Ct :=

∑ni[(Vi)t] in M ∼= M × t.

If a cycle C in M ×D has a component contained in M × t for some t, then that component

does not “propagate” through the family; we wish to eliminate such “bad” components. For any

cycle C =∑

ni[Vi] in M ×D and any analytic set W ⊆ M ×

D, let

C¬W :=∑

Vi 6⊆W

ni[Vi],

and letC∗

t :=((

C¬(M × t))· (M × t); M ×

D

)=

∑Vi 6⊆M×t

ni [(Vi)t] .

Continuity of intersections states that, if C is a cycle in M ×D with no component contained

in M × 0, and E is a cycle in M such that C0 properly intersects E in M , then there exists a

182 DAVID B. MASSEY

(possibly) smaller disk centered at the originD′ ⊆

D such that C properly intersects E ×

D′ in

M ×D′ and, for all t ∈

D′,

(continuity)(

(E ×D′) · C; M ×

D′

)∗

t

= (E · Ct; M).

A.13 We saw earlier how the projection formula allows us to reduce the calculation of inter-section multiplicities to the case where the intersection is zero-dimensional. We wish to see nowhow continuity allows us to deform in a family in order to calculate zero-dimensional intersectionmultiplicities.

We will prove a dynamic formula for intersection multiplicities; this formula is known as con-servation of number. Let E be a k-dimensional cycle in M and let f := (f1, . . . , fk) ∈ (OM )k

be such that E and V (f) intersect properly in the single point p. This implies that V (f) is purelyk-codimensional inside M at p. In what follows, we assume that we are always working in anarbitrarily small neighborhood of p.

Let g1(z, t), . . . , gk(z, t) ∈ OM×

D

be such that gi(z, 0) = fi(z) for all i. Let C be the cycle

in M ×D given by

[V (g1(z, t), . . . , gk(z, t))

]. Note that C0 =

[V (f)

](in M) and that C has no

components contained in M × 0, for otherwise V (f) would have a component of dimension atleast (dim M) + 1− k.

Applying continuity at t = 0, we find that

(E · V (f); M) =(

(E ×D′) · C; M ×

D′

)∗

0

for a smaller discD′. Note that (E ×

D′) · C is a purely 1-dimensional cycle, say

∑j mj [Wj ].

Applying continuity at general t, we find that, for all t ∈D′,

(E · Ct; M) =∑

Wj 6⊆M×0

mj [(Wj)t] .

Hence, by openness and transversality, we find that mj = (E ·Ct; M)q for sufficiently small t 6= 0and q ∈ (Wj)t. Now, since (E · V (f); M) =

∑Wj 6⊆M×0

mj [(Wj)0], we may apply our earlier

special case of dynamic intersections between curves and hypersurfaces to conclude the generalconservation of number formula

(E · V (f))p =∑

q∈

Bε∩|E∩Ct|

(E · Ct

)q,

where ε > 0 is sufficiently small,

Bε is an open ball of radius ε centered at p, |t| ε, and Ct equals[V (g1(z, t), . . . , gk(z, t))

].

A.14 Finally, we need to define the proper push-forward of cycles (see [Fu, 1.4]). Let f : X → Ybe a proper morphism of analytic spaces. Then, for each irreducible subvariety V ⊆ X, W :=f(V ) is an irreducible subvariety of Y . There is an induced embedding of rational function fields


R(W ) → R(V ), which is a finite field extension if V and W have the same dimension. Define thedegree of V over W by

deg(V/W ) :=

[R(V ) : R(W )] if dimW = dim V

0 if dim W 6= dim V,

where [R(V ) : R(W )] denotes the degree of the field extension, which equals the number of pointsin V ∩ f−1(p) for a generic choice of p ∈ W .

Define f∗(V ) byf∗(V ) = deg(V/W )[W ].

This extends linearly to a homomorphism which is called the proper push-forward of cycles:

f∗

(∑mV [V ]

)=

∑mV f∗(V ).

We will need the following special case of the more general push-forward formula (see [Fu],2.3.c). Let π : M → N be a proper map between analytic manifolds. Let f ∈ ON . Let C be acycle in M which intersects V (f π) properly in M . Then, V (f) properly intersects π∗(C) in Nand

(push-forward formula) π∗(V (f π) · C

)= V (f) · π∗(C).

We need one other formula involving the push-forward and graphs of morphisms. Suppose thatwe have an analytic map f : M → N between analytic manifolds. Then, for any irreduciblesubvariety V ⊆ M , the graph of f|V , Gr(f|V ), is isomorphic to V . Thus, one would expect thatintersecting with V in M could be identified with intersecting with Gr(f|V ) in M ×N ; this is, infact the case.

Suppose that A :=∑

mi[Vi] and B are properly intersecting cycles in M . Let Gr(A) :=∑mi[Gr(f|Vi

)] in M ×N , and let pr : M ×N → M denote the projection. Then, Gr(A) properlyintersects B ×N in M ×N , and we have the graph formula:

(graph formula) (A ·B; M) = pr∗(Gr(A) · (B ×N); M ×N).

This is easy to see: Gr(A) = (A ×N) · Gr(f), since Gr(f) determines a regular sequence in eachOVi×N (see A.4). Hence,

Gr(A) · (B ×N) = (A×N) · (B ×N) ·Gr(f) = ((A ·B)×N) ·Gr(f),

where the last equality follows from normal slicing (see A.11). The graph formula follows easilyby using A.4 again, together with the definition of the proper push-forward.

184 DAVID B. MASSEY

APPENDIX B:

THE DERIVED CATEGORY AND VANISHING CYCLES

This appendix contains a number of basic results on the derived category, perverse sheaves,and vanishing cycles. Primary sources for most of these results are [BBD], [Br], [De], [G-M3],[K-S2], [Mac2], [M-V], and [Ve].

This appendix is organized as follows:

§1. Constructible Complexes – This section contains general results on bounded, con-structible complexes of sheaves and the derived category.

§2. Perverse Sheaves – This section contains the definition and basic results on perversesheaves. Here, we also give the axiomatic characterization of the intersection cohomology com-plex. Finally in this section, we also give some results on the category of perverse sheaves. Thiscategorical information is augmented by that in section 5.

§3. Nearby and Vanishing Cycles – In this section, we define and examine the complexesof sheaves of nearby and vanishing cycles of an analytic function. These complexes contain hyper-cohomological information on the Milnor fibre of the function under consideration.

§4. Some Quick Applications – In this section, we give three easy examples of results onMilnor fibres which follow from the machinery described in the previous three sections.

§5. Truncation and Perverse Cohomology – This section contains an informal discussionon t-structures. This enables us to describe truncation functors and the perverse cohomology of acomplex. It also sheds some light on our earlier discussion of the categorical structure of perversesheaves.

§1. Constructible Complexes

Much of this section is lifted directly from Goresky and MacPherson’s paper “Intersection Ho-mology II” [G-M3].

In this appendix, we are primarily interested in sheaves on complex analytic spaces, and we makean effort to state most results in this context. However, as one frequently wishes to do such thingsas intersect with a closed ball, one really needs to consider at least the real semi-analytic case (thatis, spaces locally defined by finitely many real analytic inequalities). In fact, one can treat thesubanalytic case. Generally, when we leave the analytic category we shall do so without comment,assuming the natural generalizations of any needed results. However, the precise statements in thesubanalytic case can be found in [G-M2], [G-M3], and [K-S2].

Let R be a regular Noetherian ring with finite Krull dimension (e.g., Z,Q, or C). A complex(A•, d•) (usually denoted simply by A• if the differentials are clear or arbitrary)

· · · → A−1 d−1

−−→ A0 d0

−→ A1 d1

−→ A2 d2

−→ · · ·185

186 DAVID B. MASSEY

of sheaves of R-modules on a complex analytic space, X, is bounded if Ap = 0 for |p| large.The cohomology sheaves Hp(A•) arise by taking the (sheaf-theoretic) cohomology of the com-

plex. The stalk of Hp(A•) at a point x is written Hp(A•)x and is isomorphic to what one gets byfirst taking stalks and then taking cohomology, i.e., Hp(A•

x).

The complex A• is constructible with respect to a complex analytic stratification, S = Sα,of X provided that, for all α and i, the cohomology sheaves Hi(A•

|Sα) are locally constant and

have finitely-generated stalks; we write A• ∈ DS (X). If A• ∈ DS (X) and A• is bounded, we writeA• ∈ Db

S(X).

If A• ∈ DbS(X) for some stratification (and, hence, for any refinement of S) we say that A• is

a bounded, constructible complex and write A• ∈ Dbc(X). (Note, however, that Db

c(X) actuallydenotes the derived category and, while the objects of this category are, in fact, the bounded,constructible complexes, the morphisms are not merely maps between complexes. We shall returnto this.)

When it is important to indicate the base ring in the notation, we write DS (RX

), DbS(R

X), and

Dbc(RX

).

A single sheaf A on X is considered a complex, A•, on X by letting A0 = A and Ai = 0 fori 6= 0; thus, R•

Xdenotes the constant sheaf on X.

The shifted complex A•[n] is defined by (A•[n])k = An+k and differential dk[n] = (−1)ndk+n.

A map of complexes is a graded collection of sheaf maps φ• : A• → B• which commute withthe differentials. The shifted sheaf map φ•[n] : A•[n]→ B•[n] is defined by φk

[n] := φk+n (note thelack of a (−1)n). A map of complexes is a quasi-isomorphism provided that the induced maps

Hp(φ•) : Hp(A•)→ Hp(B•)

are isomorphisms for all p. We use the term “quasi-isomorphic” to mean the equivalence relationgenerated by “existence of a quasi-isomorphism”; this is sometimes refered to as “generalized”quasi-isomorphic.

If φ• : A• → I• is a quasi-isomorphism and each Ip is injective, then I• is called an injectiveresolution of A•. Injective resolutions always exist (in our setting), and are unique up to chainhomotopy. However, it is sometimes important to associate one particular resolution to a complex,so it is important that there is a canonical injective resolution which can be associated to anycomplex (we shall not describe the canonical resolution here).

If A• is a complex on X, then the hypercohomology module, Hp(X;A•), is defined to be thep-th cohomology of the global section functor applied to the canonical injective resolution of A•.

Note that if A is a single sheaf on X and we form A•, then Hp(X;A•) = Hp(X;A) = ordinarysheaf cohomology. In particular, Hp(X;R•

X) = Hp(X;R).

Note also that if A• and B• are quasi-isomorphic, then H∗(X;A•) ∼= H∗(X;B•).

If Y is a subspace of X and A• ∈ Dbc(X), then one usually writes H∗(Y ;A•) in place of

H∗(Y ;A•|Y ).

The usual Mayer-Vietoris sequence is valid for hypercohomology; that is, if U and V form anopen cover of X and A• ∈ Db

c(X), then there is an exact sequence

· · · → Hi(X;A•)→ Hi(U ;A•)⊕Hi(V ;A•)→ Hi(U ∩ V ;A•)→ Hi+1(X;A•)→ . . . .

Of course, hypercohomology is not a homotopy invariant. However, it is true that: if S is a realanalytic Whitney stratification of X, A• ∈ Db

S(X), and r : X → [0, 1) is a proper real analytic

APPENDIX B 187

map such that, for all S ∈ S, r|S has no critical values in (0, 1), then the inclusion r−1(0) → Xinduces an isomorphism

Hi(X;A•) ∼= Hi(r−1(0);A•).

If R is a principal ideal domain, we may talk about the rank of a finitely-generated R-module. Inthis case, if A• ∈ Db

c(X), then the Euler characteristic, χ, of the stalk cohomology is defined as thealternating sum of the ranks of the cohomology modules, i.e., χ(A•)x =

∑(−1)i rank Hi(A•)x.

If the hypercohomology modules are finitely-generated – for instance, if A• ∈ Dbc(X) and X is

compact – then the Euler characteristic χ(H∗(X;A•)

)is defined analogously.

If Hi(A•) = 0 for all but, possibly, one value of i - say, i = p, then A• is quasi-isomorphic tothe complex that has Hp(A•) in degree p and zero elsewhere. We reserve the term local systemfor a locally constant single sheaf or a complex which is concentrated in degree zero and is locallyconstant. If M is the stalk of a local system L on a path-connected space X, then L is determinedup to isomorphism by a monodromy representation π1(X,x)→ Aut(M), where x is a fixed pointin X.

For any A• ∈ Dbc(X), there is an E2 cohomological spectral sequence:

Ep,q2 = Hp(X;Hq(A•))⇒ H

p+q

(X;A•).

If A• ∈ Dbc(X), x ∈ X, and (X,x) is locally embedded in some Cn, then for all ε > 0 small, the

restriction map Hq(Bε(x);A•) → Hq(A•)x is an isomorphism

(here,

Bε(x) =

z ∈ Cn

∣∣ |z − x| <

ε )

. If, in addition, R is a principal ideal domain, the Euler characteristic χ(H∗(

Bε(x)− x;A•)

)is defined and

χ(H∗(

Bε(x)− x;A•)

)= χ

(H∗(Sε′(x);A•)

)= 0,

where 0 < ε′ < ε and Sε′(x) denotes the sphere of radius ε′ centered at x.

We now wish to say a little about the morphisms in the derived category Dbc(X). The derived

category is obtained by formally inverting the quasi-isomorphisms so that they become isomor-phisms in Db

c(X). Thus, A• and B• are isomorphic in Dbc(X) provided that there exists a complex

C• and quasi-isomorphisms A• ← C• → B•; A• and B• are then said to be incarnations of thesame isomorphism class in Db

c(X).More generally, a morphism in Db

c(X) from A• to B• is an equivalence class of diagrams ofmaps of complexes A• ← C• → B• where A• ← C• is a quasi-isomorphism. Two such diagrams,

A• f1←− C•1

g1−→ B•, A• f2←− C•2

g2−→ B•

are equivalent provided that there exists a third such diagram A• f←− C• g−→ B• and a diagramC•

1

f1 ↑ g1

A• f←−− C• g−−→ B•

f2 ↓ g2

C•2

188 DAVID B. MASSEY

which commutes up to (chain) homotopy.Composition of morphisms in Db

c(X) is not difficult to describe. If we have two representativesof morphisms, from A• to B• and from B• to D•, respectively,

A• f1←− C•1

g1−→ B•, B• f2←− C•2

g2−→ D•

then we consider the pull-back C•1×B• C•

2 (in the category of chain complexes) and the projectionsπ1 and π2 to C•

1 and C•2, respectively. As f2 is a quasi-isomorphism, so is π1, and the composed

morphism from A• to D• is represented by A• f1π1←−−− C•1 ×B• C•

2g2π2−−−→ D•.

If we restrict ourselves to considering only injective complexes, by associating to any complexits canonical injective resolution, then morphisms in the derived category become easy to describe– they are chain-homotopy classes of maps between the injective complexes.

The moral is: in Dbc(X), we essentially only care about complexes up to quasi-isomorphism.

Note, however, that the objects of Dbc(X) are not equivalence classes – this is one reason why it

is important that to each complex we can associate a canonical injective resolution. It allows usto talk about certain functors in Db

c(X) being naturally isomorphic. When we write A• ∼= B•,we mean in Db

c(X). As we shall discuss later, Dbc(X) is an additive category, but is not Abelian.

Warning: While morphisms of complexes which induce isomorphisms on cohomology sheavesbecome isomorphisms in the derived category, there are morphisms of complexes which induce thezero map on cohomology sheaves but are not zero in the derived category. The easiest example ofsuch a morphism is given by the following.

Let X be a space consisting of two complex lines L1 and L2 which intersect in a single pointp. For i = 1, 2, let CLi

denote the C-constant sheaf on Li extended by zero to all of X. There isa canonical map, α, from the sheaf CX to the direct sum of sheaves CL1 ⊕ CL2 , which on L1 − pis id⊕0, on L2 − p is 0 ⊕ id, and is the diagonal map on the stalk at p. Consider the complex,A•, which has CX in degree 0, CL1 ⊕ CL2 in degree 1, zeroes elsewhere, and the coboundary mapfrom degree 0 to degree 1 is α. This complex has cohomology only in degree 1. Nonetheless, themorphism of complexes from A• to C•

X which is the identity in degree 0 and is zero elsewheredetermines a non-zero morphism in the derived category.

We now wish to describe derived functors; for this, we will need the derived category of anarbitrary Abelian category C.

Let C be an Abelian category. Then, the derived category of bounded complexes in C is thecategory whose objects consist of bounded differential complexes of objects of C, and where the mor-phisms are obtained exactly as in the case of Db

c(X) – namely, by inverting the quasi-isomorphismsas we did above. Naturally, we denote this derived category by Db(C).

We need some more general notions before we come back to complexes of sheaves. If C isan Abelian category, then we let Kb(C) denote the category whose objects are again boundeddifferential complexes of objects of C, but where the morphisms are chain-homotopy classes of mapsof differential complexes. A triangle in Kb(C) is a sequence of morphisms A• → B• → C• → A•[1],which is usually written in the more“triangular” form

A• −→ B•

[1] C•

APPENDIX B 189

A triangle in Kb(C) is called distinguished if it is isomorphic in Kb(C) to a diagram of maps ofcomplexes

A• φ−→ B•

[1] M•

where M• is the algebraic mapping cone of φ and B• → M• → A•[1] are the canonical maps.(Recall that the algebraic mapping cone is defined by

Mk := Ak+1 ⊕ Bk −→ Ak+2 ⊕ Bk+1 =: Mk+1

(a, b) 7−→ (−∂a, φa+ δb)

where ∂ and δ are the differentials of A• and B• respectively.) Note that if φ = 0, then we have anequality M• = A•[1]⊕B• (recall that the shifted complex A•[1] has as its differential the negated,shifted differential of A•).

Now we can define derived functors. Let C denote the Abelian category of sheaves of R-moduleson an analytic space X, and let C′ be another Abelian category. Suppose that F is an additive, co-variant functor from Kb(C) to Kb(C′) such that F [1] = [1]F and such that F takes distinguishedtriangles to distinguished triangles (such an F is called a functor of triangulated categories). Sup-pose also that, for all complexes of injective sheaves I• ∈ Kb(C) which are quasi-isomorphic to 0,F (I•) is also quasi-isomorphic to 0.

Then, F induces a morphism RF – the right derived functor of F – from Db(X) to Db(C′);for any A• ∈ Db(X), let A• → I• denote the canonical injective resolution of A•, and defineRF (A•) := F (I•). The action of RF on the morphisms is the obvious associated one.

A morphism F : Kb(C) → Kb(C′) as described above is frequently obtained by starting with aleft-exact functor T : C → C′ and then extending T in a term-wise fashion to be a functor fromKb(C) to Kb(C′). In this case, we naturally write RT for the derived functor.

This is the process which is applied to:

Γ(X; ·) (global sections);

Γc(X; ·) (global sections with compact support);

f∗ (direct image);

f! (direct image with proper supports); and

f∗ (pull-back or inverse image),

where f : X → Y is a continuous map (actually, in these notes, we would need an analyticallyconstructible map; e.g., an analytic map).

If the functor T is an exact functor from sheaves to sheaves, then RT (A•) ∼= T (A•); in thiscase, we normally suppress the R. Hence, if f : X → Y , A• ∈ Db(X), and B• ∈ Db(Y ), we write:

f∗B•;

f!A•, if f is the inclusion of a subspace and, hence, f! is extension by zero;

190 DAVID B. MASSEY

f∗A•, if f is the inclusion of a closed subspace.

Note that hypercohomology is just the cohomology of the derived global section functor, i.e.,H∗(X; ·) = H∗ RΓ(X; ·). The cohomology of the derived functor of global sections with compactsupport is the compactly supported hypercohomology and is denoted H∗

c(X;A•).

If f : X → Y is the inclusion of a subset and B• ∈ Db(Y ), then the restriction of B• to X isdefined to be f∗(B•), and is usually denoted by B•

|X .

If f : X → Y is continuous and A• ∈ Dbc(X), there is a canonical map

Rf!A• → Rf∗A•.

For f : X → Y continuous, there are canonical isomorphisms

RΓ(X;A•) ∼= RΓ(Y ;Rf∗A•) and RΓc(X;A•) ∼= RΓc(Y ;Rf!A•)

which lead to canonical isomorphisms

H∗(X;A•) ∼= H∗(Y ;Rf∗A•) and H∗c(X;A•) ∼= H∗

c(Y ;Rf!A•)

for all A• in Dbc(X).

If f : X → Y is continuous, A• ∈ Dbc(X), and B• ∈ Db

c(Y ) , there are natural maps inducedby restriction of sections

B• → Rf∗f∗B• and f∗Rf∗A• → A•.

If Sα is a stratification of X, A• ∈ DbSα×Ck

(X ×Ck), and π : X ×Ck → X is the projection,then restriction of sections induces a quasi-isomorphism π∗Rπ∗A• → A•.

It follows easily that if j : X → X×Ck is the zero section, then π∗j∗A• ∼= A•. This says exactlywhat one expects: the complex A• has a product structure in the Ck directions.

An important consequence of this is the following: let S = Sα be a Whitney stratificationof X and let A• ∈ Db

S(X). Let x ∈ Sα ⊆ X. As Sα is a Whitney stratum, X has a product

structure along Sα near x. By the above, A• itself also has a product structure along Sα. Hence,by taking a normal slice, many problems concerning the complex A• can be reduced to consideringa zero-dimensional stratum.

Let A•,B• ∈ Dbc(X). Define A•⊗B• to be the single complex which is associated to the double

complex Ap ⊗Bq. The left derived functor A• L⊗ ∗ is defined by

A• L⊗B• = A• ⊗ J•,

where J• is a flat resolution of B•, i.e., the stalks of J• are flat R-modules and there exists aquasi-isomorphism J• → B•.

For all A•,B• ∈ Dbc(X), there is an isomorphism A• L

⊗B• ∼= B• L⊗A•.

APPENDIX B 191

For any map f : X → Y and any A•,B• ∈ Dbc(Y ),

f∗(A• L⊗B•) ∼= f∗A• L

⊗ f∗B•.

Fix a complex B• on X. There are two covariant functors which we wish to consider: thefunctor Hom•(B•, ∗) from the category of complexes of sheaves to complexes of sheaves and thefunctor Hom•(B•, ∗) from the category of complexes of sheaves to the category of complexes ofR-modules. These functors are given by

(Hom•(B•,A•))n =∏p∈Z

Hom(Bp,An+p)

and(Hom•(B•,A•))n =

∏p∈Z

Hom(Bp,An+p)

with differential given by[∂nf ]p = ∂n+pfp + (−1)n+1fp+1∂p

(there is an indexing error in [Iv, 12.4]). The associated derived functors are RHom•(B•, ∗) andRHom•(B•, ∗), respectively.

If P• → B• is a projective resolution of B•, then, in Dbc(X), RHom•(B•,A•) is isomorphic to

Hom•(P•,A•). For all k, RHom•(B•,A•[k]) = RHom•(B•,A•)[k].The functor RHom•(B•, ∗) is naturally isomorphic to the derived global sections functor applied

to RHom•(B•, ∗), i.e., for any A• ∈ Dbc(X),

RHom•(B•,A•) ∼= RΓ (X;RHom•(B•,A•)) .

H0(RHom•(B•,A•)) is naturally isomorphic as an R-module to the derived category homomor-phisms from B• to A•, i.e.,

H0(RHom•(B•,A•)) ∼= HomDb

c(X)(B•,A•).

If B• and A• have locally constant cohomology sheaves onX then, for all x ∈ X, RHom•(B•,A•)x

is naturally isomorphic to RHom•(B•x,A

•x).

For all A•,B•,C• ∈ Dbc(X), there is a natural isomorphism

RHom•(A• L⊗B•,C•) ∼= RHom•(A•, RHom•(B•,C•)).

Moreover, if C• has locally constant cohomology sheaves, then there is an isomorphism

RHom•(A•,B• L⊗C•) ∼= RHom•(A•,B•)

L⊗C•.

For all j, we define Extj(B•,A•) := Hj(RHom•(B•,A•)) and define

Extj(B•,A•) := Hj(RHom•(B•,A•)).

192 DAVID B. MASSEY

It is immediate that we have isomorphisms of R-modules

Extj(B•,A•) = H0(RHom•(B•,A•[j])) ∼= HomDb

c(X)(B•,A•[j]).

If X = point, B• ∈ Dbc(X), and the base ring is a PID, then B• ∼=

⊕k Hk(B•)[−k] in Db

c(X);if we also have A• ∈ Db

c(X), then

Hi(A• L⊗B•) ∼=

( ⊕p+q=i

Hp(A•)⊗Hq(B•))⊕

( ⊕r+s=i+1

Tor(Hr(A•),Hs(B•))).

If, in addition, the cohomology modules of A• are projective (hence, free), then

HomDb

c(X)(A•,B•) ∼=

⊕k

Hom(Hk(A•),Hk(B•)

).

If we have a map f : X → Y , then the functors f∗ and Rf∗ are adjoints of each other in thederived category. In fact, for all A• on X and B• on Y , there is a canonical isomorphism in Db

c(Y )

RHom•(B•, Rf∗A•) ∼= Rf∗RHom•(f∗B•,A•)

and so

HomDb

c(Y )(B•, Rf∗A•) ∼= H0 (RHom•(B•, Rf∗A•)) ∼= H0 (Y ;RHom•(B•, Rf∗A•))

∼= H0 (X;RHom•(f∗B•,A•)) ∼= H0 (RHom•(f∗B•,A•)) ∼= HomDb

c(X)(f∗B•,A•).

We wish now to describe an analogous adjoint for Rf!

Let I• be a complex of injective sheaves on Y . Then, f !(I•) is defined to be the sheaf associatedto the presheaf given by

Γ(U ; f !I•) = Hom•(f!K•U, I•),

for any open U ⊆ X, where K•U

denotes the canonical injective resolution of the constant sheafR•

U. For any A• ∈ Db

c(X), define f !A• to be f !I•, where I• is the canonical injective resolution ofA•.

Now that we have this definition, we may state:

(Verdier Duality) If f : X → Y,A• ∈ Dbc(X), and B• ∈ Db

c(Y ), then there is a canonicalisomorphism in Db

c(Y ):

Rf∗RHom•(A•, f !B•) ∼= RHom•(Rf!A•,B•)

and soHom

Dbc(X)

(A•, f !B•) ∼= HomDb

c(Y )(Rf!A•,B•).

If B• and C• are in Dbc(Y ), then we have an isomorphism

f !RHom•(B•,C•) ∼= RHom•(f∗B•, f !C•).

APPENDIX B 193

Let f : X → point. Then, the dualizing complex, D•X

, is f ! applied to the constant sheaf,i.e., D•

X= f !R•

pt. For any complex A• ∈ Db

c(X), the Verdier dual (or, simply, the dual) of A•

is RHom•(A•,D•X

) and is denoted by DXA• ( or just DA•). There is a canonical isomorphism

between D•X

and the dual of the constant sheaf on X, i.e., D•X∼= DR•

X.

Let A• ∈ Dbc(X). The dual of A•, DA•, is well-defined up to quasi-isomorphism by:

for any open U ⊆ X, there is a natural split exact sequence:

0→ Ext(Hq+1c (U ;A•), R)→ H−q(U ;DA•)→ Hom(Hq

c(U ;A•), R)→ 0.

In particular, if R is a field, then H−q(U ;DA•) ∼= Hqc(U ;A•), and so

Hq(DA•)x∼= Hq(

Bε(x);DA•) ∼= H−q

c (Bε(x);A•).

If, in addition, X is compact, H−q(X;DA•) ∼= Hq(X;A•).

Dualizing is a local operation, i.e., if i : U → X is the inclusion of an open subset and A• ∈Db

c(X), then i∗DA• ∼= Di∗A•.If L is a local system on a connected real m-manifold, N , then (DL•)[−m] is quasi-isomorphic

to a local system; if, in addition, N is smooth and oriented, and L is actually locally free withstalks Ra and monodromy representation η : π1(N,p) → Aut(Ra), then DL•[−m] is quasi-isomorphic to a local system with stalks Ra and monodromy tη : π1(N,p)→ Aut(Ra), wheretη(α) = transpose of η(α).

If A• ∈ Dbc(X), then D(A•[n]) = (DA•)[−n].

If π : X × Cn → X is projection, then D(π∗A•)[−n] ∼= π∗(DA•)[n].

The dualizing complex, DX

, is quasi-isomorphic to the complex of sheaves of singular chains onX which is associated to the complex of presheaves, C•, given by Γ(U ;C−p) := Cp(X,X − U ;R).

The cohomology sheaves of D•X

are non-zero in negative degrees only, with stalks H−p(D•X

)x =Hp(X,X − x;R).

If X is a smooth, oriented, real m-manifold, then D•X

[−m] is quasi-isomorphic to R•X

; hence,for any A• ∈ Db

c(X),

DA• ∼= RHom•(A•,R•X

[m]) =(RHom•(A•,R•

X))[m].

D•V×W

is naturally isomorphic to π∗1D•V

L⊗ π∗2D•

W, where π1 and π2 are the projections onto V

and W , respectively.

H∗(X; D•X

) ∼= homology with closed supports = Borel-Moore homology.

If X is a real, smooth, oriented m-manifold and R = R, then D•X

[−m] is naturally isomorphicto the complex of real differential forms on X.

194 DAVID B. MASSEY

D•X

is constructible with respect to any Whitney stratification of X. It follows that if S is aWhitney stratification of X, then A• ∈ Db

S(X) if and only if DA• ∈ Db

S(X).

The functor D from Dbc(X) to Db

c(X) is contravariant, and DD is naturally isomorphic to theidentity. For all A•,B• ∈ Db

c(X), we have isomorphisms

RHom•(A•,B•) ∼= RHom•(DB•,DA•) ∼= D(DB• L

⊗A•).

If f : X → Y is continuous, then we have natural isomorphisms

Rf! ∼= DRf∗D and f ! ∼= Df∗D.

If Y ⊆ X and f : X − Y → X is the inclusion, we define

Hk(X,Y ;A•) := Hk(X; f!f !A•).

Excision has the following form: if Y ⊆ U ⊆ X, where U is open in X and Y is closed in X,then

Hk(X,X − Y ;A•) ∼= Hk(U,U − Y ;A•).

If x ∈ X and A• ∈ Dbc(X), then for all ε > 0 sufficiently small,

Hqc(

Bε(x);A•) ∼= Hq(

Bε(x),

Bε(x)− x;A•) ∼= Hq(X,X − x;A•)

and so, if R is a field,

H−q(DA•)x∼= Hq(

Bε(x),

Bε(x)− x;A•) ∼= Hq(X,X − x;A•).

If f : X → Y and g : Y → Z, then there are natural isomorphisms

R(g f)∗ ∼= Rg∗ Rf∗ R(g f)! ∼= Rg! Rf!and

(g f)∗ ∼= f∗ g∗ (g f)! ∼= f ! g!.

Suppose that f : Y → X is inclusion of a subset. Then, if Y is open, f ! = f∗. If Y is closed,then Rf! = f! = f∗ = Rf∗.

If f : Y → X is the inclusion of one complex manifold into another and B• ∈ Dbc(X) has locally

constant cohomology on X, then f !B• has locally constant cohomology on Y and

f !B• ∼= f∗B•[−2 codimXY ].

APPENDIX B 195

(Here, we mean the complex codimension. There is an error here in [G-M3]; they have the negationof the correct shift.)

If π : X × Cn → X is projection and A• ∈ Dbc(X), then (π!A•)[−2n] ∼= π∗A•.

If f : X → Y is continuous, A• ∈ Dbc(X), and B• ∈ Db

c(Y ) , then dual to the canonical maps

B• → Rf∗f∗B• and f∗Rf∗A• → A•

are the canonical mapsRf!f

!B• → B• and A• → f !Rf!A•.

IfZ

f−−−−→ W

π

y yπ

Xf−−−−→ S

is a pull-back diagram (fibre square, Cartesian diagram), then for all A• ∈ Dbc(X), Rf!π∗A• ∼=

π∗Rf!A• (there is an error in [G-M3]; they have lower ∗’s, not lower !’s, but see below for whenthese agree) and, dually, Rf∗π!A• ∼= π!Rf∗A•. In particular, if f is proper (and, hence, f isproper) or π is the inclusion of an open subset (and, hence, so is π, up to homeomorphism), thenRf∗π

∗A• ∼= π∗Rf∗A•; this is also true if W = S × Cn and π : W → S is projection (and, hence,up to homeomorphism, π is projection from X × Cn to X).

If we have A• ∈ Dbc(X) and B• ∈ Db

c(W ), then we let A•L

SB• := π∗A• L

⊗ f∗B•, assumingthat the maps π and f are clear. If S is a point, so that Z ∼= X ×W , then we omit the S in the

notation and write simply A•L B•.

There is a Kunneth formula, which we now state in its most general form, in terms of maps overa base space S. Suppose that we have two maps f1 : X1 → Y1 and f2 : X2 → Y2 over S, i.e., thereare commutative diagrams

X1f1−→ Y1 X2

f2−→ Y2

r1 t1 and r2 t2S S .

Then, there is an induced map f = f1 ×Sf2 : X1 ×S

X2 → Y1 ×SY2. If A• ∈ Db

c(X1) andB• ∈ Db

c(X2), there is the Kunneth isomorphism

Rf!(A• L

S

B•) ∼= Rf1!A• L

SRf2!B

•.

Using the above notation, if S is a point and F• ∈ Dbc(Y1) and G• ∈ Db

c(Y2), then there is anatural isomorphism (the adjoint Kunneth isomorphism)

f !(F• L

G•) ∼= f1!F• L

f2!G•.

196 DAVID B. MASSEY

If we let q1 and q2 denote the projections from Y1 × Y2 onto Y1 and Y2, respectively, then theadjoint Kunneth formula can be proved by using the following natural isomorphism twice

DF• L G• ∼= RHom•(q∗1F

•, q!2G•).

Let Z be a locally closed subset of an analytic space X. There are two derived functors,associated to Z, that we wish to describe: the derived functors of restricting-extending to Z, andof taking the sections supported on Z. Let i denote the inclusion of Z into X.

If A is a (single) sheaf on X, then the restriction-extension of A to Z, (A)Z , is given by i!i∗(A).Thus, up to isomorphism, (A)Z is characterized by ((A)Z)|Z ∼= A|Z and ((A)Z)|X−Z

= 0. Thisfunctor is exact, and so we also denote the derived functor by ()Z .

Now, we want to define the sheaf of sections of A supported by Z, ΓZ(A). If U is an open

subset of X which contains Z, then we define

ΓZ(U ;A) := ker

Γ(U ;A)→ Γ(U − Z;A)

.

Up to isomorphism, ΓZ(U ;A) is independent of the open set U (this uses that A is a sheaf, not

just a presheaf). The sheaf ΓZ(A) is defined by, for all open U ⊆ X, Γ(U ; Γ

Z(A)) := ΓU∩Z

(U ;A).One easily sees that suppΓ

Z(A) ⊆ Z. It is also easy to see that, if Z is open, then Γ

Z(A) =

i∗i∗(A).

The functor ΓZ() is left exact; of course, we denote the right derived functor by RΓ

Z().

There is a canonical isomorphism i! ∼= i∗ RΓZ. It follows that, if Z is closed, then RΓ

Z∼= i!i

!.In addition, if Z is open, then RΓ

Z∼= Ri∗i

∗.

Avoiding Injective Resolutions:

To calculate right derived functors from the definition, one must use injective resolutions. How-ever, this is inconvenient in many proofs if some functor involved in the proof does not take injectivecomplexes to injective complexes. There are (at least) four “devices” which come to our aid, andenable one to prove many of the isomorphisms described earlier; these devices are fine resolu-tions, flabby resolutions, c-soft resolutions, and injective subcategories with respect toa functor.

If T is a left-exact functor on the category of sheaves on X, then the right derived functor RT isdefined by applying T term-wise to the sheaves in a canonical injective resolution. The importanceof saying that a certain subcategory of the category of sheaves on X is injective with respect to Tis that one may take a resolution in which the individual sheaves are in the given subcategory, thenapply T term-wise, and end up with a complex which is canonically isomorphic to that producedby RT .

Recall that a single sheaf A on X is:

fine, if partitions of unity of A subordinate to any given locally finite open cover of X exist;

flabby, if for every open subset U ⊆ X, the restriction homomorphism Γ(X;A) → Γ(U ;A) is asurjection;

c-soft, if for every compact subset K ⊆ X, the restriction homomorphism Γ(X;A)→ Γ(K;A) is asurjection;

APPENDIX B 197

Injective sheaves are flabby, and flabby sheaves are c-soft. In addition, fine sheaves are c-soft.

The subcategory of c-soft sheaves is injective with respect to the functors Γ(X; ∗), Γc(X; ∗), andf!. The subcategory of flabby sheaves is injective with respect to the functor f∗.

If A• is a bounded complex of sheaves, then a bounded c-soft resolution of A• is given byA• → A• ⊗ S•, where S• is a c-soft, bounded above, resolution of the base ring (which alwaysexists in our context).

Triangles:

Dbc(X) is an additive category, but is not an Abelian category. In place of short exact sequences,

one has distinguished triangles, just as we did in Kb(C). A triangle of morphisms in Dbc(X)

A• −→ B•

[1] C•

(the [1] indicates a morphism shifted by one, i.e., a morphism C• → A•[1]) is called distinguishedif it is isomorphic in Db

c(X) to a diagram of sheaf maps

A• φ−→ B•

[1] M•

where M• is the algebraic mapping cone of φ and B• →M• → A•[1] are the canonical maps.

The “in-line” notation for a triangle is A• → B• → C• → A•[1] or A• → B• → C• [1]−→.

Any short exact sequence of complexes becomes a distinguished triangle in Dbc(X). Any edge of

a distinguished triangle determines the triangle up to (non-canonical) isomorphism in Dbc(X); more

specifically, we can “turn” the distinguished triangle: A• α−→ B• β−→ C• γ−→ A•[1] is a distinguishedtriangle if and only if

B• β−→ C• γ−→ A•[1]−α[1]−−−→ B•[1]

is a distinguished triangle.Given two distinguished triangles and maps u and v which make the left-hand square of the

following diagram commute

A• → B• → C• → A•[1]

↓ u ↓ v ↓ u[1]

A• → B• → C• → A•[1],

there exists a (not necessarily unique) w : C• → C• such that

A• → B• → C• → A•[1]

↓ u ↓ v ↓ w ↓ u[1]

A• → B• → C• → A•[1]

198 DAVID B. MASSEY

also commutes. We say that the original commutative square embeds in a morphism of distin-guished triangles.

We will now give the octahedral lemma, which allows one to realize an isomorphism betweenmapping cones of two composed maps. Suppose that we have two distinguished triangles

A• f−→ B• g−→ C• h−→ A•[1]

andB• β−→ E• γ−→ F• δ−→ B•[1].

Then, there exists a complex M• and two distinguished triangles

A• βf−−→ E• τ−→M• ω−→ A•[1]

andC• σ−→M• ν−→ F• g[1]δ−−−→ C•[1].

such that the following diagram commutes

A• f−−−→ B• g−−−→ C• h−−−→ A•[1]

id ↓ β ↓ σ ↓ id ↓

A• βf−−−→ E• τ−−−→M• ω−−−→ A•[1]

f ↓ id ↓ ν ↓ f [1] ↓

B• β−−−−→ E• γ−−−→ F• δ−−−→ B•[1]

g ↓ τ ↓ id ↓ g[1] ↓

C• σ−−−→M• ν−−−→ F• g[1]δ−−−→ C•[1].

It is somewhat difficult to draw this in its octahedral form (and worse to type it); moreover, itis no easier to read the relations from the octahedron. However, the interested reader can give ita try: the octahedron is formed by gluing together two pyramids along their square bases. Onepyramid has B• at its top vertex, with A•, C•, E•, and F• at the vertices of its base, and hasthe original two distinguished triangles as opposite faces. The other pyramid has M• at its topvertex, with A•, C•, E•, and F• at the vertices of its base, and has the other two distinguishedtriangles (whose existence is asserted in the lemma) as opposite faces. The two pyramids are joinedtogether by matching the vertices of the two bases, forming an octahedron in which the faces arealternately distinguished and commuting.

A distinguished triangle determines long exact sequences on cohomology and hypercohomology:

· · · → Hp(A•)→ Hp(B•)→ Hp(C•)→ Hp+1(A•)→ · · ·

· · · → Hp(X;A•)→ Hp(X;B•)→ Hp(X;C•)→ Hp+1(X;A•)→ · · · .

If f : X → Y and F• ∈ Dbc(X), then the functors Rf∗, Rf!, f∗, f !, and F• L

⊗ ∗ all takedistinguished triangles to distinguished triangles (with all arrows in the same direction and theshift in the same place).

APPENDIX B 199

As for RHom•, if F• ∈ Dbc(X) and A• → B• → C• → A•[1] is a distinguished triangle in

Dbc(X), then we have distinguished triangles

RHom•(F•,A•) −→ RHom•(F•,B•) RHom•(A•,F•)←− RHom•(B•,F•)

[1] and [1] RHom•(F•,C•) RHom•(C•,F•).

By applying the right-hand triangle above to the special case where F• = D•X , we find that

the dualizing functor D also takes distinguished triangles to distinguished triangles, but with areversal of arrows, i.e., if we have a distinguished triangle A• → B• → C• → A•[1] in Db

c(X),then, by dualizing, we have distinguished triangles

DA• ←− DB• DC• −→ DB•

[1] or [1] DC• DA•.

There are (at least) six distinguished triangles associated to the functors ()Z and RΓZ. Let F•

be in Dbc(X), U1 and U2 be open subsets of X, Z1 and Z2 be closed subsets of X, Z be locally

closed in X, and Z ′ be closed in Z. Then, we have the following distinguished triangles:

RΓU1∪U2(F•)→ RΓU1

(F•)⊕RΓU2(F•)→ RΓU1∩U2

(F•)[1]−→

RΓZ1∩Z2

(F•)→ RΓZ1

(F•)⊕RΓZ2

(F•)→ RΓZ1∪Z2

(F•)[1]−→

(F•)U1∩U2→ (F•)U1

⊕ (F•)U2→ (F•)U1∪U2

[1]−→

(F•)Z1∪Z2

→ (F•)Z1⊕ (F•)

Z2→ (F•)

Z1∩Z2

[1]−→

RΓZ′ (F

•)→ RΓZ(F•)→ RΓ

Z−Z′ (F•)

[1]−→

(F•)Z−Z′ → (F•)

Z→ (F•)

Z′[1]−→ .

If j : Y → X is the inclusion of a closed subspace and i : U → X the inclusion of the opencomplement, then for all A• ∈ Db

c(X), the last two triangles above give us distinguished triangles

Ri!i!A• −→ A• Rj!j

!A• −→ A•

[1] and [1] Rj∗j

∗A• Ri∗i∗A•,

200 DAVID B. MASSEY

where the second triangle can be obtained from the first by dualizing. (Note that Ri! = i!,Rj∗ = j∗ = j! = Rj!, and i! = i∗.) The associated long exact sequences on hypercohomology arethose for the pairs H∗(X,Y ;A•) and H∗(X,U ;A•), respectively.

By applying these two triangles to Ri∗i∗A• and Ri!i

!A•, respectively, we obtain a naturalisomorphism

Rj!j!Ri!i

!A•[1] ∼= Rj∗j∗Ri∗i

∗A•.

As in our earlier discussion of the octahedral lemma, all of the morphisms of the last twoparagraphs fit into the fundamental octahedron of the pair (X,Y ). The four distinguished trianglesmaking up the fundamental octahedron are the top pair

Ri!i!A• → A• → Rj∗j

∗A• → Ri!i!A•[1]

andA• → Ri∗i

∗A• → Rj!j!A•[1]→ A•[1]

and the bottom pairRi!i

!A• → Ri∗i∗A• →M• → Ri!i

!A•[1]

andRj∗j

∗A• →M• → Rj!j!A•[1]→ Rj∗j

∗A•[1],

where M• ∼= Rj!j!Ri!i

!A•[1] ∼= Rj∗j∗Ri∗i

∗A•.

§2. Perverse Sheaves

Suppose that P• ∈ Dbc(X). There are two non-equivalent definitions of what it means for P• to

be perverse. The first one (which is actually the definition of perverse) is a purely local definitionand, when the base ring is a field, is symmetric with respect to dualizing. This definition isdefinitely the more elegant of the two, but it gives cohomology groups only in negative dimensions;this seems non-intuitive from the topologist’s point of view.

The second definition of perverse - which differs from the first only by a shift - has the advantagethat the cohomology groups appear in non-negative dimensions only. Also, the constant sheaf ona local complete intersection is such a sheaf and, with this definition of perverse, the nearby andvanishing cycles (see §3) of a perverse sheaf are again perverse. Finally, if one wants intersectioncohomology with its usual indexing (that is, the indexing that gives cohomology in non-negativedimensions) to be a perverse sheaf, then one must use this second definition of perverse.

Despite these advantages of this second definition of perverse, the fact that it does not localizewell on non-pure-dimensional spaces complicates general statements in almost every case. State-ments tend to be much cleaner using the first definition. Hence, below, we use the term perversesheaf for this first definition, and use positively perverse sheaf for the second definition.

We shall give most statements in terms of perverse sheaves only; the reader may do the necessaryshifts to obtain the positively perverse statements. The exceptions to this are those few statementswhich seem cleaner using positively perverse.

Definition: Let X be a complex analytic space, and for each x ∈ X, let jx : x → X denote theinclusion.

If F• ∈ Dbc(X), then the support of Hi(F•) is the closure in X of

x ∈ X| Hi(F•)x 6= 0 = x ∈ X| Hi(j∗xF•) 6= 0;

APPENDIX B 201

we denote this by suppi F•.The i-th cosupport of F• is the closure in X of

x ∈ X| Hi(j!xF•) 6= 0 = x ∈ X| Hi(

Bε(x),

Bε(x)− x; F•) 6= 0;

we denote this by cosuppi F•.

If the base ring, R, is a field, then cosuppi F• = supp-iDF•.

Definition: Let X be a complex analytic space (not necessarily pure dimensional). Then, P• ∈Db

c(X) is perverse provided that for all i:

(support) dim(supp-i P•) 6 i;

(cosupport) dim(cosuppi P•) 6 i,

where we set the dimension of the empty set to be −∞.

This definition is equivalent to: let Sα be any Whitney stratification of X with respect towhich P• is constructible, and let sα : Sα → X denote the inclusion. Then,

(support) Hk(s∗αP•) = 0 for k > −dimCSα;

(cosupport) Hk(s!αP•) = 0 for k < −dimCSα.

(There is a missing minus sign in [G-M2, 6.A.5].)

If X is an n-dimensional space, then P• is positively perverse if and only if P•[n] is perverse.

From the definition, it is clear that being perverse is a local property.

If the base ring R is, in fact, a field, then the support and cosupport conditions can be writtenin the following form, which is symmetric with respect to dualizing:

(support) dim(supp-i P•) 6 i;

(cosupport) dim(supp-iDP•) 6 i.

Suppose that P• is perverse on X, (X,x) is locally embedded in Cn, S is a stratum of a Whitneystratification with respect to which P• is constructible, and x ∈ S. Let M be a normal slice ofX at x; that is, let M be a smooth submanifold of Cn of dimension n− dim S which transverselyintersects S at x. Then, for some open neighborhood U of x in X, P•

|X∩M∩U[−dim S] is perverse

on X ∩M ∩ U .

Let P• ∈ Dbc(X); one can use this normal slicing proposition to prove:

if P• is perverse, then Hi(P•) = 0 for all i < −dim X;and so,

if P• is positively perverse, then Hi(P•) = 0 for all i < 0.

202 DAVID B. MASSEY

A converse to the normal slicing proposition is:

if π : X×Cs → X is projection and P• is positively perverse on X, then π∗P• is also positivelyperverse. Thus, if P• is perverse on X, then π∗P•[s] is perverse.

Suppose P• ∈ Dbc(X

n). Let Σ = supp H∗(P•). Then, P• is perverse on X if and only if P•|Σ

is perverse on Σ. Hence, P• is positively perverse on X if and only if P•|Σ [codimXΣ] is positively

perverse on Σ.Another way of saying this is: if j : Σ → X is the inclusion of a closed subspace, then Q• is

perverse on Σ if and only if j!Q• is perverse on X; hence, Q• is positively perverse on Σ if andonly if j!Q•[−codimXΣ] is positively perverse on X.

It follows that if P• is perverse, then Hi(P•) = 0 unless −dim Σ 6 i 6 0; and so, if P• ispositively perverse, then Hi(P•) = 0 unless codimXΣ 6 i 6 n. In particular, on an n-dimensionalspace, a positively perverse sheaf which is supported only at isolated points has cohomology onlyin dimension n (i.e., the middle dimension).

The constant sheaf R•X

is positively perverse provided that X is a pure-dimensional local com-plete intersection. More generally, if X is a pure-dimensional local complete intersection, and Mis a locally free sheaf of R-modules, then M• is a positively perverse sheaf on X.

The other basic example of a perverse sheaf that we wish to give is that of intersection coho-mology with local coefficients (with the perverse indexing, i.e., cohomology in degrees less than orequal to zero). Note that the definition below is shifted by −dimC X from the definition in [G-M3], and yields a perverse sheaf which has possibly non-zero cohomology only in degrees between−dimC X and −1, inclusive.

Let X be a n-dimensional complex analytic set, let X(n) = X1 ∪ . . . Xk be the union of then-dimensional components of X, and let L be a local system on a smooth, open dense subset,X, of X(n). Then, in Db

c(X), there is an object, IC•X

(L), called the intersection cohomology withcoefficients in L which is uniquely determined up to quasi-isomorphism by:

0) IC•X

(L)|X−X(n)

= 0;

1) IC•X

(L)| X

= L•[n];

2) Hi(IC•

X(L)

)= 0 for i < −n;

3) dim supp−i(IC•

X(L)

)< i for all i < n;

4) dim cosuppi(IC•

X(L)

)< i for all i < n.

Note the strict inequalities in 3) and 4).

The uniqueness assertion implies that

IC•X

(L) ∼= j1! IC•X1

(L|X1∩

X

)⊕ · · · ⊕ jk! IC•

Xk(L|

Xk∩X

),

where jm denotes the inclusion of Xm into X.

In many sources, IC•X

(L) is only defined if X is pure-dimensional. We find it convenient to havethe intersection cohomology complex defined in the general situation – though, condition 0) above

APPENDIX B 203

says that our intersection cohomology complex is precisely the intersection cohomology complex onthe pure-dimensional space X(n) extended by zero to all of X. See section 5 for more on IC•

X(L).

The uniqueness assertion which accompanied our axioms for intersection cohomology impliesthat IC•

X(L) is semi-simple. More precisely, suppose that X is a n-dimensional complex analytic

set, let be the union of the n-dimensional components of X, and let L be a local system on a

smooth, open dense subset,X, of X(n).

The Category of Perverse Sheaves (see, also, section 5)

The category of perverse sheaves on X, Perv(X), is the full subcategory of Dbc(X) whose objects

are the perverse sheaves. Given a Whitney stratification, S, of X, it is also useful to consider thecategory PervS (X) := Perv(X)∩Db

S(X) of perverse sheaves which are constructible with respect

to S.Perv(X) and PervS (X) are both Abelian categories in which the short exact sequences

0→ A• → B• → C• → 0

are precisely the distinguished triangles

A• −→ B•

[1] C• .

If we have complexes A•,B•, and C• in Dbc(X) (resp. Db

S(X)), a distinguished triangle A• →

B• → C• → A•[1], and A• and C• are perverse, then B• is also in Perv(X) (resp. PervS (X)).

If the Whitney stratification S has a finite number of strata, then PervS (X) is actually anArtinian category, which means that every perverse sheaf which is constructible with respect to Shas a finite composition series in PervS (X) with uniquely determined simple subquotients. If Xis compact, then Perv(X) is also Artinian.

The simple objects in Perv(X) (resp. PervS (X)) are extensions by zero of intersection co-homology sheaves on irreducible analytic subvarieties (resp. connected components of strata) ofX with coefficients in irreducible local systems. To be precise, let M be a connected analyticsubmanifold (resp. a connected component of a stratum) of X and let LM be an irreducible localsystem on M ; then, the pair (M,LM ) is called an irreducible enriched subvariety of X (where Mdenotes the closure of M). Let j : M → X denote the inclusion. Then, the simple objects ofPerv(X) (resp. PervS (X)) are those of the form j!IC•

M(LM ), where (M,LM ) is an irreducible

enriched subvariety (again, we are indexing intersection cohomology so that it is non-zero only innon-positive dimensions).

Finally, we wish to state the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber.For this statement, we must restrict ourselves to R = Q. We give the statement as it appears in[Mac2], except that in [Mac2] intersection cohomology is defined as a positively perverse sheaf,and we must adjust by shifting. Note that, in [Mac2], the setting is algebraic; the analytic versionappears in [Sai].

204 DAVID B. MASSEY

An algebraic map f : X → Y is called projective if it can be factored as an embedding X →Y × Pm (for some m) followed by projection Y × Pm → Y .

The Decomposition Theorem [BBD, 6.2.5]: If f : X → Y is proper, then there exists aunique set of irreducible enriched subvarieties (Mα,Lα) in Y and Laurent polynomials φα =· · ·+ φα

−2t−2 + φα

−1t−1 + φα

0 + φα1 t+ φα

2 t2 + . . . such that there is a quasi-isomorphism

Rf∗IC•X

(Q•X

) ∼=⊕α,i

IC•Mα

(Lα)[−i]⊗Qφαi ,

(here, IC•Mα

(Lα) actually equals jα!IC•Mα

(Lα), where jα : Mα → Y is the inclusion).

Moreover, if f is projective, then the coefficients of φα are palindromic around 0 (i.e., φα(t−1) =φα(t)) and the even and odd terms are separately unimodal (i.e., if i 6 0, then φα

i−2 6 φαi ).

Applying hypercohomology to each side, we obtain:

IHk(X; Q) =⊕α,i

(IHk−i(Mα;Lα))φαi .

We now wish to describe the category of perverse sheaves on a one-dimensional space; this is aparticularly nice case of the results obtained in [M-V]. Unfortunately, we will use the notions ofvanishing cycles and nearby cycles, which are not covered until the next section. Nonetheless, itseems appropriate to place this material here.

We actually wish to consider perverse sheaves on the germ of a complex analytic space X at apoint x. Hence, we assume that X is a one-dimensional complex analytic space with irreducibleanalytic components X1, . . . , Xd which all contain x, such that Xi is homeomorphic to a complexline and Xi − x is smooth for all i. We wish to describe the category, C, of perverse sheaveson X with complex coefficients which are constructible with respect to the stratification X1 −x, . . . , Xd − x, x.

Since perverse sheaves are topological in nature, we may reduce ourselves to considering exactlythe case where X consists of d complex lines through the origin in some CN . Let L denote a linearform on CN such that X ∩ L−1(0) = 0.

Suppose now that P• is in C, i.e., P• is perverse on X and constructible with respect tothe stratification which has 0 as the only zero-dimensional stratum. Then P•

|X−0consists of a

collection of local systems, L1, . . .Ld, in degree −1. These local systems are completely determinedby monodromy isomorphisms hi : Cri → Cri representing looping once around the origin in Xi. Interms of nearby cycles, the monodromy automorphism on H0(ψLP•[−1])0 ∼=

⊕i Cri is given by⊕

i hi.The vanishing cycles φLP•[−1] are a perverse sheaf on a point, and so have possibly non-zero

cohomology only in degree 0; say, H0(φLP•[−1])0 ∼= Cλ. We have the canonical map

r : H0(ψLP•[−1])0 → H0(φLP•[−1])0

and the variation mapvar : H0(φLP•[−1])0 → H0(ψLP•[−1])0,

and var r = id−⊕

i hi.

APPENDIX B 205

Thus, an object in C determines a vector space W := H0(φLP•[−1])0, a vector space Vi := Cri

for each irreducible component Xi, an automorphism hi on Vi, and two linear maps α :⊕

i Vi →Wand β : W →

⊕i Vi such that β α = id−

⊕i hi. This situation is nicely represented by a

commutative triangle

⊕iViid−

⊕i hi−−−−−−−−−→ ⊕iVi

α β

W .

The category C is equivalent to the category of such triangles, where a morphism of triangles isdefined in the obvious way: a morphism is determined by linear maps τi : Vi → V ′

i and η : W →W ′

such that⊕iVi

α−−→Wβ−−→ ⊕iVi

⊕i τi ↓ η ↓ ⊕iτi ↓

⊕iV′i

α′−−→W ′ β′−−→ ⊕iV′i

commutes.

§3. Nearby and Vanishing Cycles

Historically, there has been some confusion surrounding the terminology nearby (or neighboring)cycles and vanishing cycles; now, however, the terminology seems to have stabilized. In the past,the term “vanishing cycles” was sometimes used to describe what are now called the nearby cycles(this is true, for instance, in [A’C], [BBD], and [G-M1].)

The two different indexing schemes for perverse sheaves also add to this confusion in statementssuch as “the nearby cycles of a perverse sheaf are perverse”. Finally, a new piece of confusionhas been added in [K-S2], where the sheaf of vanishing cycles is shifted by one from the usualdefinition (we will not use this new, shifted definition).

The point is: one should be very careful when reading works on nearby and vanishing cycles.

Let S = Sα be a Whitney stratification of X and suppose F• ∈ DbS(X). Given an analytic

map f : X → C, define a (stratified) critical point of f (with respect to S) to be a point x ∈ Sα ⊆ Xsuch that f|Sα

has a critical point at x; we denote the set of such critical points by ΣSf .We wish to investigate how the cohomology of the level sets of f with coefficients in F• changes

at a critical point (which we normally assume lies in f−1(0)).Consider the diagram

E −−−−→ C∗

π

y yπ

X − f−1(0)f−−−−→ C∗

i ↓f−1(0) →

jX

where:

206 DAVID B. MASSEY

j : f−1(0) → X is inclusion;

i : X − f−1(0) → X is inclusion;

f = restriction of f ;

C∗ = cyclic (universal) cover of C∗;

and E denotes the pull-back.

The nearby (or neighboring) cycles of F• along f are defined to be

ψfF• := j∗R(i π)∗(i π)∗F•.

Note that this is a sheaf on f−1(0).As ψf

(F•[k]

)=

(ψfF•)[k], we may write ψfF•[k] unambiguously. In fact, it is frequently

useful to consider the functor where one first shifts the complex by k and then takes the nearbycycles; thus, we introduce the notation ψf [k] to be the functor such that ψf [k]F• = ψfF•[k] (andwhich has the corresponding action on morphisms). The functor ψf takes distinguished trianglesto distinguished triangles.

If P• is a perverse sheaf on X, then ψf [−1]P• is perverse on f−1(0). (Actually, to concludethat ψf [−1]P• is perverse, we only need to assume that P•

|X−f−1(0)is perverse.)

Because ψf [−1] takes perverse sheaves to perverse sheaves, it is useful to include the shift by −1in many statements about ψf . Consequently, we also want to shift j∗F• by −1 in many statements,and so we write j∗[−1] for the functor which first shifts by −1 and then pulls-back by j.

As there is a canonical map F• → Rg∗g∗F• for any map g : Z → X, there is a map

F• → R(i π)∗(i π)∗F•

and, hence, a canonical map, called the comparison map:

j∗[−1]F• c−−→ j∗[−1]R(i π)∗(i π)∗F• = ψf [−1]F•.

For x ∈ f−1(0), the stalk cohomology of ψfF• at x is the cohomology of the Milnor fibre of fat x with coefficients in F•, i.e., for all ε > 0 small and all ξ ∈ C∗ with |ξ| << ε,

Hi(ψfF•)x∼= Hi(

Bε(x) ∩X ∩ f−1(ξ);F•),

where the open ballBε(x) is taken inside any local embedding of (X,x) in affine space. The sheaf

ψfF• only depends on f and F•|X−f−1(0)

.

While the above definition of the nearby cycles treats all angular directions equally, it is perhapsmore illuminating to fix an angle θ and describe the nearby cycles in terms of moving out slightlyalong the ray eiθ[0,∞). Consider the three inclusions kθ : f−1(eiθ(0,∞)) → f−1(eiθ[0,∞)),mθ : f−1(0) → f−1(eiθ[0,∞)), and lθ : f−1(eiθ[0,∞)) → X.

APPENDIX B 207

Then, one can define the nearby cycles at angle θ to be ψθfF

• := m∗θRkθ∗k

∗θ l∗θF

•.

For each θ, there is a canonical isomorphism ψfF• ∼= ψθfF

•. By letting θ travel around afull circle, we obtain isomorphisms ψθ

fF• ∼= ψθ+2π

f F•. These isomorphisms correspond to themonodromy automorphism Tf : ψf [−1]F• → ψf [−1]F•, which comes from the deck transformationobtained in our definition of ψfF• (and, hence, ψf [−1]F•) by traveling once around the origin in C.Actually, Tf is a natural automorphism from the functor ψf [−1] to itself; thus, strictly speaking,when we write Tf : ψf [−1]F• → ψf [−1]F•, we should include F• in the notation for Tf – however,we shall normally omit the explicit reference to F• if the complex is clear.

There is a natural distinguished trianglej∗[−1]Ri∗i∗F• −→ ψf [−1]F•

[1] Tf − idψf [−1]F•.

The associated long exact sequences on stalk cohomology are the Wang sequences.

The comparison map j∗[−1]F• c−−→ ψf [−1]F• is Tf -equivariant, i.e., c = Tf c.

Since we have a map c[1] : j∗F• → ψfF•, the third vertex of a distinguished triangle is definedup to quasi-isomorphism. We define the sheaf of vanishing cycles, φfF• , of F• along f to be thisthird vertex, i.e., there is a distinguished triangle

j∗F• −→ ψfF•

[1] φfF•.

Letting φf [−1] denote the functor which first shifts by −1 and then applies φf , we can writethe triangle above as

j∗[−1]F• c−−−→ ψf [−1]F•

[1] φf [−1]F•.

Note that this is a triangle of sheaves on f−1(0). Note also that, by replacing F• with i!i!F•, we

conclude that there is a natural isomorphism ψf [−1]F• ∼= φf [−1](i!i!F•). There is another naturalisomorphism ψf [−1]F• ∼= φf [−1](Ri∗i∗F•).

The functor φf takes distinguished triangles to distinguished triangles.

If P• is a perverse sheaf on X, then φf [−1]P• is a perverse sheaf on f−1(0).

For x ∈ f−1(0), the stalk cohomology of φfF• at x is the relative cohomology of the Milnorfibre of f at x with coefficients in F• and with a shift by one, i.e., for all ε > 0 small and all ξ ∈ C∗

with |ξ| << ε,

Hi(φfF•)x∼= Hi+1(

Bε(x) ∩X,

Bε(x) ∩X ∩ f−1(ξ);F•).

208 DAVID B. MASSEY

As an example, if X = Cn+1 and F• = C•X , then for all x ∈ f−1(0), Hi(ψfC•

X)x = i-thcohomology of the Milnor fibre of f at x (with C coefficients) = Hi(Ff,x; C), while Hi(φfC•

X)x =reduced i-th cohomology of the Milnor fibre of f at x = Hi(Ff,x; C).

Just as we defined the nearby cycles at angle θ to be ψθfF

• := m∗θRkθ∗k

∗θ l∗θF

•, we can definethe vanishing cycles at angle θ to be φθ

fF• := m∗

θmθ !m!θl∗θF

•[1] = m!θl∗θF

•[1]. Then, φθfF

• ∼= φfF•,and again there is a monodromy automorphism Tf : φf [−1]F• → φf [−1]F•. The monodromy Tf

is actually a natural automorphism of the functor φf [−1].If we let Zθ := z ∈ X | Re

(eiθf(z)

)6 0, then there is a canonical isomorphism

φθfF

• ∼=(RΓ

Zθ(F•)

)|f−1(0)

[1].

Thus, there is a monodromy automorphism on the distinguished triangle

j∗[−1]F• c−−→ ψf [−1]F• r−−→ φf [−1]F• −→ j∗F•

given by (id, Tf , Tf ), i.e., a commutative diagram

j∗[−1]F• c−−→ ψf [−1]F• r−−→ φf [−1]F• −→ j∗F•

id ↓ Tf ↓ Tf ↓ id ↓

j∗[−1]F• c−−→ ψf [−1]F• r−−→ φf [−1]F• −→ j∗F•.

From this, it follows formally that there exists a variation morphism, var : φf [−1]F• →ψf [−1]F• such that r var = id−Tf and var r = id−Tf . (Note that, if we are not using fieldcoefficients, then the variation morphism does not necessarily exist on the level of chain complexes– the derived category structure is necessary here.)

The monodromy isomorphisms Tf and Tf are natural automorphisms of the (shifted) nearbycycle and vanishing cycle functors, respectively, and the maps c, r, and var above are all naturalmaps.

The variation map can be described in a more concrete fashion. There is the canonical mapfrom F• to Ri∗i

∗F•. Applying the shifted vanishing cycle functor, we obtain a natural mapfrom φf [−1](F•) to φf [−1](Ri∗i∗F•), and as we mentioned above, there is a natural isomorphismφf [−1](Ri∗i∗F•) ∼= ψf [−1]F•. The variation map is the composition of these two natural maps.To make this more clear, we will describe the variation map on the stalk cohomology; this shouldalso help clarify how one obtains the isomorphism φf [−1](Ri∗i∗F•) ∼= ψf [−1]F•.

We follow the construction in [G-M1]. Let x be a point in f−1(0), let N denote the intersectionof X with a sufficiently small open ball around x (for some Reimannian metric), and let Dη be acomplex disk of sufficiently small radius, η, centered at the origin so that, for all ξ with 0 < ξ 6 η,N ∩ f−1(∂Dξ)

f−−→ ∂Dξ represents the Milnor fibration of f at x with coefficients in F•. LetW := N∩f−1 (v ∈ Dη | Re v > 0 − 0), let Z := N∩f−1 (v ∈ Dη | Re v 6 0 − 0), let A :=N ∩ f−1 (v ∈ Dη | Re v = 0, Im v > 0), and let B := N ∩ f−1 (v ∈ Dη | Re v = 0, Im v < 0).

Then, we have isomorphisms:

Hi(φfF•)x∼= Hi+1(N ∩ f−1(Dη), N ∩ f−1(η);F•) ∼= Hi+1(N ∩ f−1(Dη),W ;F•);

APPENDIX B 209

the map induced by inclusion of pairs:

Hi+1(N ∩ f−1(Dη),W ;F•)→ Hi+1(N ∩ f−1(Dη − 0),W ;F•);

and isomorphisms:

Hi+1(N ∩ f−1(Dη − 0),W ;F•) ∼= Hi+1(Z,A ∪B;F•) ∼= Hi(ψfF•)x,

where the first isomorphism is by excision, and the second is from the long exact sequence of thepair.

The map induced by the (shifted) variation on the stalk cohomology is the composition of theabove maps.

Applying the shifted vanishing cycle functor to the distinguished triangle

j!j!F• −→ F•

[1] Ri∗i

∗F•,

noting that φf [−1](j!j!F•) ∼= j!F•, and using the natural isomorphism

ψf [−1]F• ∼= φf [−1](Ri∗i∗F•),

we obtain the distinguished triangle

j!F• −→ φf [−1]F•

[1] varψf [−1]F•.

Starting with the two distinguished triangles

φf [−1]F• var−−−→ ψf [−1]F• −→ j![1]F• −→ φfF•

andψf [−1]F• c−−−→ φf [−1]F• −→ j∗F• −→ ψfF•,

we may apply the octahedral lemma to conclude that there exists a complex wfF• and two distin-guished triangles

φf [−1]F• id−Tf−−−−−→ φf [−1]F• −→ wfF• −→ φfF•

andj![1]F• −→ wfF• −→ j∗F• τ−−→ j![2]F•.

We refer to the morphism ωf := τ [−1] from j∗[−1]F• to j![1]F• as the Wang morphism off . The application of the octahedral lemma above tells us that the mapping cone of id−Tf isisomorphic to the mapping cone of ωf . Note that, while j∗[−1]F• and j![1]F• depend only onf−1(0) (and F•), ωf may change if f (or some factor of f) is raising to a power.

210 DAVID B. MASSEY

For any Whitney stratification, S, with respect to which F• is constructible, the support ofH∗(φfF•) is contained in the stratified critical locus of f , ΣSf . In addition, if S is a Whitneystratification with respect to which F• is constructible and such that f−1(0) is a union of strata,then – by [BMM] and [P2] – it follows that S also satisfies Thom’s af condition; by Thom’s secondisotopy lemma, this implies that the entire situation locally trivializes over strata, and hence bothψfF• and φfF• are constructible with respect to S ∈ S | S ⊆ f−1(0).

Suppose we have X π−→ Yf−→ C where π is proper and π : π−1f−1(0)→ f−1(0) is the restriction

of π. Then, for all A• ∈ Dbc(X),

Rπ∗(ψfπA•) ∼= ψf (Rπ∗A•) and Rπ∗(φfπA•) ∼= φf (Rπ∗A•).

The Sebastiani-Thom Isomorphism

Let f : X → C and g : Y → C be complex analytic functions. Let π1 and π2 denote theprojections of X × Y onto X and Y , respectively. Let A• and B• be bounded, constructible

complexes of sheaves of R-modules on X and Y , respectively. In this situation, A•L B• :=

π∗1A• L⊗ π∗2B

•. Let us adopt the similar notation f g := f π1 + g π2.

Let p1 and p2 denote the projections of V (f)× V (g) onto V (f) and V (g), respectively, and letk denote the inclusion of V (f)× V (g) into V (f g).

Theorem (Sebastiani-Thom Isomorphism). There is a natural isomorphism

k∗φfg

[−1](A• L

B•) ∼= φf[−1]A• L

φg[−1]B•,

and this isomorphism commutes with the corresponding monodromies.Moreover, if we let p := (x,y) ∈ X × Y be such that f(x) = 0 and g(y) = 0, then, in an open

neighborhood of p, the complex φfg

[−1](A•

L B•) has support contained in V (f) × V (g), and,

in any open set in which we have this containment, there are natural isomorphisms

φfg

[−1](A• L

B•) ∼= k!(φf[−1]A• L

φg[−1]B•) ∼= k∗(φf

[−1]A• L φ

g[−1]B•).

If Q• is perverse on X − f−1(0) and i : X − f−1(0)→ X is the inclusion, then it is easy to seethat Ri∗Q• satisfies the cosupport condition; moreover, by combining the fact that ψf (Ri∗Q•)[−1]is perverse on f−1(0) with the Wang sequences on stalk cohomology, one can prove that Ri∗Q•

also satisfies the support condition - hence, Ri∗Q• is perverse. In an analogous fashion, one obtainsthat Ri!Q• is perverse (if the base ring is a field, this can be obtained by dualizing).

If R is a field, then the operators ψf [−1] and D commute, as do φf [−1] and D; i.e.,

D(ψfA•[−1]) ∼= ψf (DA•)[−1] and D(φfA•[−1]) ∼= φf (DA•)[−1].

APPENDIX B 211

These isomorphisms in Dbc(X) are non-canonical.

Let A• ∈ Dbc(X) and f : X → C. The monodromy automorphism ψf [−1]A• Tf−→ ψf [−1]A•

induces a map on cohomology sheaves which is quasi-unipotent, i.e., letting Tf also denote themap on cohomology, this means that there exist integers k and j such that (id−T k

f )j = 0.

Suppose that the base ring is a field; if mx denotes the maximal ideal of X at x and f ∈ m2x,

then the Lefschetz number of the map H∗(ψf [−1]A•)xTf−→ H∗(ψf [−1]A•)x equals 0, i.e.,∑

i

(−1)i TraceHi(ψf [−1]A•)xTf−→ Hi(ψf [−1]A•)x = 0.

If ψf [−1]A• is a perverse sheaf, then we may use the Abelian structure of the category Perv(X)

to investigate the map ψf [−1]A• Tf−→ ψf [−1]A•. This morphism can be factored into Tf =F · (1 + N), where F has finite order and N is nilpotent. It follows that there is a uniqueincreasing filtration W i on ψf [−1]A• such that N sends W i to W i−2 and N i takes Griψf [−1]A•

isomorphically to Gr−iψf [−1]A•, where Gri is the associated graded to the filtration W •. This iscalled the nilpotent filtration of ψf [−1]A•. (The existence of such a filtration is just linear algebra;the interesting result is the following theorem, due to Gabber.)

Theorem: If we have f : X → C, S a Whitney stratification of X with a finite number of strata,and A• ∈ Db

S(X) such that

A•|X−f−1(0)

∼= IC•X−f−1(0)

(C•X

),

then the graded pieces of the nilpotent filtration of ψf [−1]A• are semi-simple in PervS (f−1(0)),i.e., they are direct sums of intersection cohomology sheaves of irreducible enriched subvarieties off−1(0) (extended by zero).

In particular, if X = Cn+1, then each Griψf [−1]C•X [n+ 1] is semi-simple.

§4. Some Quick Applications

The applications of perverse sheaves are widespread and are frequently quite deep - particularlyfor those applications which rely on the decomposition theorem. For beautiful discussions ofthese applications, we highly recommend [Mac1] and [Mac2]. We shall not describe any of theseapplications here; rather we shall give some fairly easy results on general Milnor fibres. Theseresults are “easy” now that we have all the machinery of the first three sections at our disposal.While the applications below could undoubtedly be proved without the general theory of perversesheaves, with this theory in hand, the results and their proofs can be presented in a unified mannerand, what is more, the proofs become mere exercises.

Consider the classical case of the Milnor fibre of a non-zero map f : (Cn+1,0) → (C, 0). LetX = Cn+1 and let s = dim Σf . Then, as X is a manifold, C•

X is a positively perverse sheafand so φfC•

X is positively perverse on f−1(0) with support only on Σf . It follows that the stalkcohomology of φfC•

X is non-zero only for dimensions i with n − s 6 i 6 n; that is, we recoverthe well-known result that the reduced cohomology of the Milnor fibre is non-zero only in thesedimensions.

212 DAVID B. MASSEY

A much more general case is just as easy to derive from the machinery that we have. Supposethat X is a purely (n+1)-dimensional local complete intersection with arbitrary singularities. LetS be a Whitney stratification of X. Let p ∈ X be such that dimpf

−1(0) = n, and let Ff,p

denotethe Milnor fibre of f at p. Then, as X is a local complete intersection, C•

X is a positively perversesheaf and so φfC•

X is positively perverse on f−1(0) with support only on ΣSf . It follows that thestalk cohomology of φfC•

X is non-zero only for dimensions i with n− dimpΣSf 6 i 6 n. Hence,the reduced cohomology of F

f,pis non-zero only in these dimensions.

While this general statement could no doubt be proved by induction on hyperplane sections, theabove proof via general techniques avoids the re-working of many technical lemmas on privilegedneighborhoods and generic slices.

Another application relates to the homotopy-type of the complex link of a space at a point; forinstance, for an s-dimensional local complete intersection, the complex link has the homotopy-typeof a bouquet of spheres of real dimension s− 1. In terms of vanishing cycles and perverse sheaves,we only obtain this result up to cohomology: let (X,x) be a germ of an analytic space embeddedin some Cn, and assume s := dim X = dimxX. Suppose that we have a positively perverse sheaf,P•, on X (e.g., the constant sheaf, if X is a local complete intersection). Let l be a generic linearform, and consider φl−l(x)P•; this is a perverse sheaf on an s − 1 dimensional space and, as l isgeneric, it is supported at the single point x (because the hyperplane slice l = l(x) can be chosen totransversely intersect all the strata of any stratification with respect to which P• is constructible- except, possibly, the point-stratum x itself). Hence, H∗(φl−l(x)P•)x is (possibly) non-zero onlyin dimension s − 1. In the case of the constant sheaf on a local complete intersection, this givesthe desired result.

For our final application, we wish to investigate functions with one-dimensional critical loci; wemust first set up some notation.

Let U be an open neighborhood of the origin in Cn+1and suppose that f : (U ,0)→ (C, 0) has

a one-dimensional critical locus at the origin, i.e., dim0Σf = 1. The reduced cohomology of theMilnor fibre, F

f,0, of f at the origin is possibly non-zero only in dimensions n− 1 and n. We wish

to show that the n − 1-st cohomology group embeds inside another group which is fairly easy todescribe; thus, we obtain a bound on the n− 1-st Betti number of the Milnor fibre of f .

For each component ν of Σf , one may consider a generic hyperplane slice, H, at points p ∈ ν−0close to the origin; then, the restricted function, f|H , will have an isolated critical point at p. Byshrinking the neighborhood U if necessary, we may assume that the Milnor number of this isolatedsingularity of f|H at p is independent of the point p ∈ ν − 0; denote this value by

µν . As ν − 0 is

homotopy-equivalent to a circle, there is a monodromy map from the Milnor fibre of f|H at p ∈ ν−0

to itself, which induces a map on the middle dimensional cohomology, i.e., a map hν : Zµν → Z

µν .

We wish to show that Hn−1(Ff,0

) (with integer coefficients) injects into ⊕νker(id− hν).Let j denote the inclusion of the origin into X = V (f), let i denote the inclusion of X − 0 into

X, and let K• denote φf (Z•U). As Z•

Uis positively perverse, φf (Z•

U) is positively perverse with

one-dimensional support (as we are assuming a one-dimensional critical locus). Also, we alwayshave the distinguished triangle

Rj∗j!K• −→ K•

[1] Ri∗i

∗K•

We wish to examine the associated stalk cohomology exact sequence at the origin.

APPENDIX B 213

First, we have that Hn−1((Rj∗j!K•)0) = Hn−1(j!K•) and so, by the cosupport condition forperverse sheaves, Hn−1((Rj∗j!K•)0) = 0.

Now, we need to look more closely at the sheaf Ri∗i∗K•. i∗K• is the restriction of K• to X−0;near the origin, this sheaf has cohomology only in degree n−1 with support on Σf −0. Moreover,the cohomology sheaf Hn−1(i∗K•) is locally constant when restricted to Σf − 0. It follows thati∗K• is naturally isomorphic in the derived category to the extension by zero of a local system ofcoefficients in dimension n− 1 on Σ− 0.

To be more precise, let p denote the inclusion of the closed subset Σf − 0 into X − 0. Then,there exists a locally constant (single) sheaf, L, on Σf − 0 such that when L is considered as acomplex, L•, we have that p!L•[−(n− 1)] ∼= p∗L•[−(n− 1)] is naturally isomorphic to i∗K•. For

each component ν of Σf , the restriction of L to ν − 0 is a local system with stalks Zµν which is

completely determined by the monodromy map hν : Zµν → Z

µν .

Therefore, inside a small open ballB,

H0((Ri∗i∗L•)0) ∼= ⊕νH0(B ∩ (ν − 0);L)

and these global sections are well-known to be given by ker(id− hν). It follows that

Hn−1((Ri∗i∗K•)0) ∼= ⊕νker(id− hν).

Thus, when we consider the long exact sequence on stalk cohomology associated to our distin-guished triangle, we find – starting in dimension n− 1 – that it begins

0→ Hn−1(Ff,0

)→ ⊕νker(id− hν)→ . . . .


§5. Truncation and Perverse Cohomology

This section is taken entirely from [BBD], [G-M3], and [K-S2].

There are (at least) two forms of truncation associated to an object F• ∈ Dbc(X) – one form

of truncation is related to the ordinary cohomology of the complex, while the other form leads tosomething called the perverse cohomology or perverse projection. These two types of truncationbear little resemblance to each other, except in the general framework of a t-structure on Db

c(X).

Loosely speaking, a t-structure on Dbc(X) consists of two full subcategories, denoted D

60(X)

and D>0

(X), such that for any F• ∈ Dbc(X), there exist E• ∈ D

60(X), G• ∈ D

>0(X), and a

distinguished triangle

E• −→ F•

[1] ;G•[−1]

moreover, such E• and G• are required to be unique up to isomorphism in Dbc(X).

214 DAVID B. MASSEY

Given a t-structure as above, and using the same notation, we write E• = τ60F• (the truncationof F• below 0) and G• = τ>0 (F•[1]) (the truncation of F•[1] above 0); these are the basic truncationfunctors associated to the t-structure.

In addition, we write D6n

(X) for

D60

(X)[−n] :=F•[−n] | F• ∈ D

60(X)

,

and we analogously write D>n

(X) for D>0

(X)[−n].Also, we define τ6nF• by

τ6nF• = (τ60(F•[n])) [−n] = ([−n] τ60 [n])F•,

and we analogously define τ>nF• as ([−n] τ>0 [n])F•.

Note that τ6nF• ∈ D6n

(X), τ>nF• ∈ D>n

(X) and, for all n, we have a distinguished triangle

τ6nF• −→ F•

[1] .τ>n+1F•

Writing ' to denote natural isomorphisms between functors: for all a and b,

τ6b τ>a ' τ>a τ6b,

τ6b τ6a ' τ6a τ6b,

andτ>b τ>a ' τ>a τ>b.

If a > b, thenτ6b τ6a ' τ6b,

andτ>a τ>b ' τ>a.

Also, if a > b, thenτ6b τ>a = τ>a τ6b = 0.

The heart of the t-structure is defined to be the full subcategory C := D60

(X) ∩D>0

(X); thisis always an Abelian category. We wish to describe the kernels and cokernels in this category.

Let E•,F• ∈ C and let f be a morphism from E• to F•. We can form a distinguished trianglein Db

c(X)

E• f−−−→ F•

[1] ,G•

where G• need not be in C. Then, up to natural isomorphism,

coker f = τ>0G• and ker f = τ60(G•[−1]).

APPENDIX B 215

We define cohomology associated to a t-structure as follows. Define tH0(F•) to be τ>0τ60F•;this is naturally isomorphic to τ60τ

>0F•. Now, define tHn(F•) to betH0(F•[n]) =

(τ>nτ6nF•) [n].

Note that this cohomology does not give back modules or even sheaves of modules, but rathergives back complexes which are objects in the heart of the t-structure.

If F• ∈ Dbc(X), then the following are equivalent:

1) F• ∈ D60

(X) (resp. D>0

(X));

2) the morphism τ60F• → F• is an isomorphism (resp. the morphism F• → τ>0F• is anisomorphism);

3) τ>1F• = 0 (resp. τ6−1F• = 0));

4) τ>iF• = 0 for all i > 1 (resp. τ6iF• = 0 for all i 6 −1);

5) there exists a such that F• ∈ D6a

(X) and tHi(F•) = 0 for all i > 1 (resp. there exists a suchthat F• ∈ D

>a

(X) and tHi(F•) = 0 for all i 6 −1).

It follows that, if F• ∈ Dbc(X), then the following are equivalent:

1) F• ∈ C;

2) tH0(F•) is isomorphic to F•;

3) there exist a and b such that F• ∈ D>a

(X), F• ∈ D6b

(X), and tHn(F•) = 0for all n 6= 0.

As the heart is an Abelian category, we may talk about exact sequences in C. Any distinguishedtriangle in Db

c(X) determines a long exact sequence of objects in the heart of the t-structure; if

E• −→ F•

[1] G•

is a distinguished triangle in Dbc(X), then the associated long exact sequence in C is

· · · → tH−1(G•)→ tH0(E•)→ tH0(F•)→ tH0(G•)→ tH1(E•)→ . . . .

We are finished now with our generalities on t-structures and wish to, at last, give our twoprimary examples.

The “ordinary” t-structure

The “ordinary” t-structure on Dbc(X) is given by

D60

(X) = F• ∈ Dbc(X) | Hi(F•) = 0 for all i > 0

216 DAVID B. MASSEY

andD

>0(X) = F• ∈ Db

c(X) | Hi(F•) = 0 for all i < 0.

The associated truncation functors are the ordinary ones described in [G-M3]. If F• ∈ Dbc(X),

then

(τ6pF•)n =

Fn if n < p

ker dp if n = p

0 if n > p

and

(τ>pF•)n

=

0 if n < p

coker dp−1 if n = p

Fn if n > p.

These truncated complexes are naturally quasi-isomorphic to the complexes

(τ6pF•)n =

Fn if n 6 p

Im dp if n = p+ 10 if n > p+ 1

and

(τ>pF•)n

=

0 if n < p− 1Im dp−1 if n = p− 1Fn if n > p.

If A•,B• ∈ Dbc(X), then

1. (τ6pA•)x = τ6p (A•x);

2. Hk (τ6pA•)x =

Hk (A•)x if k 6 p

0 for k > p.

3. If φ : A• → B• is a morphism of complexes of sheaves which induces isomorphisms on theassociated cohomology sheaves

φ∗ : Hn(A•) ∼= Hn(B•) for all n 6 p,

then τ6pφ : τ6pA• → τ6pB• is a quasi-isomorphism.

4. If f : X → Y is a continuous map and C• is a complex of sheaves on Y , then

τ6pf∗(C•) ∼= f∗τ6p(C•).

APPENDIX B 217

5. If R is a field and A• is a complex of sheaves of R-modules on X with locally constantcohomology sheaves, then there are natural quasi-isomorphisms

τ>−pRHom•(A•,R•X)→ τ>−pRHom•(τ6pA•,R•

X)← RHom•(τ6pA•,R•X).

The heart of this t-structure consists of those complexes which have non-zero cohomology sheavesonly in degree 0; such complexes are quasi-isomorphic to complexes which are non-zero only indegree 0.

The t-structure cohomology of a complex F• is essentially the sheaf cohomology of F•; tHn(F•)is quasi-isomorphic to a complex which has Hn(F•) in degree 0 and is zero in all other degrees.With this identification, the t-structure long exact sequence associated to a distinguished triangleis merely the usual long exact sequence on sheaf cohomology.

We are now going to give the construction of the intersection cohomology complexes as it ispresented in [G-M3]. Our indexing will look different from that of [G-M3] for several reasons.

First, we are dealing only with complex analytic spaces, X, and we are using only middleperversity; this accounts for some of the indexing differences. In addition, in this setting, theintersection cohomology complex defined in [G-M3] would have possibly non-zero cohomologyonly in degrees between −2 dimC X and −(dimC X)− 1, inclusive. The definition below is shiftedby −dimC X from the [G-M3] definition, and yields a perverse sheaf which has possibly non-zerocohomology only in degrees between −dimC X and −1, inclusive.

Let X be a complex analytic n-dimensional space with a complex analytic Whitney stratificationS = Sα. While we do not explicitly require that X is pure-dimensional, it will follow from theconstruction that components of X of dimension less than n will essentially be ignored.

For all k, let Xk denote the union of the strata of dimension less than or equal to k. Byconvention, we set X−1 = ∅. Hence, we have a filtration

∅ = X−1 ⊆ X0 ⊆ X1 ⊆ · · · ⊆ Xn−1 ⊆ Xn = X.

For all k, let Uk := X − Xn−k, and let ik denote the inclusion Uk → Uk+1. Let L•U1

be a localsystem on the top-dimensional strata.

Then, the intersection cohomology complex on X with coefficients in L•U1

, as described in section2, is given by

IC•X(L•

U1) := τ6−1Rin∗ . . . τ61−n

Ri2∗τ6−nRi1∗

(L•U1

[n]).

Up to quasi-isomorphism, this complex is independent of the stratification. Note that thecohomology sheaves of IC•

X(L•U1

) are supported only in degrees k for which −n 6 k 6 −1 (unlessX is 0-dimensional, and then IC•

X(L•U1

) ∼= L•U1).

Also note that it follows from the construction that there is always a canonical map from theshifted constant sheaf R•

X [n] to IC•X(R•

U1) which induces an isomorphism when restricted to U1.

To see this, consider the canonical morphism R•Uk+1

[n] → Rik∗i∗kR

•Uk+1

[n] for each k > 1. Asi∗kR

•Uk+1

[n] ∼= R•Uk

[n], we have a canonical map R•Uk+1

[n] → Rik∗R•Uk

[n] and, hence, a canonical

map between the truncations τ6k−n−1

(R•

Uk+1[n]

)→ τ6k−n−1Rik∗

(R•

Uk[n]

). But,

τ6k−n−1

(R•

Uk+1[n]

) ∼= R•Uk+1

[n]

218 DAVID B. MASSEY

and so we have a canonical map R•Uk+1

[n] → τ6k−n−1Rik∗(R•

Uk[n]

). By piecing all of these maps

together, one obtains the desired morphism.

The perverse t-structure

The perverse t-structure (with middle perversity µ) on Dbc(X) is given by

µ

D60

(X) = F• ∈ Dbc(X) | dim supp-j F• 6 j for all j

andµ

D>0

(X) = F• ∈ Dbc(X) | dim cosuppj F• 6 j for all j.

Note that the heart of this t-structure is precisely Perv(X). Thus, every distinguished triangle inDb

c(X) determines a long exact sequence in the Abelian category Perv(X).

We naturally call the t-structure cohomology associated to the perverse t-structure the perversecohomology or perverse projection and denote it in degree n by µHn(F•).

Let d be an integer, and let f : Y → X be a morphism of complex spaces such that dim f−1(x) 6d, for all x ∈ X. Let dimY/X := dimY − dimX. Then,

1) f∗ sendsµ

D60

(X) toµ

D6d

(Y ), and sendsµ

D>0

(X) toµ

D>dim Y/X

(Y );

2) f ! sendsµ

D>0

(X) toµ

D>−d

(Y ), and sendsµ

D60

(X) toµ

D6− dim Y/X

(Y );;

3) if F• ∈ µ

D60

(Y ) and Rf!F• ∈ Dbc(X), then Rf!F• ∈ µ

D6d

(X);

4) if F• ∈ µ

D>0

(Y ) and Rf∗F• ∈ Dbc(X), then Rf∗F• ∈ µ

D>−d

(X).

Let f : Y → X be a morphism of complex spaces such that each point in X has an openneighborhood U such that f−1(U) is a Stein space (e.g., an affine map between algebraic varieties).Then,

1) if F• ∈ µ

D60

(Y ) and Rf∗F• ∈ Dbc(X), then Rf∗F• ∈ µ

D60

(X);

2) if F• ∈ µ

D>0

(Y ) and Rf!F• ∈ Dbc(X), then Rf!F• ∈ µ

D>0

(X).

If f : X → C is an analytic map, then the functors ψf [−1] and φf [−1] are t-exact with respectto the perverse t-structures; this means that if E• ∈ µ

D60

(X) and F• ∈ µ

D>0

(X), then ψfE•[−1]and φfE•[−1] are in

µ

D60

(f−1(0)), and ψfF•[−1] and φfF•[−1] are inµ

D>0

(f−1(0)).In particular, ψf [−1] and φf [−1] take perverse sheaves to perverse sheaves and, for any F• ∈

Dbc(X),

µHn(ψfF•[−1]) ∼= ψfµHn(F•)[−1] and µHn(φfF•[−1]) ∼= φf

µHn(F•)[−1].

If the base ring is a field, then the functor µH0 also commutes with Verdier dualizing; that is,there is a natural isomorphism

D µH0 ∼= µH0 D.

APPENDIX B 219

Let F• be a bounded complex of sheaves on X which is constructible with respect to a connectedWhitney stratification Sα of X, and let dα := dimSα. Then, µH0(F•) is also constructible withrespect to S, and

(µH0(F•)

)|Nα

[−dα] is naturally isomorphic to µH0(F•|Nα

[−dα]), where Nα denotesa normal slice to Sα.

Let Smax be a maximal stratum contained in the support of F•, and let m = dimSmax. Then,(µH0(F•)

)|Smax

is isomorphic (in the derived category) to the complex which has (H−m(F•))|Smax

in degree −m and zero in all other degrees.In particular, suppF• =

⋃i supp µHi(F•), and if F• is supported on an isolated point, q, then

H0(µH0(F•))q ∼= H0(F•)q. From this, and the fact that perverse cohomology commutes withnearby and vanishing cycles shifted by −1, one easily concludes that, at all points x ∈ X,

χ(F•)x =∑

k

(−1)kχ(µHk(F•)

)x.

Switching Coefficients

Suppose that the base ring R is a p.i.d. For each prime ideal p of R, let kp denote the field offractions of R/p, i.e., k0 is the field of fractions of R, and for p 6= 0, kp = R/p. There are the obvious

functors δp : Dbc(RX

)→ Dbc((kp)

X), which sends F• to F• L

⊗ (kp)•X , and εp : Dbc((kp)

X)→ Db

c(RX),

which considers kp-vector spaces as R-modules.If A• is a complex of kp-vector spaces, we may consider the perverse cohomology of A•,

µHikp

(A•), or the perverse cohomology of ε(A•), which we denote by µHiR(A•). If A• ∈ Db

c((kp)X

)and Smax is a maximal stratum contained in the support of A•, then there is a canonical isomor-phism

ε((µHi

kp(A•))|Sα

) ∼= (µHiR(A•))|Sα

;

in particular, supp µHikp

(A•) = supp µHiR(A•).

If F• ∈ Dbc(RX

), Smax is a maximal stratum contained in the support of F•, and x ∈ Smax,then for some prime ideal p ⊂ R and for some integer i, Hi(F•)x ⊗ kp 6= 0; it follows that Smax is

also a maximal stratum in the support of F• L⊗ (kp)

•X

. Thus,

suppF• =⋃p

supp(F• L⊗ (kp)

•X

)

and sosuppF• =

⋃i,p

supp µHikp

(F• L⊗ (kp)

•X

),

where the boundedness and constructibility of F• imply that this union is locally finite.

220 DAVID B. MASSEY

APPENDIX C:

PRIVILEGED NEIGHBORHOODS ANDLIFTING MILNOR FIBRATIONS

In this appendix, we prove a number of very technical results. These results tell us when we canuse certain types of “nice” neighborhoods to define the Milnor fibre (at least, up to homotopy),and give conditions under which Milnor fibrations remain constant in a parameterized family. Lenumbers and cycles do not appear here, though we will use the relative polar curve.

Throughout, for convenience, we concentrate our attention at the origin. Let U be an openneighborhood of the origin in some Cn+1 and let h : (U ,0) → (C, 0) be an analytic function.

In what sense the Milnor fibre and Milnor fibration of h are well-defined has been discussed ina number of places (see, for instance, [Se-Th]). If one is primarily interested in the ambient, localtopology of the hypersurface V (h) defined by h, then “the” Milnor fibre is only well-defined up tohomotopy-type [Le6]. Thus, we may make the weakest possible definition of the Milnor fibre of hat the origin as a homotopy-type:

Definition/Proposition C.1. A system of Milnor neighborhoods for h at the origin is a fundamen-tal system of neighborhoods, Cα, at the origin in U such that for all Cα ⊆ Cβ , there exists ε > 0such that for all complex ξ with 0 < |ξ| < ε, we have that the inclusion Cα∩V (h−ξ) → Cβ∪V (h−ξ)is a homotopy-equivalence. The standard system of Milnor neighborhoods for h at the origin is justthe set of closed balls of sufficiently small radius centered at the origin (this system is independentof h except for how small the radii must be).

If Cα is a system of Milnor neighborhoods for h at the origin, then for each Cα there existsε > 0 such that the homotopy-type of Cα∩V (h− ξ) is independent of the complex number ξ chosenas long as 0 < |ξ| < ε. Moreover, this homotopy-type is independent of the choice of the particularCα and is, in fact, independent of the choice of the system of Milnor neighborhoods.

Proof. The proof is standard. Let Cα be a system of Milnor neighborhoods for h at the origin.We shall compare it with the standard system. Select any Cβ . Now, pick Cα, Bη, and Bδ such thatBη and Bδ are in the standard system of Milnor neighborhoods for h at the origin and such thatBδ ⊆ Cα ⊆ Bη ⊆ Cβ . We may certainly pick ε > 0 such that, for all complex ξ with 0 < |ξ| < ε,the inclusion Cα ∩ V (h − ξ) → Cβ ∩ V (h − ξ) and the inclusion Bδ ∩ V (h − ξ) → Bη ∩ V (h − ξ)are both homotopy-equivalences. It follows that the inclusion Cα ∩ V (h − ξ) → Bη ∩ V (h − ξ) isa homotopy-equivalence for all small ξ 6= 0 and thus, as the homotopy-type of Bη ∩ V (h − ξ) isindependent of ξ, so is that of Cα ∩ V (h− ξ). The conclusion follows immediately.

It is sometimes more convenient to prove that Cα ∩ V (h− ξ) → Cβ ∩ V (h− ξ) is a homotopy-equivalence whenever Cα is contained in the interior of Cβ . It is easy to see by the proof abovethat this is enough to show that the system is a system of Milnor neighborhoods.

A system of Milnor neighborhoods allows one to discuss the Milnor fibre up to homotopy.However, one frequently wishes to use stratified, differential techniques to study the Milnor fibreand, hence, one would like for the Milnor fibre to have the structure of a smooth, compact manifoldwith (stratified) boundary and would also like to have some control over what happens on theboundary as one moves through a family of singularities. Furthermore, one would like to have a

221

222 DAVID B. MASSEY

notion of the Milnor fibration – at least up to fibre-homotopy-type.To gain this additional structure, we will use two types of (complex analytic) stratifications. One

is the well-known Whitney stratification [G-M2], [Mat], [Th]. The second is a good stratification,as defined in 1.24. Note that any refinement of a good stratification which does not refine thesmooth stratum is automatically a good stratification. This fact will be very useful when combinedwith the following proposition, which is Theorem 18.11 of [W] (or just a small portion of Theorem1.7 of [G-M2]).

Proposition C.2. Let X be an analytic subset of CN and let Y be an analytic subset of X.Suppose that D and F are analytic stratifications for X and Y , respectively. Then, there exists ananalytic stratification, L, of X which is a common refinement of both D and F , i.e. every stratumof L is contained in a stratum of D, Y is a union of strata of L, and every stratum of L which iscontained in Y is contained in a stratum of F .

The L above is sometimes referred to as a refinement of D adapted to Y .

(The reader should note that when Goresky and MacPherson use the term “stratification”, theymean that the Whitney conditions are satisfied. Hence, their Theorem 1.7 actually allows us topick a common analytic, Whitney refinement.)

We now generalize the notion of a privileged polydisc as given in [Le3]. This definition shouldbe compared with [L-T1, 2.2.3].

Definition C.3. Let G be a good stratification for h at the origin. A fundamental system ofneighborhoods, Cα, at the origin in U is a system of privileged neighborhoods for h at 0 withrespect to G if and only if

i) Cα is a system of compact, Milnor neighborhoods for h at the origin;

and, for each Cα, there is an associated Whitney stratification, Sα, of Cα such that

ii) the interior of Cα,Cα, in U is a stratum in Sα;

iii) Cα equals the closure ofCα in U ;

By ii) and iii) and the condition of the frontier, the boundary of Cα, ∂Cα, is a union of Whitneystrata, and we make the final requirement:

iv) the boundary strata of each Cα transversely intersect all the strata of G.

A fundamental system of neighborhoods, C = Cα, is a system of privileged neighborhoods forh at 0 if and only if there exists a good stratification, G, for h at 0 such that C is a system ofprivileged neighborhoods for h at 0 with respect to G.

A fundamental system of neighborhoods, C = Cα, satisfying i), ii), and iii) above is a systemof weakly privileged neighborhoods for h at 0 if and only if for each Cα, for all small ξ 6= 0, V (h−ξ)transversely intersects the boundary strata of Cα. We shall see below that a system of privilegedneighborhoods is automatically a system of weakly privileged neighborhoods.

A fundamental system of neighborhoods, C = Cα, is a universal system of privileged neigh-borhoods for h at 0 if and only if for every good stratification, G, for h at 0, there exists an openneighborhood W of the origin such that Cα ∈ C | Cα ⊆ W is a system of privileged neighborhoodsfor h at 0 with respect to G.

APPENDIX C 223

One should note that the set of closed balls centered at the origin is a universal system ofprivileged neighborhoods for h, regardless of the function h – this is a very “universal” system,and this may seem like the more natural notion. This, however, seems to be too restrictive.Universal for h simply means that, locally, the fundamental system is privileged independent ofthe choice of good stratification for the particular function h.

Proposition C.4. Suppose that C = Cα is a system of privileged neighborhoods for h at 0.Then, C = Cα is a system of weakly privileged neighborhoods for h at 0 and, hence, for all Cα,for all small δ > 0, Cα ∩ h−1(∂Dδ)

h−→ ∂Dδ is a proper, stratified submersion and is thus a locallytrivial fibration. The fibre-homotopy-type of this fibration is independent of the choice of the systemof weakly privileged neighborhoods, C, for h, the choice of Cα, and the choice of small δ > 0.

Proof. The proof is essentially that of Le in [Le 4]. Let G be a good stratification for h at theorigin with respect to which C is a system of privileged neighborhoods. Pick a Cα in C. We shall

actually show that there exists ε > 0 such that Cα ∩ h−1(Dε − 0) h−→

Dε − 0 is a proper, stratified

submersion. It follows that, for all δ with 0 < δ < ε, Cα ∩ h−1(∂Dδ)h−→ ∂Dδ is a proper, stratified

submersion and, hence, a locally trivial fibration with fibre-homotopy-type independent of thechoice of δ. This certainly shows that C is a system of weakly privileged neighborhoods for h at 0.

Suppose to the contrary that no matter how small we choose ε > 0 it is not the case that

Cα ∩ h−1(Dε − 0) h−→

Dε − 0 is a proper, stratified submersion. As each Cα is compact, clearly

this map is always proper. So, by the local finiteness of the stratification, there must exist asingle Whitney stratum, S, of Cα and a sequence of points pi ∈ S such that the pi convergeto some point p ∈ V (h), Tpi

S converges to some T , TpiV (h − h(pi)) converges to some T , and

TpiS ⊆ Tpi

V (h− h(pi)). Let G denote the good stratum of G which contains p and let R denotethe Whitney stratum of Cα which contains p.

As TpiS ⊆ Tpi

V (h− h(pi)), we must have that T ⊆ T . By the Thom condition, TpG ⊆ T . ByWhitney’s condition a), TpR ⊆ T . Hence, TpR and TpG are both contained in T – a contradictionas R and G intersect transversely.

Thus, there exists ε > 0 such that for all δ with 0 < δ < ε, Cα ∩ h−1(∂Dδ)h−→ ∂Dδ is a proper,

stratified submersion and, hence, a locally trivial fibration with fibre-homotopy-type independentof the choice of δ. To see that the fibre-homotopy-type is independent of the choice of C and thechoice of Cα, one may once again compare with the standard system of Milnor neighborhoodsand then use the theorem of Dold [Hu, p.209], since we know that the inclusion of each fibre is ahomotopy-equivalence by the proof of C.1. We leave the details to the reader.

Note that we have the implications: Cα is a universal system ⇒ for all good stratifications G,Cα is a privileged system with respect to G ⇒ Cα is a privileged system ⇒ Cα is a weaklyprivileged system ⇒ Cα is a Milnor system.

Definition C.5. If C = Cα is a system of Milnor neighborhoods for h at 0, then a Milnor pair

for h at 0 is a pair (Cα,Dδ) such that for all ξ ∈

Dδ − 0, Cα ∩ V (h − ξ) has the homotopy-type

of the Milnor fibre. If, in addition, C is a system of weakly privileged neighborhoods, then we alsomake the requirement that Cα ∩ h−1(∂Dδ)

h−→ ∂Dδ is a proper, stratified submersion.

224 DAVID B. MASSEY

We now wish to consider an analytic function f : (D × U ,

D × 0) → (C, 0) where

D is an open

complex disc centered at the origin and U ⊆ Cn+1. We use the coordinates (t, z0, . . . , zn) forD×U .

We distinguish the t-coordinate because we will either be considering the particular hyperplaneslice V (t) or because we will be interested in the family ft(z0, . . . , zn) := f(t, z0, . . . , zn).

Proposition C.6. Suppose that V (t) is prepolar for f at the origin with respect to a good stratifi-cation G, and let Cα be a system of privileged neighborhoods with respect to the good stratificationG ∩ V (t) for f|V (t)

. Then, there exits an open neighborhood, W , of the origin in V (t) such that,for all Cα ⊆ W , there exists τα > 0 such that

i) there exists ω > 0 such that

Dτα

× ∂Cα ∩ Ψ−1((

Dω − 0)×

Dτα

)yΨ := (f, t)(

Dω − 0)×

Dτα

is a proper, stratified submersion;

ii) for all δ with 0 < δ < τα, there exists ξ > 0 such that

Dδ × Cα ∩ f−1(Dξ − 0)yf

Dξ − 0

is a proper, stratified submersion, where the strata are the cross-product strata ofDδ ×Cα together

with those of ∂Dδ × Cα; and

iii) Dδ × Cα | 0 < δ < τα is a system of Milnor neighborhoods for f at the origin and hence,by ii), is in fact a system of weakly privileged neighborhoods.

Proof. There exists an open neighborhood of the origin inD × U of the form

Dη ×W such that

(Dη × W ) ∩ Σf ⊆ V (f). As V (t) is prepolar, we may assume that G is defined inside

Dη × W

and that V (t) transversely intersects all strata of G, other than the origin, insideDη×W . Finally,

as V (t) is prepolar, we may use Theorem 1.28 to conclude that γ1f,t(0) exists and, hence, we may

selectDη ×W so that (0×W ) ∩ Γ1

f,t ⊆ 0. Let Cα ⊆ W .

i) This follows the proof of Proposition 2.1 of [Le1], applied to each stratum of ∂Cα. Supposethe contrary. Then, we would have a stratum S of Cα and a sequence of points pi not in V (f) butin C× S such that pi = (ti,qi) → p = (0,q) ∈ V (t) ∩ V (f) and such that

(*) TpiV (f − f(pi), t− ti) + Tpi

(C× S) 6= Cn+2.

APPENDIX C 225

(That TpiV (f−f(pi), t−ti) exists is not completely trivial – it follows from the assumptions madein the preceding paragraph.) Let G denote the good stratum of V (f) containing p. Note that Gcannot be the point-stratum 0 as p is contained in 0× ∂Cα. Let R denote the stratum of ∂Cα

containing q.By taking a subsequence if necessary, we may assume that Tpi

V (f − f(pi)) converges to someT and that Tqi

S converges to some T . By the Thom condition, TpG ⊆ T and, by Whitney’scondition a), TqR ⊆ T . Furthermore, as V (t) is prepolar, V (t) transversely intersects G at p.

Thus,Tpi

(C× S) → C× T

andTpiV (f − f(pi), t− ti) → T ∩ TpV (t).

Also, we have thatTp(G ∩ V (t)) = TpG ∩ TpV (t) ⊆ T ∩ TpV (t)

and we know thatTp(G ∩ V (t)) + Tp(0× S) = 0× Cn+1,

as Cα is a system of privileged neighborhoods with respect to G ∩ V (t). It follows at once that

T ∩ TpV (t) + C× T = Cn+2,

but this contradicts (∗). This proves i).

ii) That f can be made a submersion onDδ ×

Cα follows from the fact that

Dδ ×

Cα ⊆

Dη ×W

and (Dη ×W ) ∩ Σf ⊆ V (f).

That f can be made a stratified submersion onDδ × ∂Cα and on ∂Dδ × ∂Cα is exactly the

argument of i).

Thus, what remains to be shown is that f can be made a submersion on the stratum ∂Dδ×Cα.

By Theorem 1.28 and Proposition 1.23, dim0(Γ1f,t ∩ V (f)) 6 0 and thus we may assume that

Γ1f,t ∩ V (f) ∩ (∂Dδ × Cα) is empty.As Γ1

f,t ∩ (∂Dδ × Cα) is compact, |f | obtains a minimum, ξ > 0, on Γ1f,t ∩ (∂Dδ × Cα). Now,

consider the critical points of f restricted to ∂Dδ ×Cα that occur in f−1(

Dξ − 0). These points

occur precisely on

Γ1f,t ∩ (∂Dδ ×

Cα) ∩ f−1(

Dξ − 0)

which we know is empty. This proves ii).

iii) We first need two results.

a) for all ω1, ω2 with 0 < ω1 < ω2 < τα, there exists ξ > 0 such that

C× Cα ∩ Φ−1((Dξ − 0)× [ω2

1 , ω22 ])yΦ := (f, |t|2)

(Dξ − 0)× [ω2

1 , ω22 ]

226 DAVID B. MASSEY

is a proper, stratified submersion and thus, for all η ∈Dξ − 0, the inclusion

(Dω1 × Cα) ∩ V (f − η) → (Dω2 × Cα) ∩ V (f − η)

is a homotopy-equivalence; and

b) if Cα ⊆Cβ , then there exist τ, ξ > 0 such that, for all δ ∈

Dτ−0 and η ∈

Dξ−0, the inclusion

(Dδ × Cα) ∩ V (f − η) → (Dδ × Cβ) ∩ V (f − η)

is a homotopy-equivalence.

Assuming a) and b) for the moment, we proceed with the proof. Suppose that Dσ×Cα ⊆ Dρ×Cβ .

By b), for all small, non-zero δ and η,

(Dδ × Cα) ∩ V (f − η) → (Dδ × Cβ) ∩ V (f − η)

is a homotopy-equivalence. If we select δ so small that Dδ is contained in both Dσ and Dρ, thenwe may apply a) twice to obtain that, for all small, non-zero η,

(Dδ × Cα) ∩ V (f − η) → (Dσ × Cα) ∩ V (f − η)

and(Dδ × Cβ) ∩ V (f − η) → (Dσ × Cβ) ∩ V (f − η)

are homotopy-equivalences. The conclusion that

(Dδ × Cα) ∩ V (f − η) → (Dρ × Cβ) ∩ V (f − η)

is a homotopy-equivalence now follows immediately by combining the three previous homotopy-equivalences.

We now prove a) and b).

Proof of a): That Φ is a stratified submersion on C× ∂Cα is once again exactly the proof of i).

That Φ is a submersion on C×Cα is similar to our argument in ii): as dim0(Γ1

f,t ∩ V (f)) 6 0, wemay assume that

Γ1f,t ∩ V (f) ∩ ((Dω2 −

Dω1)× Cα)

is empty. Therefore, by compactness, |f | obtains a minimum, ξ > 0, on Γ1f,t ∩ ((Dω2 −

Dω1)×Cα).

Now, consider the critical points of Φ restricted to C×Cα that occur in Φ−1((

Dξ − 0)× [ω2

1 , ω22 ]).

These points occur precisely in

Γ1f,t ∩ f−1(

Dξ − 0) ∩ ((Dω2 −

Dω1)× Cα)

which we know is empty. This proves a).

Proof of b): Let Cα ⊆Cβ . Let τ be so small that inside Dτ × Cβ all points of Γ1

f,t occur in

Dτ ×Cα. Further, choose τ < minτα, τβ so that we may apply i) in both cases. Choose ξ so

APPENDIX C 227

small that (Cα, Dξ) and (Cβ , Dξ) are Milnor pairs for f|V (t)and so small that we may apply i) to

both Dτ × ∂Cα and Dτ × ∂Cβ over (Dξ − 0)×

Dτ . Fix some δ ∈

Dτ − 0 and η ∈

Dξ − 0.

By i) or ii), V (f − η) transversely intersects all the strata of C × ∂Cα and C × ∂Cβ , so mayWhitney stratify (C × Cβ) ∩ V (f − η) by taking as strata the intersection of V (f − η) with each

of C×Cα, C× (

Cβ − Cα), and the strata of C× ∂Cα and C× ∂Cβ .

As Cα ∩ V (f|V (t)− η) → Cβ ∩ V (f|V (t)

− η) is a homotopy-equivalence and V (f|V (t)) − η)

transversely intersects ∂Cα and ∂Cβ , for all small Dµ we must have that

(Dµ × Cα) ∩ V (f − η) → (Dµ × Cβ) ∩ V (f − η)

is also a homotopy-equivalence. We wish to pass from Dµ to Dδ by considering the function |t|2on the stratified space (C× Cβ) ∩ V (f − η) (with the stratification given above).

By i), |t|2 has no critical points on the strata of (C×∂Cα)∩V (f − η) and (C×∂Cβ)∩V (f − η)

when |t| < δ. In addition, the critical points on the interior strata, (C ×Cα) ∩ V (f − η) and

(C× (Cβ −Cα))∩V (f − η), occur on the polar curve and, hence, by our earlier requirement, these

critical points all occur in C×Cα. Therefore, using stratified Morse theory [G-M2] together with

the homotopy-equivalence lemma 3.7 of [Mi2], we find that the inclusion

(Dδ × Cα) ∩ V (f − η) → (Dδ × Cβ) ∩ V (f − η)


For a family of analytic functions ft : (U ,0) → (C, 0), we are interested in how the Milnor fibreand fibration “jump” as we move from small non-zero t to t = 0. Hence, we make the followingdefinition.

Definition C.7. If we are considering the family ft : (U ,0) → (C, 0), we refer to i) of C.6 bysaying that the family satisfies the conormal condition with respect to Cα.

The point of this condition is that it says that the Milnor fibration of f0 lifts trivially in thefamily ft on the boundary of the neighborhoods Cα.

Definition C.8. The Thom set at the origin, Tf , is the set of (n+1)-planes which occur as limitsat the origin of the tangent spaces to level hypersurfaces of f , i.e. T ∈ Tf if and only if there exists

a sequence of points pi inD× U − Σf such that pi → 0 and T = lim Tpi

V (f − f(pi)).Equivalently, Tf is the fibre over the origin in the Jacobian blow-up of f (see [H-L]). Tf is thus

a closed algebraic subset of the Grassmanian Gn+1(Cn+2) = the projective space of (n + 1)-planesin Cn+2.

Proposition C.9. Suppose that V (t) is a prepolar slice for f at 0 or that V (t) = T0V (t) 6∈ Tf .Then,

i) dim0Γ1f,t 6 1, and

ii) the family ft satisfies the conormal condition with respect to any universal system of privilegedneighborhoods, C, for f0 at 0.

228 DAVID B. MASSEY

Moreover, whenever i) and ii) are satisfied, there is an inclusion of the Milnor fibre Fft0 ,0 intothe Milnor fibre Ff0,0 for all small non-zero t0; the homotopy-type of this inclusion is independentof the choice of t0 and the choice of the universal system of privileged neighborhoods, C.

Proof. That there is such an inclusion whenever i) and ii) are satisfied is standard. One considersthe map Ψ := (f, t) and its restriction

(Dτ × C) ∩Ψ−1((

Dξ − 0)× Dτ

)yΨ(

Dξ − 0)× Dτ

for appropriately small choices of C ∈ C, ξ, and τ . By the conormal condition, this is a stratifiedsubmersion on the boundary. As dim0Γ1

f,t 6 1, the discriminant of Ψ, Ψ(Γ1

f,t

), is also at most

one-dimensional. Thus, we may lift a path in the base which avoids the discriminant to get adiffeomorphism between the Milnor fibre of f0 and C ∩V (ft0 −η) for all small t0 and for all η with0 < |η| |t0|. And, though we do not know that C is a system of privileged neighborhoods forft0 , we may still take a small enough ball inside C to obtain the desired inclusion, which is clearlyindependent of the choice of t0.

That the inclusion is independent of the choice of privileged neighborhoods follows similarly.Suppose that C′ is second universal system of privileged neighborhoods for f0. Let C ∈ C and let

C ′ ∈ C′ be such that C ′ ⊆C, and such that C and C ′ are small enough to give the Milnor fibre, i.e.

for all small non-zero ξ, the inclusion of C ′∩V (f0−ξ) into C∩V (f0−ξ) is a homotopy-equivalencewhere both spaces are homotopy-equivalent to the Milnor fibre of f0 at the origin. Then, as above,over a curve which avoids the discriminant, we have a proper, stratified submersion – where thestrata are those of Dτ × ∂C together with those of Dτ × ∂C ′ plus the interior.

Hence, the homotopy-equivalence C ′∩V (f0−ξ) → C∩V (f0−ξ) lifts to a homotopy-equivalenceC ′ ∩ V (ft0 − ξ) → C ∩ V (ft0 − ξ). The independence statement now follows easily.

We must still show that if V (t) is a prepolar slice for f at 0 or V (t) = T0V (t) 6∈ Tf , then i) andii) hold.

If V (t) is prepolar for f at 0, then i) follows from Theorem 1.28 and ii) follows from C.6.i. IfV (t) 6∈ Tf , then clearly Γ1

f,t is empty near the origin. It remains for us to show that if V (t) 6∈ Tf ,then the family ft satisfies the conormal condition with respect to any universal system of privilegedneighborhoods, C, for f0.

If V (t) 6∈ Tf , then V (t) certainly transversely intersects the smooth part of V (f) in a neigh-borhood of the origin. Hence, we may use Proposition C.2 to conclude that there exists a goodstratification, G, for f at the origin such that the strata of G which are contained in V (t) form agood stratification for f0 at the origin. The proof now proceeds like that of C.6.i.

Suppose to the contrary that, for arbitrarily small Cα in C, there exists a stratum S of ∂Cα anda sequence of points pi not in V (f) but which are in C× S such that pi = (ti,qi) → p := (0,q) ∈V (t) ∩ V (f) and such that

(*) TpiV (f − f(pi), t− ti) + Tpi

(C× S) 6= Cn+2.

Let G denote the good stratum of V (f) which contains p. Note that G is contained in V (t) by thenature of our good stratification and that G cannot be simply the stratum consisting of the originsince p is contained in 0× ∂Cα. Let R denote the stratum of ∂Cα containing q.

APPENDIX C 229

By taking a subsequence if necessary, we may assume that TpiV (f − f(pi)) converges to someT and that TqiS converges to some T . By the Thom condition, TpG ⊆ T and, by Whitney’scondition a), TqR ⊆ T . Furthermore, as V (t) 6∈ Tf , we may assume that p is close enough to theorigin that T 6= V (t).

Thus, Tpi(C × S) → C × T and Tpi

V (f − f(pi), t − t − i) → T ∩ TpiV (t). Also, we have

that TpG = TpG ∩ TpV (t) ⊆ T ∩ TpV (t), and we know that TpG + Tp(0 × S) = 0 × Cn+1, asCα is a system of privileged neighborhoods with respect to G ∩ V (t). It follows at once thatT ∩ TpV (t) + C× T = Cn+2 – which contradicts (∗).

If the polar curve, Γ1f,t, is empty, then the map Ψ which appears in the proof of Proposition

C.9 is a stratified submersion over the entire base space and so, for all small t0 6= 0, we havea fibre-prerserving inclusion of the total space of the Milnor fibration of ft0 into the total spaceof the Milnor fibration of f0. Moreover, exactly as above, this inclusion is independent – up tohomotopy – of all of the choices made. By the theorem of Dold (see [Hu, p. 209]), this inclusion isa fibre homotopy-equivalence if and only if the inclusion of each fibre is a homotopy-equivalence.Therefore, we make the following definitions.

Definition C.10 Whenever i) and ii) of C.9 hold, we say that the family, ft, satisfies the universalconormal condition.

If ft satisfies the universal conormal condition, we say that ft has the homotopy Milnor fibrelifting property if and only if the inclusion of C.9 is a homotopy-equivalence.

If ft satisfies the universal conormal condition, we say that ft has the homology Milnor fibrelifting property if and only if the inclusion of C.9 induces isomorphisms on all integral homologygroups.

The family, ft, has the homotopy Milnor fibration lifting property if and only if ft has thehomotopy Milnor fibre lifting property and Γ1

f,t = ∅ in a neighborhood of the origin. This definitionmakes sense in light of our above discussion concerning the result of Dold.

One may also discuss the Milnor fibre and Milnor fibration up to diffeomorphism if one is willingto restrict consideration to the standard universal system of Milnor neighborhoods, namely the setof closed balls centered at the origin. In this case, we may use the h-cobordism Theorem and thepseudo-isotopy result of Cerf [Ce] to translate the homotopy information into smooth information– provided that we are in a sufficiently high dimension and that the Milnor fibre and its boundaryare sufficiently connected. More specifically, if U is an open neighborhood of the origin in Cn+1,ft : (U ,0) → (C, 0) has the homotopy Milnor fibration lifting property, n > 3, and the Milnor fibreand its boundary are simply-connected for each ft for all small t, then the diffeomorphism-type ofthe Milnor fibrations is constant in the family near t = 0. This connectedness condition can berealized by requiring n− dim0Σf0 > 3 (see [K-M] and [Ra]).

We wish to state the diffeomorphism results discussed above precisely. First, we give withoutproof Cerf’s pseudo-isotopy result in the form that we shall need it.

Lemma C.11. Let X be a smooth manifold with boundary ∂X = X0∪X1 and let π : X → S1 bea smooth locally trivial fibration over a circle with fibre diffeomorphic to M × [0, 1], where M is aclosed, simply-connected, smooth manifold of dimension > 5.

Then, the restriction of π to X0 is a smooth locally trivial fibration with fibre diffeomorphic to

230 DAVID B. MASSEY

M , and there exists a commutative diagram

(X, X0)∼=−−−−→

diffeo.(X0 × [0, 1], X0 × 0)

π π|X0 pr1

S1

where the diffeomorphism is the identity on X0 = X0 × 0.

Proposition C.12. Let U be an open neighborhood of the origin in Cn+1. Suppose that the familyft : (U ,0) → (C, 0) has the homotopy Milnor fibration lifting property and n−dim0Σf0 > 3. Then,the diffeomorphism-type of the Milnor fibrations of ft at the origin is independent of t for all smallt.

Proof. We shall use the notation from the proof of Proposition C.9. We fix the universal systemof privileged neighborhoods to be the collection of closed balls centered at the origin.

As ft has the homotopy Milnor fibration lifting property, the polar curve Γ1f,t is empty and so the

map Ψ in the proof of Proposition C.9 is a proper stratified submersion. Hence, for 0 < ξ, |t0| ε,

the Milnor fibration of f0 is diffeomorphic to Bε ∩ f−1t0 (∂Dξ)

ft0−−→ ∂Dξ. The problem, of course, isthat Bε may be too large a ball in which to define the Milnor fibration of ft0 . Let F and E denotethe fibre and the total space, respectively, of this previous fibration.

Let F ′ denote the Milnor fibre of ft0 at the origin, where we again use closed balls for the Milnorneighborhoods. Let E′ denote the total space of the Milnor fibration ft0 .

As ft has the homotopy Milnor fibration lifting property, the inclusion of E′ into E induces aninclusion F ′ → F which is a homotopy-equivalence.

Since n − dim0Σf0 > 3, F , F ′, ∂F , and ∂F ′ are simply-connected (see [Ra]). Combining thiswith the fact that F ′ → F is a homotopy-equivalence, we may duplicate the argument of Le and

Ramanujam [L-R] to conclude that ∆T := E−

E′ is the total space of a differentiable fibration over

∂Dξ with projection ft0 and fibre F −F ′ which is diffeomorphic to ∂F × [0, 1] via the h-cobordism

theorem.Now, by Lemma C.11, ∆T

ft0−−→ ∂Dξ is diffeomorphic to

∂E′ × [0, 1]ft0pr1−−−−−→ ∂Dξ × 0

by a diffeomorphism which is the identity on ∂E′ = ∂E′×0. Combining this with a fibred collar

of ∂E′ in E′, we conclude that E′ ft0−−→ ∂Dξ is diffeomorphic to Eft0−−→ ∂Dξ, which we already know

is diffeomorphic to the Milnor fibration of f0 at 0.

We now wish to prove a fundamental result – namely, that if we have a family ft in which theMilnor fibrations of a hyperplane slice are independent of t and the number of handles attachedin passing from the Milnor fibre of the hyperplane slice to the entire Milnor fibre is constant, thenthe Milnor fibrations are constant in the family. Despite the fact that the dimension of the criticalloci is allowed to be arbitrary, the argument is exactly that which we used in [Mas3] where thecritical loci were all one-dimensional.

APPENDIX C 231

Theorem C.13. Let W be an open neighborhood of the origin in Cn+2 and let gt : (W,0) → (C, 0)be an analytic family. Let s denote dim0Σg0. Assume that gt satisfies the universal conormal con-dition and that L is a linear form such that V (L) is prepolar for ft at the origin for all small t andsuch that gt|V (L)

satisfies the universal conormal condition. Suppose further that(Γ1

gt,L· V (gt)

)0

is constant for all small t.Under the above assumptions, if gt|V (L)

has the homology Milnor fibre lifting property, then gt

has the homology Milnor fibre lifting property.Moreover, if s 6 n − 1 and gt|V (L)

has the homotopy Milnor fibre lifting property, then gt hasthe homotopy Milnor fibre lifting property.

Proof. This is actually quite trivial. Let F0 and Ft0 denote the Milnor fibre of g0 and gt0 forsmall non-zero t0, respectively. The Milnor fibres of g0|V (L)

and gt0 |V (L)are then F0 ∩ V (L) and

Ft0 ∩ V (L), respectively. Let γ denote the constant value of(Γ1

gt,L· V (gt)

)0.

As gt and gt|V (L)satisfy the universal conormal condition, we may repeat the argument of

Proposition C.9 – lifting a path in the base which avoids the discriminants of both (gt, t) and(gt|V (L)

, t) – to obtain compatible inclusions Ft0 → F0 and Ft0 ∩ V (L) → F0 ∩ V (L).Suppose that gt|V (L)

has the homology Milnor fibre lifting property, i.e. Ft0 ∩V (L) → F0∩V (L)induces isomorphisms on homology. We wish to show that Ft0 → F0 induces isomorphisms onhomology. We will accomplish this by showing that H∗(F0, Ft0) = 0.

By considering the homology long exact sequence of the triple (F0, F0 ∩ V (L), Ft0 ∩ V (L)), wefind that Hi(F0, Ft0 ∩V (L)) ∼= Hi(F0, F0 ∩V (L)) for all i. By Le’s attaching result (Theorem 0.9)or Theorem 3.1, Hi(F0, F0 ∩ V (L)) = 0 unless i = n + 1 and Hn+1(F0, F0 ∩ V (L)) ∼= Zγ .

Now, we are going to consider the homology long exact sequence of the triple (F0, Ft0 , Ft0∩V (L)).From the last paragraph, we know that Hi(F0, Ft0 ∩ V (L)) = 0 unless i = n + 1. In addition,Hi(Ft0 , Ft0 ∩ V (L)) = 0 unless i = n + 1. Moreover,

Hn+1(F0, F0 ∩ V (L)) ∼= Hn+1(Ft0 , Ft0 ∩ V (L)) ∼= Zγ .

Thus, in the long exact sequence of the triple (F0, Ft0 , Ft0 ∩V (L)), all terms are zero except in theportion

0 → Zγ → Zγ → Hn+1(F0, Ft0) → 0.

But, as in the proof of the result of Le and Ramanujam [L-R], Hn+1(F0, Ft0) is free Abelian,since F0 is obtained from Ft0 by attaching handles of index less than or equal to n + 1. (Oneconsiders the function distance squared from the origin and lets the function grow from the smallball used to define Ft0 out to the ball used to define F0. One hits no critical points of index greaterthan or equal to n + 2.) Thus, Hn+1(F0, Ft0) = 0 and we have proved the first claim.

The second claim follows from the first, since s 6 n− 1 guarantees that F0 and Ft0 are simply-connected, and then we apply the Whitehead Theorem.

There are two more big results which we need to prove in this appendix – both deal withsuspending singularities (see Chapter I.8). The first result is that there exists a universal system ofprivileged neighborhoods of a particularly nice form for the function h + wj , where w is a variabledisjoint from those of h. The second result says, with a few extra assumptions, that the constancyof the Milnor fibrations in the family ft implies the constancy of the Milnor fibrations in the familyft + wj , where, again, w is disjoint from the variables of ft. This second result seems reasonable

232 DAVID B. MASSEY

since the result of Proposition II.8.1 is that the Milnor fibre of ft + wj at the origin is homotopy-equivalent to one-point union of j − 1 copies of the Milnor fibre of ft at the origin. However, bothof these results are technical nightmares.

Let U be an open neighborhood of the origin in Cn+1 and let h : (U ,0) → (C, 0) be an analyticfunction. Let j > 2 and define h(w, z) := h(z)+wj . We wish to show that the set Dω×B2n+2

ε | 0 <

ω ε is a universal system of privileged neighborhoods for h at the origin. Note that we maynot use C.6 to conclude that Dω × B2n+2

ε | 0 < ω ε is even weakly privileged since the sliceV (w) contains the entire critical locus of h + wj and, hence, is certainly not prepolar for h + wj .Of course, the actual argument is very similar to the proof of Proposition C.6.

Proposition C.14. The set Dω × B2n+2ε | 0 < ω ε is a universal system of privileged

neighborhoods for h + wj at the origin.

Proof. As j > 2, Σ(h + wj) = 0×Σh. Fix any good stratification, G, for h + wj at the origin inCn+2.

Let ε0 > 0 be so small that the critical locus of the map h inside B2n+2ε0 is contained in V (h)

and so small that, for all ε with 0 < ε 6 ε0,

0 × ∂B2n+2ε transversely intersects all strata of 0 × ΣV (h) inside 0 × Cn+1,

∂B2n+4ε transversely intersects all strata of G (we write ∂B2n+4

ε t G), and

∂B2n+2ε transversely intersects all strata of some good stratification for h at 0.

This last condition guarantees, for all small non-zero ζ, that

(*) ∂B2n+2ε t V (h− h(ζ)).

Now fix an ε between 0 and ε0. We wish to show that there exists ωε > 0 such that, for all ωwith 0 < ω 6 ωε, we have:

a)Dω × ∂B2n+2

ε t G;

b) ∂Dω ×B2n+2

ε t G;

c) ∂Dω × ∂B2n+2ε t G.

After we show this, it will still remain to show that, if Dω1 ×Bε1 ⊆ Dω2 ×Bε2 , then, for all smallnon-zero t, the inclusion

Dω1 ×Bε1 ∩ V (h + wj − t) → Dω2 ×Bε2 ∩ V (h + wj − t)


Proof of a): Clearly, as 0 × ∂B2n+2ε transversely intersects all strata of 0 × ΣV (h) inside

0×Cn+1, C×∂B2n+2ε transversely intersects all singular strata of V (h+wj). Suppose, however,

that no matter how small we pick ω > 0, we still have a point in the smooth stratum, S :=

V (h + wj)− 0 × ΣV (h), where S does not transversely intersectDω × ∂B2n+2

ε .Then, we would have a sequence pi := (wi,qi) ∈ C × ∂Bε contained in S such that pi →

p := (0,q) ∈ 0 × ∂Bε, TpiS ⊆ Tpi(C × ∂Bε), TpiS converges to some T , and Tpi(C × ∂Bε) →C× Tq(∂Bε). Let S′ denote the stratum of G containing p.

APPENDIX C 233

By the Thom condition, TpS′ ⊆ T (this is true because T comes from the smooth stratum –we are not assuming Whitney conditions hold between the strata). Hence,

TpS′ ⊆ T ⊆ C× Tq(∂Bε) = Tp(∂B2n+4ε ),

where this last equality is true because the w-coordinate of p is 0. But, this contradicts the factthat ∂B2n+4

ε t G. This proves a).

Before we prove b) and c), note that if w ∈ ∂Dω, then w 6= 0 and, hence, the only stratum of G

which ∂Dω×B2n+2

ε and ∂Dω×∂B2n+2ε intersect is the smooth stratum S := V (h+wj)−0×ΣV (h).

Proof of b): Actually, we show, regardless of the size of ω > 0, that ∂Dω ×B2n+2

ε t S.

For if not, we would have p := (w,q) ∈ S such that w 6= 0 and TpV (h+wj) ⊆ Tp(∂D|w|×B2n+2

ε ).This implies that

∂h

∂z0 |q= · · · = ∂h

∂zn |q= 0,

i.e. that q ∈ Σh. Recalling that we chose ε such that Bε ∩ Σh ⊆ V (h), we see that h(q) = 0.However, this contradicts that h(q) = −wj 6= 0. This proves b).

Proof of c): Suppose not. Then, we would have a sequence pi := (wi,qi) ∈ S ∩ (C × ∂B2n+2ε )

with wi 6= 0, pi → p = (0,q) ∈ 0 × ∂Bε, and such that

TpiV (h + wj) + Tpi

(∂D|wi| × ∂Bε) 6= Cn+2.

This implies that TqiV (h−h(qi)) ⊆ Tqi

(∂B2n+2ε ), while h(qi) = −wj

i approaches – but is unequalto – zero. This, however, is impossible by (∗). This proves c).

We must still prove the homotopy-equivalence statement. In a manner completely similar to theproofs of a), b), and c) above, one can easily show, using the Thom condition, that the followingstatements are true:

d) for all ε with 0 < ε 6 ε0, if ω1 is between 0 and ωε, then for all ω2 with 0 < ω2 6 ω1, thereexists ξ > 0 such that (

C×B2n+2ε

)∩Ψ−1

((Dξ − 0)× [ω2

2 , ω21 ]

)y Ψ := (h + wj , |w|2)

(Dξ − 0)× [ω22 , ω2

1 ]

is a proper, stratified submersion and therefore

d′)(Dω2 ×Bε

)∩ (h + wj)−1(Dξ − 0) →

(Dω1 ×Bε

)∩ (h + wj)−1(Dξ − 0)

is a fibre-homotopy equivalence between total spaces (where the projection in each case is theobvious map h + wj).

e) if 0 < ε2 < ε1 6 ε0, then for all small, non-zero ω, there exists ξ > 0 such that

234 DAVID B. MASSEY(Dω × Cn+1

)∩ Φ−1

((Dξ − 0)× [ε22, ε

21]

)y Φ := (h + wj , |z|2)

(Dξ − 0)× [ε22, ε21]

is a proper, stratified submersion and therefore

e′)(Dω ×Bε2

)∩ (h + wj)−1(Dξ − 0) →

(Dω ×Bε1

)∩ (h + wj)−1(Dξ − 0)

is a fibre-homotopy equivalence.

Now, suppose that we have Dω1 ×Bε1 ⊆ Dω1 ×Bε1 , where 0 < ε2 < ε1 6 ε0, 0 < ω2 < ω1 6 ωε1 ,and ω2 < ωε2 . We shall show that, for all small ξ > 0,(

Dω2 ×Bε2

)∩ (h + wj)−1(Dξ − 0) →

(Dω1 ×Bε1

)∩ (h + wj)−1(Dξ − 0)

is a fibre-homotopy equivalence.By applying e), we know that, for all small ω > 0, there exists ξ 6= 0 such that e′) holds. On

the other hand – by applying d) twice – for all small ω > 0, there exists ξ 6= 0 such that(Dω ×Bε2

)∩ (h + wj)−1(Dξ − 0) →

(Dω2 ×Bε2

)∩ (h + wj)−1(Dξ − 0)

and (Dω2 ×Bε1

)∩ (h + wj)−1(Dξ − 0) →

(Dω1 ×Bε1

)∩ (h + wj)−1(Dξ − 0)

are fibre-homotopy equivalences.The desired conclusion follows from the two homotopy-equivalences above together with e′).

For the final results of this appendix, we return to the setting of families of analytic functions.Again, U will denote an open neighborhood of the origin in Cn+1 and ft : (U ,0) → (C, 0) will bean analytic family. We continue with w being a variable disjoint from those of ft and with j > 2.Recall from C.8 that Tf denotes the Thom set of f at the origin.

We need the following easy lemma:

Lemma C.15. If V (t) 6∈ Tf , then V (t) 6∈ Tf+wj .

Proof. This is completely trivial. We leave it as an exercise.

Proposition C.16. Suppose that V (t) 6∈ Tf and that the family ft + wj has the homology Milnorfibre lifting property. Then, Γ1

f,t = ∅ near the origin and ft has the homology Milnor fibre liftingproperty.

Moreover, if dim0Σf0 6 n−2, V (t) 6∈ Tf , and the family ft +wj has the homotopy Milnor fibrelifting property, then ft has the homotopy Milnor fibration lifting property.

Proof. The second claim follows immediately from the first claim, since the condition dim0Σf0 6n− 2 implies that the Milnor fibres are simply-connected. Also, since V (t) 6∈ Tf , we immediately

APPENDIX C 235

have that Γ1f,t = ∅ near the origin. What we need to prove is that ft has the homology Milnor

fibre lifting property.Fix a good stratification G for f0 at the origin. We must now make many choices.

1) Let (Bε0 , Dλ0) be a Milnor pair for f0 such that2) Bε0 ∩ Σf0 ⊆ V (f0), and3) ∂Bε0 transversely intersects the strata of G.

From C.9, we know that the conormal condition holds, and so we may pick η, τ > 0 such that

4) the map G := (f, t) restricted to C× ∂Bε0 has no critical values in (Dη − 0)× Dτ .

Using C.14, we may also choose ω0, ξ0 > 0 such that

5) (Dω0 ×Bε0 , Dξ0) is a Milnor pair for f0 + wj , where6) ωj

0 < η, and7) all of the obvious Whitney strata of Dω0 × Bε0 transversely intersect all of the strata in thegood stratification for f0 + wj which is induced by G (as given in Proposition 8.3).

Now, as V (t) 6∈ Tf , Lemma C.15 tells us that V (t) 6∈ Tf+wj . Hence, ft + wj satisfies the universalconormal condition and so, for all small ν 6= 0 and all small t1,

8)(Dω0 ×Bε0

)∩ V (ft1 + wj − ν) is diffeomorphic to Ff0+wj ,0.

We select t1 so that

9) t1 is in Dτ ,10) Γ1

f,t ∩(D|t1| ×Bε0

)= ∅, and

11) Σf ∩(D|t1| ×Bε0

)⊆ V (f).

As t1 is in Dτ , there exists λ′0 such that

12) for all γ with 0 < γ < λ′0, Bε0 ∩ V (ft1 − γ) is diffeomorphic to Ff0,0.13) Now, let (Bε, Dλ) be a Milnor pair for ft1 with14) ε < ε0 and λ < λ′0.

Then, there exist ω, ξ > 0 such that

15) (Dω ×Bε, Dξ) is a Milnor pair for ft1 + wj , where we assume that16) ωj < minλ, λ′0, ω

j0 and

17) ξ < minξ0, ωj , η − ωj

0, where η − ωj0 > 0 by 6).

Finally, we select ν in 8) so small that

18) 0 < |ν| < minη − ωj0, λ− ωj , ξ.

Now that we have made all of these choices, we are ready to begin the intuitive part of theproof.

We have the inclusions

Fft1+wj ,0∼= (Dω ×Bε) ∩ V (ft1 + wj − ν) i−→ (Dω ×Bε0) ∩ V (ft1 + wj − ν)

l−→ (Dω0 ×Bε0) ∩ V (ft1 + wj − ν) ∼= Ff0+wj ,0,

236 DAVID B. MASSEY

where we are assuming that l i induces isomorphisms on homology. We will first show thatl induces isomorphisms on homology and, hence, so does i. Actually, we will show that l is ahomotopy-equivalence.

We accomplish this by showing that

(*)((Dω0 −

Dω)×Bε0

)∩ V (ft1 + wj − ν)y w

Dω0 −Dω

is a proper, stratified submersion.

Critical points of the map in (Dω0 −Dω) ×

Bε0 occur where grad(ft1) = 0; that is, at points

(w, t1, z) such that (t1, z) is in Γ1f,t or in Σf . By 10), Γ1

f,t ∩ (Dt1 × Bε0) is empty and, by 11),Σf ∩ (Dt1 × Bε0) ⊆ V (f). But, if ft1 = 0, then wj − ν = 0. However, this is impossible since

w ∈ Dω0 −Dω and thus we would have to have |wj | > ωj – but we know that ωj > ξ > |ν| by 17)

and 18).

Now, we consider critical points of (∗) which occur on (Dω0−Dω)×∂Bε0 . These occur at points

(w,p) where TpV (ft1 − ft1(p)) ⊆ Tp∂Bε0 . However, 0 < |ft1(p)| = |wj − ν| 6 |w|j + |ν|, where0 < |wj−ν| by the argument of the preceding paragraph. But, w ∈ Dω0 and so |w|j + |ν| 6 ωj

0 + |ν|which is 6 ωj

0 + η−ωj0 by 18). Hence, 0 < |ft1(p)| 6 η, t1 ∈ Dτ , and TpV (ft1 − ft1(p)) ⊆ Tp∂Bε0 ;

this contradicts 4).Therefore, the map (∗) is a proper, stratified submersion and, hence, is a locally trivial fibration.

It follows at once that the inclusion, l, is a homotopy-equivalence and, thus, it follows that ourearlier map

(Dω ×Bε) ∩ V (ft1 + wj − ν) i−→ (Dω ×Bε0) ∩ V (ft1 + wj − ν)

induces isomorphisms on homology.

We wish now to show that i is obtained up to homotopy by wedging together j − 1 copies ofthe suspension of the inclusion map Bε ∩ V (ft1 − ν) → Bε0 ∩ V (ft1 − ν) which, by 12), 13), and18), is nothing more than the inclusion Fft1 ,0 → Ff0,0. It would then follow that Fft1 ,0 → Ff0,0

induces isomorphisms on homology since i does.But, since |wj − ν| 6 |w|j + |ν| 6 ωj + |ν| 6 minξ, λ′0 by 18) and 14), we may proceed as in

Proposition II.8.1 and find that projection by w realizes, up to homotopy:

(Dω ×Bε) ∩ V (ft1 + wj − ν) as the wedge of j − 1 copies of the suspension of Fft1 ,0,

(Dω ×Bε0) ∩ V (ft1 + wj − ν) as the wedge of j − 1 copies of the suspension of Ff0,0, and

the map i as the wedge of j − 1 copies of the suspension of the map

Fft1 ,0∼= Bε ∩ V (ft1 − ν) → Bε0 ∩ V (ft1 − ν) ∼= Ff0,0.

The conclusion follows.

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Subject Index

Additivity result . . . . . . . . . . . . . . . . . . . . . 150Agreeable reorganization . . . . . . . . . . . . . . 14Aligned good stratification . . . . . . . . . . . 97Aligned singularity . . . . . . . . . . . . . . . . . . . 97Aligning coordinates . . . . . . . . . . . . . . . . . 97Analytic cycle . . . . . . . . . . . . . . . . . . . . . . . 177Characteristic cycle . . . . . . . . . . . . . . . . . 128Characteristic polar cycle . . . . . . . . . . . 172Conormal condition . . . . . . . . . . . . . . . . . 227Conormal Jacobian tuple . . . . . . . . . . . . 160Conormal polar cycle . . . . . . . . . . . . . . . . 161Conormal Le-Vogel cycle . . . . . . . . . . . . . 161Continuous family of sheaves . . . . . . . . .147Coordinate planes example . 36, 37, 86-90Correct dimension . . . . . . . . . . . . . . . . . . . . 12Critical locus

algebriac . . . . . . . . . . . . . . . . . . . . . . . . . . 118C− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116canonical stratified. . . . . . . . . . . . . . . . .119conormal-regular . . . . . . . . . . . . . . . . . . .118F•− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Nash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118regular . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118relative differential . . . . . . . . . . . . . . . . . 119stratified. . . . . . . . . . . . . . . . . . . . . . . . . . .119

Derived category . . . . . . . . . . . Appendix BEssential arrangement . . . . . . . . . . . . . . . . 88Exceptional pair . . . . . . . . . . . . . . . . . . . . . 28Flat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75, 85FM cone singularity . . . . . . . . . . . . . . 61, 62Gap cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Gap ratio . . . . . . . . . . . . . . . . . . . . . . . . . 25, 26Gap sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Gap varieties . . . . . . . . . . . . . . . . . . . . . . . . 7, 8Generic linear reorganization . . . . . . . . . . 14Generic arrangement . . . . . . . . . . . . . . . . . 86

Generic Le number . . . . . . . . . . . . . . . . . . 110Global Le number . . . . . . . . . . . . . . . . . . . . 80Good stratification . . . . . . . . . . . . . . 54, 165Handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Hyperplane arrangement . . . . . . . . . . 75, 85Intersection theory . . . . . . . . . Appendix AKunneth isomorphism . . . . . . . . . . . . . . . . 195Le’s attaching result . . . . . . . . . . . . . . 38, 39Le cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Le-Iomdine formulas . . . . . . . . . 73, 169-170Le-Iomdine-Vogel formulas . . . . . . . . . . . . 27Le number . . . . . . . . . . . . . . . . . . . . . . . 43, 44Le-Ramanujam result . . . . . . . . . . . . . . . . 39Le-Saito result . . . . . . . . 39, 40, 91-96, 170Le-Vogel cycle . . . . . . . . . . . . . . . . . . 161, 166Le-Vogel number . . . . . . . . . . . . . . . . . . . . .166Le-Vogel stratification . . . . . . . . . . . . . . . 163Le-Vogel tuple . . . . . . . . . . . . . . . . . . . 163-164Milnor fibration . . . . . . . . . . . . . . . . . . 35, 36Milnor fibre . . . . . . . . . . . . . . . . . . . . . . . . . . 36Milnor fibre lifting property . . . . . . . . . 229Milnor neighborhood . . . . . . . . . . . . . . . . 221Milnor number . . . . . . . . . . . . . . . . . . 36, 138Milnor pair . . . . . . . . . . . . . . . . . . . . . . . . . 223Mobius function . . . . . . . . . . . . . . . . . . . . . . 88Morse inequalities . . . . . . . . . . . . . . . . . . . . 68Nearby cycles . . . . . . . . . . . . . . . . . . . 205-206Non-reduced plane curve example . . . . 64Perverse cohomology . . . . . . . . . . . . . . . . . 218Perverse sheaf . . . . . . . . . . . . . . . . . . . 200-201Plucker formula . . . . . . . . . . . . . . . . . . . . . . 74Polar curve, relative . . . . . . 39, 41, 123-127Polar cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Polar number . . . . . . . . . . . . . . . . . . . . . 42-43Polar ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Polar variety . . . . . . . . . . . . . . . . . . . . . . . . . 41

243

Positively perverse sheaf . . . . . . . . . . . . 201Pre-aligning coordinates . . . . . . . . . . . . . . 98Prepolar coordinates . . . . . . . . . . . . . . . . . 54Prepolar deformation . . . . . . . . . . . . . . . 103Prepolar slice . . . . . . . . . . . . . . . . . . . . . . . . 54Prepolar tuple . . . . . . . . . . . . . . . . . . . . . . . 54Privileged neighborhood . . . . . . . . . . . . 222

universal system of . . . . . . . . . . . . . . . 222weakly . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Pseudo-isotopy result . . . . . . . . . . . 229-230Pseudo-Zariski topology . . . . . . . . . . . .13-14Sebastiani-Thom result . . . . . . . . . . . 38, 210Segre-Vogel relation . . . . . . . . . . . . . . . . . . . 22Semi-continuity of Le numbers . . . . . . . 82Σ∗f . . . . . . . . . . . . . . . . . . . . . see Critical locusStability of Continuity . . . . . . . . . . . . . . . 148Super aligned singularity . . . . . . . . . . . . . 99Suspension result . . . . . . . . . . . . 37, 38, 102Swallowtail singularity . . . . . . . . . . . . 76-79Thom’s af condition . . . . . . . . . . . . . 91, 142Thom reduction . . . . . . . . . . . . . . . . . . . . . 145Thom set . . . . . . . . . . . . . . . . . . . . . . . . . . . 227Total exceptional divisor . . . . . . . . . . . . . 130Uniform Le-Iomdine formulas . . . . . . . . 81Unifying reorganization . . . . . . . . . . . . . . . 23Universal conormal condition . . . . . . . . 229Upper-semicontinuity result . . . . . . . . . . 150Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . 207Vanishing Mobius function . . . . . . . . . . . . . 86Visible stratum . . . . . . . . . . . . . . . . . . . . . . . 136Vogel cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Vogel reorganization . . . . . . . . . . . . . . . . . . . 23Vogel set . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 10Weighted homogeneouspolynomial . . . . . . . . . . . . . . . . . . . 36, 74-76

Whitney umbrella . . . . . . . . . . . . . . . . 37, 38Zariski multiplicity conjecture . . . . . . . 100

244

numerical control over complex analytic singularities...be diﬃcult to exaggerate the importance of...

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