Isolated singularities in analytic spaces

Jose Seade

These are notes for my lectures at the meeting “On the Geometry andTopology of Singularities”, in honor of the 60th Birthday of Le Dung Trang.The aim is to give an introduction to some aspects of singularity theory.The topics selected for this discussion were chosen having also the purposeof explaining to the participants of the School some of the material that willbe used in other lecture courses.

The notes are divided into four sections, corresponding to my 4 lectures.Section one discusses the local conical structure of analytic sets. Section

two looks more carefully at the case of complex surface singularities, andexplains their relationship with plumbing and Waldhausen manifolds.

Section three discusses Milnor’s fibration theorem for holomorphic func-tions.

Finally, in section four we look at additional geometric structures onehas on isolated singularity-germs, namely the contact, Spinc and (with somerestrictions) spin structures on the link. We begin by explainning brieflywhat is the meanning of having these structures.

1 The local conical structure of analytic sets

Let us begin with an example; consider the Pham-Brieskorn polynomial

f(z) = za0

0 + · · ·+ zan

n , ai > 1 ,

where z = (z0, ..., zn) ∈

n+1. It is clear that the origin 0 ∈

n+1 is the onlycritical point of f , so the fibres Vt = f−1(t) are all complex n-manifolds fort 6= 0 and V = f−1(0) is a complex hypersurface with an isolated singularityat 0.

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We want to study the topology of V and of the V ′t s. For this let d be

the least common multiple of the ai and define an action Γ of the non-zerocomplex numbers

∗ on

n+1 by:

λ · (z0, · · · , zn) 7→ (λd/a0z0, · · · , λd/anzn) .

Notice this action satisfies:

f(λ · (z0, · · · , zn)) = λd · f(z0, · · · , zn) .

Hence V is an invariant set of the action and one has the following property:

Property 1.1 Restricting the action to t ∈ + we get a real analytic flowΦt (or a vector field) on

n+1 whose orbits are real lines (arcs) which

converge to 0 when t tends to 0, they escape to ∞ when t → ∞, beingtransversal to all spheres around 0, and they leave V invariant (i.e., V isunion of orbits).

Notice that this flow defines a 1-parameter group of diffeomorphisms ofn+1 that preserve V . In fact we can do more: consider the unit sphere

S2n+1, and let Sr be some other sphere around the origin, of positive radiusr < 1. Now define a map

S2n+1 φr

−→ Sr ,

as follows: given a point x ∈ S2n+1, let it flow by Φt until it reaches the

sphere Sr. This is obviously a diffeomorphism. Furthermore, since V is unionof Γ-orbits, the intersection K1 = V ∩S

2n+1 is mapped diffeomorphically ontoKr = V ∩ Sr. Thus we get a diffeomorphism of pairs (S2n+1, K1) ∼= (Sr, Kr).Doing this for all r within (0, 1) we get a 1-parameter family of diffeomor-phisms that shrinks the sphere more and more, converging to the origin, andthis family preserves the intersections of V with the various spheres.

Thus we arrive to the following:

Theorem 1.2 The variety V intersects transversally every (2n+1)-sphere Sr

around the origin; hence the intersection Kr = V ∩ Sr is a smooth manifoldof real dimension 2n − 1 embedded as a codimension 2 submanifold of Sr.Furthermore, for each r, 0 < r < 1, we have a diffeomorphism of pairs(S2n+1, K1) ∼= (Sr, Kr). In fact, if is the unit ball in

n+1, then the pair

( \ 0, V \ 0 ∩ ) is diffeomorphic to the cylinder over (S2n+1, K1), and( , V ∩ ) is homeomorphic to the cone over (S2n+1, K1), with vertex at 0.

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PSfrag replacements

V ∩ S2N−1r

Figure 1: The conical structure

In particular, the diffeomorphism type of the manifoldKr does not dependon the choice of the sphere. This manifold is called the link of the singularity.It determines the topology of V ; and the way Kr is embedded in the spheredetermines the embedding of V in

n+1. These manifolds are nowadays

know as Brieskorn manifolds, because E. Brieskorn [9, 10] studied themthoroughly, obtaining remarkable results about their topology when n > 3.There are also important results in this direction by Hirzebruch, Milnor andothers (see for instance Chapter I in [58] for more about the topology of thesemanifolds).

However, the essence of 1.2 was already known to K. Brauner (1928) in

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the case of two variables, and we briefly explain his results below in thissection.

Theorem 1.2 is the paradigm of the following general result:

Every (real or complex) analytic space is locally a cone around each of itspoints.

The rest of this section is devoted to discussing this statement.Let us consider first the isolated singularity case. The arguments in gen-

eral are essentially the same, just more complicated technically.Let V be a real or complex analytic space, of real dimension d > 0,

and let 0 be an isolated singular point of V . For simplicity we assume Vis equidimensional and locally irreducible at 0. We assume also that V isembedded in some affine space n and the singular point is the origin.

The aim is to construct a vector field on a sufficiently small ball r around0 ∈ n having the same properties as the above vector field (or flow), namely:

i) It is everywhere transversal to all the spheres in r centered at 0. Thusits integral lines are transversal to all the spheres and converge to 0.

ii) It has V as an invariant set, i.e., V is union of orbits (integral lines).

We do it in two steps. First we consider the function r : n → [0,∞)defined by x 7→ ‖x‖2, whose gradient vector field ∇r satisfies condition (i)above. The next step is to “adapt” this vector field to V in a neighbourhoodof 0, so that it becomes tangent to V \ 0. For this we use the CurveSelection Lemma of Milnor:

Lemma 1.3 (Curve Selection Lemma) Let U be an open neighbourhood of0 in n and let f1, ..., fk, g1, ..., gs be real analytic functions on U such that0 is in the closure of the semi-analytic set:

Z := x ∈ U | f1(z) = ... = fk(x) ; and gi(x) > 0, ∀i = 1, ..., s

Then there exists a real analytic curve γ : [0, δ) → U with γ(0) = 0 andγ(t) ∈ Z for all t ∈ (0, δ).

In practice, f1, ..., fk are the functions that define V , and g1, ..., gs definea semianalytic set H in V , containing 0 in its closure. The lemma says thatif that happens, then there is a whole analytic curve in H converging to 0.

This lemma is extremely useful, and using it one can easily prove thefollowing lemma.

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Lemma 1.4 Let g : V ∩U → [0,∞) be the restriction of a real analytic func-tion g defined on U , such that g−1(0) = 0. Then 0 is not an accumulationpoint of critical values of g on V ∩ U \ 0.

We refer to Milnor’s book [38] for the proof of the Curve Selection Lemma;he does it in the algebraic category, but his proof works in general with minor(obvious) modifications. And we refer to Looijenga’s book [34, p. 22] for adirect proof lemma 1.4 using the Curve Selection Lemma. In the complexanalytic context, Lemma 1.4 is a particular case of the Bertini-Sard theorem.Milnor, in his book, proved this lemma for algebraic maps using a general (say“finiteness”) result of Whitney about the number of components of algebraicsets.

Now we return to our function r : n → [0,∞) defined by x 7→ ‖x‖2.The lemma above implies that there are no critical points of its restrictionr = r|V sufficiently near 0. Notice that the critical points of r on V \ 0 arethe points where V \ 0 is tangent to a level surface of r, i.e., tangent to asphere around 0. Thus we get that the gradient vector field ∇r of r has nocritical points on r ∩ (V \ 0), for a sufficiently small ball r around 0.Hence ∇r is transversal to all the spheres contained in r and it is tangentto V \ 0.

Let us now extend (parallel extension) ∇r to a neighbourhood U1 of

V ∩ r \ 0 in r. We denote this extension by ∇r, and take a partition of

unity to glue the vector fields ∇r on U1 and ∇r on r \ V .The resulting vector field is singular only at 0 and satisfies properties (i)

and (ii) above. We denote this vector field by vrad and call it a radial vectorfield for V at 0.

Using this vector field we can now easily conclude that V has a localconical structure near 0. The manifold K = V ∩ Sε, for ε > 0 sufficientlysmall, is called the link of the singularity. One has Milnor’s theorem:

Theorem 1.5 The link K is a smooth manifold of dimension d− 1, whosediffeomorphism type is independent of the choice of ε , and the pair( ε, ε ∩ V ) is homeomorphic to the cone over (Sε, Sε ∩ V ).

Now we consider the general case of (possibly) non-isolated singularities.For this we need to use Whitney stratifications, that will be studied in DavidTrottman’s course (in this School). Here we recall briefly what this means(see for instance [33] for more on the topic). Let V be as above, a real

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Figure 2: The link determines the local topology of the singular variety

analytic space. A stratification of V means a locally finite partition of Vinto subsets Vα, called strata, such that:

i) Each Vα is a non-singular subanalytic space (in particular a manifold).ii) The closure V α of each stratum in V , and the set V α\Vα are subanalytic

spaces.iii) If Vβ ∩ V α 6= ∅ then Vβ ⊂ V α.The stratification is said to be Whitney regular, or simply a Whitney

stratification, if it further satisfies the,

Whitney condition: for all strata Vα, Vβ such that Vβ ⊂ V α, and foreach sequence (xn, yn) of points in Vα × Vβ converging to a point (y, y) inVβ × Vβ, for which exists the limit T of the tangent spaces Txn

(tangent tothe stratum), and also exists the limit L of the secant lines xnyn, one has aninclusion L ⊂ T .

A theorem of Whitney (in the complex analytic case) and Hironaka saysthat given V as above, and an arbitrary locally finite family Aβ of sub-analytic sets of V , there is a Whitney stratification of V for which each Aβ

is union of strata. In this article, Aβ is the singular set of V , possiblyintersected with a small ball around the singularity we are looking at.

Let us now return to our study of the local conical structure of analyticsets. We need the following definitions.

azer

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Definition 1.6 Let V be a real analytic space equipped with a Whitney strat-ification Vα. A stratified vector field on V means a continuous section v ofthe tangent bundle T n|V such that at each point x ∈ V , the vector v(x) iscontained in the stratum that contains x.

Definition 1.7 A stratified vector field, defined on a neighbourhood of apoint p ∈ V , is radial at p if it is transversal to the intersection of V with allsufficiently small spheres in n centered at p.

One has the following lemma; the first part of it is due to M. H. Schwartz[53, 54] in the complex analytic case, but her arguments (the radial extensiontechnique) also work in the real analytic setting.

Lemma 1.8 Let V be as above, equipped with a Whitney stratification Vα,and let p be a point in V . Then there exist a stratified radial vector field ona neighbourhood of p in the ambient space, whose solutions are transversal toall sufficiently small spheres around p, and converge to p.

The M. H. Schwartz’ technique of radial extension allows us to constructa stratified, radial vector field on a neighbourhood of each point of V . To getit to be integrable and so that its solutions are transversal to all sufficientlysmall spheres around p (and converge to p), one needs some extra work: asin the isolated singularity case, the Curve Selection lemma allows us to provethis claim on each stratum; then one uses the Whitney conditions to glue allthese vector fields into a single one, having the properties we want.

We summarize this discussion in the following well-known theorem (seefor instance [33]).

Theorem 1.9 Let V be a real analytic space, p a point in V , and Vα aWhitney stratification of V . Then:

i) Every sufficiently small sphere Sr(p) centered at p meets every Whitneystratum transversally.

ii) For r > 0 sufficiently small, the pair ( r(p), r(p)∩ V ) is homeomorphicto the cone over the pair (Sr(p), Sr(p) ∩ V ). Thus,

iii) the homeomorphism type of the intersection V ∩ Sr(p) does not dependon the choice of the sphere (provided this is small enough).

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As before, the variety Kr = Sr(p) ∩ V is called the link of p in V . Itdetermines the topology of V near p, and its embedding in Sr(p) determines(locally) the embedding of V in the amient space.

To finish this section, let us envisage briefly the cases of low dimensionalcomplex analytic varieties.

Suppose first V has complex dimension 1. Then its link K is a union ofcircles, one for each branch of V . For instance, if V is defined by a singleequation f :

2 →

and

f =s∏

i=1

fni

i ,

is its decomposition into irreducible factors, then V has s irreducible com-ponents, or branches. The link K is then a union of s knots in a 3-sphereSε ⊂

2. The study of this type of knots is the topic of the course given

by Anne Pichon in this School. For instance, if f is the polynomial zp1 + zq

2

with p, q being relative prime, then V is irreducible and its link K is a torusknot of type (p, q), i.e. it is contained in a torus S

1 × S1 wrapped so that

it goes around the torus giving p turns in one direction and q turns in theother direction. The cases (2, 5) and (3, 4) are depicted in Figure 3.

Figure 3: Toral knots of types (2, 5) and (3, 4).

If V has complex dimension 1 and it has an isolated singularity at a point0, then the link K is an oriented 3-manifold, which is a graph manifold. This

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situation will be the topic of next lecture, and much more on the subject willbe said in the course of Walter Neumann, next week in this School.

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2 On the Topology of Complex Surface Sin-

gularities

In this section we study germs of complex analytic surfaces with an isolatedsingularity.

Consider first a singular (reduced) complex curve C in some smooth com-plex surface X. An important result for plane curves, due to Max Noether(1883), is that by a finite sequence of blowing ups we can always resolve thesingularities of C. Let us explain this with a little more care (see [5] for aclear account on the subject). Given a smooth point x in a complex surfaceX, take local coordinates so that we identify the germ of X at x with thatof

2 at 0. Let us take a small disc U around x and consider the map:

γ : U − x →P 1 which associates to each y ∈ U − x the point in

P 1

represented by the line determined by x and y. The graph of γ is an analyticsubset of (U − x) ×

P 1, whose closure

X = graph(γ) ⊂ (U − x) ×P 1

turns out to be a smooth complex surface. Notice that X is obtained byremoving x from X and replacing it by the limits of lines converging to x.Thus we have replaced x by a copy of

P 1. There is a projection map

π : X → X which is biholomorphic away from E ∼=P 1 = π−1(x). This

transformation is called the blow up of X at x (in the literature this is calledsometimes a σ-process or a monoidal transformation).

Now, given the reduced (maybe reducible) singular curve C ⊂ X, with Xa smooth complex surface, let x be a point in Csing, the singular set of C, and

look at the blow up of X at x, π : X → X. The closure π−1(C − x) in X iscalled the proper (or strict) transform of C under the blow up, and denoted

C. Notice that C is obtained by removing x from C and replacing it by thelimits of lines which are tangent to C − x. This curve C is analytic in Xand projects to C under π; this curve may still be singular, but somehow itssingularities are simpler. We may now repeat the process, choosing a singularpoint in C, blowing up X at this point to get π2 : X2 → X and then considerthe proper transform of C in X2, which is the closure of (π2 π)−1(C − x),and so on. The theorem is (see [3, II.7.1] for a short proof):

Theorem 2.1 Let X be a smooth complex surface and C ⊂ X an embeddedreduced curve. Then there is a smooth complex surface Y and a proper map

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τ : Y → X obtained by a finite sequence of blow ups, such that the propertransform C of C in Y is smooth.

The curve E = τ−1(C), is called the total transform of C in Y . It consists

of the proper transform C and the divisor τ−1(Csing). The theorem abovecan be refined to the following theorem, which will be used later in the text.

Theorem 2.2 Let X and C be as above. Then by performing finitely manyblow ups more, if necessary, we can assume that the whole total transformE = τ−1(C) of C in Y has only ordinary normal double points as singulari-

ties, and these are all away from the proper transform C of C.

We recall that an ordinary normal double point is locally defined by theequation xy = 0.

Now consider the germ (V, 0) of a normal complex surface singularity. Thefollowing important theorem has a long history. This was first stated by Jungbut his proof was not complete, and was completed later by Hirzebruch. Thefirst complete proof of 2.3 is due to O. Zariski; this was done using blowingups and it was based on a previous proof by R. Walker (1935) which was notcorrect either. We refer to [3, Ch. III] for a proof. This result was latergeneralized by Hironaka to all dimensions and for (real or complex) analyticspaces with arbitrary singularities.

Theorem 2.3 Let (V, 0) be a normal complex surface singularity. For sim-plicity assume it is defined in a sufficiently small ball around the origin insome

n, so that V ∗ = V − 0 is non-singular. Then there exists a non-singular complex surface V and a proper analytic map π : V → V, suchthat:

i) the inverse image of 0, E = π−1(0), is a (connected, reduced) divisor in

V , i.e. a union of 1-dimensional compact curves in V ; and

ii) the restriction of π to π−1(V ∗) is a biholomorphic map between V − Eand V ∗.

The surface V is called a resolution of the singularity of V , and π : V → Vis the resolution map. Sometimes these are called desingularizations of thesingularities instead of resolutions. The divisor E is called the exceptionaldivisor.

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Notice that “the” resolution of (V, 0) is not unique: given a resolution

V we can obtain new resolutions by performing blow ups at points in E.By Theorem 2.2 above, given a resolution, we can make blow ups on it, ifnecessary, so that the divisor E in Theorem 2.3 is good, i.e.:

iii) each irreducible component Ei of E is non-singular; and

iv) E has normal crossings, i.e. Ei intersects Ej, i 6= j, in at most one point,where they meet transversally, and no three of them intersect.

Definition 2.4 A resolution π : V → V is good if its exceptional divisor isgood, i.e. if it satisfies conditions (iii) and (iv) above.

Some authors allow good resolutions to have irreducible components of Eintersecting transversally in more than one point, and they reserve the namevery good for resolutions as in 2.4. This makes no big difference and we preferto keep the notation of 2.4.

We recall that given a non-singular complex surface V and a Riemannsurface S in it, the self-intersection of S, usually denoted by S · S or simplyby S2, is the Euler class of its normal bundle ν(S) in V (which coincides withits Chern class) evaluated in the fundamental cycle [S]. Equivalently, S ·S isthe number of zeroes, counted with signs, of a generic section of the normalbundle ν(S). It is an exercise to see that every time we make a blow up on asmooth complex 2-manifold, we get a copy of

P 1 with self-intersection -1.

It is remarkable that the converse is true: recall that a non-singular curve ina smooth complex surface is said to be exceptional of the first kind if it is acopy of

P 1 embedded with self-intersection -1, one has:

Theorem 2.5 (Castelnuovo’s criterium) Let X be a non-singular complexsurface and C an exceptional curve of the first kind. Then S can be blowndown analytically and we still get a non-singular surface X.

This result is in fact a special case of a more general theorem of Castel-nuovo for exceptional divisors of the first kind. We refer to [19, Ch. 3 ] fora proof. Notice that 2.4 has the very important consequence of giving us aminimal model:

Definition 2.6 A resolution Xπ→ V is minimal if given any other resolution

X ′ π′

→ V , there is a proper analytic map X ′ p→ X such that π′ = π p.

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One has (see [3, III.6.2]):

Theorem 2.7 Up to isomorphism, there exists a unique minimal resolutionof V , and this is characterized by not containing non-singular rational curveswith self intersection -1.

We remark that these statements are false in dimensions more than 2:there are not minimal resolutions in general.

Notice that the minimal resolution may not be good. For instance (c.f. [15]),for the Brieskorn singularity

z21 + z3

2 + z73 = 0 ,

the minimal resolution has an exceptional divisor consisting of three non-singular rational curves meeting at one point, so it is not good; making oneblow up at that point we obtain a good resolution, which has now a centralcurve which is a 2-sphere with self-intersection -1, and three other spheresmeeting each the central curve in one point and with self intersections -2, -3,-7.

Something similar happens in general: we can make the minimal resolu-tion good by performing blow ups, if necessary, and there is a unique (up toisomorphism) minimal good resolution.

Consider now a divisor E =⋃r

i=1Ei in a complex 2-manifold X, whoseirreducible components Ei are non-singular, they all meet transversally andno three of them intersect. To such a divisor we can associate an r×r integralmatrix A = ((Eij)), called the intersection matrix of E, as follows: on thediagonal ∆ of A we put the self intersection numbers E2

i ; and if a curve Ei

meets Ej at Eij points, we put this number as the corresponding coefficientof A. So this is necessarily a symmetric matrix, whose coefficients awayfrom the diagonal ∆ are integers ≥ 0 and in ∆ we have the self-intersectionnumbers of the Ei, called the weights of these curves.

We have the following remarkable theorems of Mumford and Grauert(see [3, III.2]):

Theorem 2.8 If E is the exceptional divisor of a resolution Xπ→ V , where

V is a normal surface, then the intersection matrix A is negative definite(and the weights of the Ei are all negative numbers).

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Theorem 2.9 Conversely, if the divisor E in X is such that the intersectionmatrix A is negative definite, then we can blow down E analytically; we get anormal complex surface V , in general with a singularity at the image 0 of E,and the projection π : X → V is a good resolution of (V, 0) with exceptionaldivisor E.

A divisor E as above is usually called an exceptional divisor, meaning bythis that it can be blown down. It is said to be of the first kind when theblow down is smooth.

Notice that we can associate a weighted graph G = G(E) to a good ex-ceptional divisor E in a complex 2-manifold X as follows: to each irreduciblecomponent Ei of E we associate a vertex vi, and if the curves Ei and Ej

meet, then we join the vertices vi and vj by an edge. Each vertex has twointegers attached to it: one is the genus gi ≥ 0 of the corresponding Riemannsurface Ei; the other is the weight wi = E2

i ∈ , which is the self-intersectionnumber of Ei in X. This weighted graph is called the dual graph of the ex-ceptional divisor E, or the dual graph of the resolution when E is regardedas the exceptional set of a good resolution of a normal singularity.

We observe that every finite graph Σ has associated a matrix I(Σ) calledthe matrix of adjacencies of the graph: it has zeroes in the diagonal andif a vertex vi is joined to vj by δij edges, then we put this number in thecorresponding place of I(Σ). It follows that the intersection matrix of theexceptional divisor E is the result of taking the matrix of adjacencies of thedual graph and replacing its diagonal by the vector of weights w1, · · · , wm,wi = Ei · Ei.

A beautiful thing of these constructions is that the dual graph of a resolu-tion allows us to re-construct the topology of the resolution, and hence thatof the link of the singularity. For this we need to introduce a constructionknown as plumbing. This was used already by Milnor to construct his firstexamples of exotic spheres and by Von Randow (1962) in relation with Seifertmanifolds, though it was Hirzebruch who made this construction systematic.The plumbing construction is very nicely explained in [25] (see also [18]), wejust recall it here briefly.

Let E be a real 2-dimensional oriented vector bundle over a Riemannsurface S, and denote by D(E) its unit disc bundle for some metric. Thetotal space of D(E), that we denote by the same symbol, is a 4-dimensionalsmooth manifold with boundary the unit sphere bundle S(E). Notice thatrestricted to a small disc ε in S the manifold D(E) is a product of the

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form 2 × 2, where the first disc is ε ⊂ S and the second disc is inthe fibres of E. Now suppose we are given two such bundles Ei, Ej, overRiemann surfaces Si, Sj. To perform plumbing on them we consider thetotal spaces of the corresponding unit disc bundles D(Ei), D(Ej), we choosesmall discs i,ε, j,ε in Si, Sj, and take the restriction of D(Ei), D(Ej)to these discs. Each of them is of the form 2 × 2 as above. We nowidentify each point (x, y) ∈ i,ε × 2 ⊂ D(Ei) with the corresponding point(y, x) ∈ j,ε× 2 ⊂ D(Ej), i.e. interchanging base points in one of them withfibre points in the other. The result is a 4-dimensional, oriented manifoldwith boundary and with corners, which can be smoothed off in a uniqueway up to isotopy. We denote this manifold by P (Ei, Ej). One says thatP (Ei, Ej) is obtained by plumbing the bundles Ei and Ej over the Riemannsurfaces Si and Sj.

Figure 4: Plumbing line bundles over circles.

The boundary S(Ei, Ej) = ∂P (Ei, Ej) of this 4-manifold is obtained byplumbing the corresponding sphere bundles Si(E) and Sj(E): we removefrom Si(E) and Sj(E) the interior of the solid tori ∂Di × 2 and similarlyfor Ej. Thus we get two 3-manifolds with boundary a torus S

1 × S1 in each;

we then identify these boundaries by glueing the meridians in one torus tothe parallels in the other. The result is a 3-manifold with corners, whichcan be smoothed off in a unique way up to isotopy. The surfaces Si, Sj arenaturally embedded in P (Ei, Ej) as the zero-sections of the correspondingbundles, and they meet transversally in one point.

Notice that the manifolds one gets in this way are entirely described, upto diffeomorphism, by the genera of the Riemann surfaces Si, Sj, and by theEuler classes of the corresponding bundles, since these classes determine theisomorphism class of the bundles.

Definition 2.10 A plumbing graph is a triple (Σ, w, g) consisting of a finite

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graph Σ with vertices v1, · · · , vr, r ≥ 1 and with no loops, a vector w ofweights, w = (w1, · · · , wr), wi ∈ , and a vector g = (g1, · · · , gr) of genera,gi ∈ .

In this definition by a loop we mean an arrow that begins an ends in thesame vertex, and we do not allow this (geometrically this means a singularcurve in the exceptional divisor that has a double crossing). There can becycles, i.e. a chain of vertices and edges that returns to itself after a certaintime. In [42] Neumann considers a more general situation of plumbing graphsthan the one envisaged here, which is important for the plumbing calculusdeveloped there, but it does not really make a difference for the present work.

So the previous results say that dual graph of a good resolution of anormal singularity is a plumbing graph with negative definite intersectionmatrix. For instance, Figure 5 depicts the resolution corresponding to thesurface singularity z2

1 + z22 + zr

3 = 0.

Figure 5: Dynkin diagramm Ar.

More generally, the classical Dynkin diagramms Ar, Dr, E6, E7 and E8

are the dual graphs of the minimal resolutions of surface singularities of theform Γ/

2, where Γ is a finite subgroup of SU(2) (see for instance [58] for

details).Now, given a plumbing graph we may perform plumbing according to the

graph: for each vertex vi take a Riemann surface Si of genus gi and an oriented2-plane bundle Ei over Si with Euler class wi. If there is an edge between thevertices vi and vj, we plumb the corresponding bundles as above. If a vertexvi is joined with other vertices, we choose pairwise disjoint small discs ineach surface, as many as adjacent vertices one has, and perform plumbing bypairs as above. The result is a 4-dimensional manifold P(E) with boundaryS(E). It follows from the construction that the manifold P(E) contains theunion E =

⋃Si as a deformation retract, and these surfaces are contained

in P(E) with self-intersection wi. Hence the homology of P(E) is that of E,and the intersection form on P(E) is given by the intersection matrix of itscorresponding graph.

A manifold obtained in this way is known as a plumbed manifold, andthis term may refer either to the 4-manifold P(E) with boundary, or to

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Figure 6: Dynkin diagrams.

its boundary, which is a 3-manifold. Notice that if the plumbing graph(Σ, w, g) is the dual graph of a resolution π : V → V , then the manifoldP(E) is diffeomorphic to a regular neighbourhood of the exceptional set Ein the resolution, which may be taken to be of the form π−1(V ∩ ε), where

ε is a small closed disc in the ambient space

n with centre at 0. Sincethe resolution map is a biholomorphism away from E, it follows that theboundary S(E) is diffeomorphic to the link of V , a fact that we state as atheorem:

Theorem 2.11 Let π : V → V be a good resolution of (V, 0), a normalsurface singularity. Then the irreducible components E1, · · · , Er of the ex-ceptional divisor E determine a plumbing graph G(E), called the dual graphof the resolution, and performing plumbing according to this graph we obtaina 4-manifold homeomorphic to π−1(V ∩ ε) ⊂ V , whose boundary is the linkM , where ε is a small closed disc in the ambient space

n with centre at

0.

Hence we know that a plumbing graph is the dual graph of a resolution ifand only if its intersection matrix is negative definite. This gives a necessary

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and sufficient condition for an oriented 3-manifold to be the link of a surfacesingularity. Let us finish this section by stating a well-known open problem.

Problem 1. Which 3-manifolds arise as links of isolated hypersurface sin-gularities in

3?

In other words, we know from theorems 2.8 and 2.9 that an oriented 3-manifold is a singularity-link if and only if it is a plumbing manifold withnegative definite intersection matrix. So the question is, among these mani-folds, which ones arise as links of hypersurfaces?

This interesting question was studied by various authors in the 1980s,most notably by A. Durfee and St. Yau, obtaining some partial results. Butthe problem is, at present, far from being solved.

We notice that every closed oriented 3-manifold embeds in the 5-spherewith trivial normal bundle, and bounding a simply connected parallelizablemanifolds. This somehow says that the problem is not only topological, butone needs to look for more subtle (presumably geometric) structures in orderto characterize the links of hypersurface singularities.

18

3 On Milnor’s Fibration Theorem

Milnor’s Fibration Theorem is a result about the topology and geometry ofthe fibers of a holomorphic function in a neighbourhood of a critical point.

As in Section 1, we begin with the example of the Pham-Brieskorn sin-gularities,

f(z) = za0

0 + · · ·+ zan

n , ai > 1 ,

where z = (z0, ..., zn) ∈

n+1. The fibres Vt = f−1(t) are all complex n-manifolds for t 6= 0 and V = f−1(0) is a complex hypersurface with anisolated singularity at 0.

Recall one has a natural

∗-action on

n+1 defined by:

λ · (z0, · · · , zn) 7→ (λd/a0z0, · · · , λd/anzn) ,

where d is the least common multiple of the ai, and this action satisfies:

f(λ · (z0, · · · , zn)) = λd · f(z0, · · · , zn) .

Hence V is an invariant set of the action. In Section 1 we restricted thisaction to the real numbers in

∗, and we used the corresponding flow on V

to deduce that V has a conical structure.Now we look at the restriction of the above

∗-action to the unit complex

numbers. One has:

Property 3.1 Restricting the action to the unit circle eiθ we get an S1-

action on

n+1 such that each sphere around 0 is invariant and:

f(eiθ · (z0, ..., zn)) = eidθf(z0, · · · , zn) ,

that is, if we set ζ = f(z0, · · · , zn), then multiplication by eiθ in

n+1 trans-ports the fibre f−1(ζ) into the fibre over eidθ · ζ.

In other words, this action is by isometries of

n+1, its orbits are transver-sal to the fibers f−1(t) for t 6= 0 and move the points of

n+1 \ V taking

fibers diffeomorphically into fibers. Thus, for every disc ∆ in

centered at0 one has that,

f |f−1(∆\0) : f−1(∆ \ 0) −→ ∆ \ 0 ,

is a C∞ fibre bundle. The same statement holds if we consider only theboundary ∂∆ of the disc.

19

Moreover, fix a small ball ε around 0 of arbitrary positive radius, andlet δ > 0 be sufficiently small with respect to ε, so that all the fibers f−1(t)with δ ≥ |t| > 0 meet transversally the boundary sphere Sε = ∂ ε. LetN(ε, δ) be defined by:

N(ε, δ) = ε ∩ (f−1(∂∆δ) ,

where ∆δ is the disc in

with center at 0 and radius δ. Then

f : N(ε, δ) → ∂∆δ , (3.2)

is a C∞ fibre bundle.The Manifold N(ε, δ) is known as a Milnor tube for f . This is the union

of the portion of the fibers f−1(t), |t| = δ, contained within the ball ε.

Now observe one also has:

Property 3.3 The real analytic flow defined by restricting the

∗-action tot ∈ + has the additional property that for points in

n+1 \V , the argument

of the complex number f(z) is constant on each orbit, i.e., f(z)/|f(z)| =f(tz)/|f(tz)| for t ∈ +, and the norm of f(z) is an strictly increasingfunction of t.

We may thus use this flow to “push” the tube N(ε, δ) to the boundarysphere. That is, for each point x in N(ε, δ), consider its orbit under the realflow defined by the

∗-action (restricted to +). The point moves so that its

orbit is transversal to all the spheres in ε centered at 0, and also transversalto all the tubes f−1(∂∆r). Furthermore, at each point of this trajectory onehas that the argument of f(z) is constant. Now let x move along this trajec-tory until it meets the boundary sphere Sε. This defines a diffeomorphism Ψbetween N(ε, δ) and the sphere Sε minus a tubular neighborhood N(K) ofthe link K. One thus has a fiber bundle,

ψ : Sε \N(K) −→ ∂∆δ ,

given by Ψ−1 followed by f . Normalizing, we get a fiber bundle,

φ =f

|f |: Sε \N(K) −→ S

1 , (3.4)

With some more work one can show that this fibration actually extends toall of Sε \K.

20

These two fibrations, 3.2 and 3.4, are the two different versions one usuallyhas of the Milnor fibration associated to the function f . In the sequel we willsee that one has these two fibrations in general.

In fact the classical Fibration Theorem of Milnor [38] says:

Theorem 3.5 Let f : (

n+1, 0) → (, 0) be a holomorphic map-germ, ε > 0

small enough and let K = Sε ∩ V be its link. Then:

φ =f

|f |: Sε \K −→ S

1 ,

is a (locally trivial) C∞ fiber bundle.

Notice that in 3.5 there are actually two statements melted into a singleone: the first is that Sε \ K is a fibre bundle over S

1, the other is that theprojection map can be taken to be the obvious one φ = f

|f |. One has (see

for instance [58] that the first statement actually extends to a very generalsetting, while the second statement is in fact more subtle.

The idea of Milnor’s proof is simple: first show that the map φ hasno critical points at all, so the fibres of φ are all smooth, codimension-1submanifolds of (Sε \ Kε) where K = Kε is the link. Then construct atangent vector field on (Sε\Kε) which is transversal to the fibres of φ and thecorresponding flow moves at constant speed with respect to the argument ofthe complex number φ(z), so it carries fibres of φ into fibres of φ. This provesone has a product structure around each fibre of φ. For this, to begin, Milnor

shows that the critical points of (Sε \ Kε)φ→ S

1 , if there were such points,are exactly the points z = (z0, . . . , zn) where the vector

(i grad(log(f(z)))

)

is a real multiple of z. To prove this, set

φ(z) =f

|f |(z) := eiθ(z) ,

so one has:θ(z) = Re (−i log f(z)) .

An easy computation shows that given any curve z = p(t) in

n+1 \ f−1(0),the chain rule implies:

dθ(p(t))/dt = Re 〈dp

dt(t), i grad log f(z)〉 , (3.6)

21

where 〈·, ·〉 denotes the usual Hermitian product in

n+1. Hence, givena vector v(z) in

n+1 based at z, the directional derivative of θ(z) in the

direction of v(z) is:Re 〈v(z), i grad log f(z)〉 .

Since the real part of the hermitian product is the usual inner product in 2n,it follows that if v(z) is tangent to the sphere S

2n−1ε , then the corresponding

directional derivative vanishes whenever(i grad(log(f(z)))

)is orthogonal to

the sphere, i.e., when it is a real multiple of z; conversely, if this inner productvanishes for all vectors tangent to the sphere, then z is a critical point of φ,and the claim follows.

Once we know how to characterize the critical points of φ and how theargument of the complex number φ(z) varies as z moves along paths in Sε\K,Milnor uses his Curve Selection Lemma (1.3) to conclude that φ has no criticalpoints at all. This part is a little technical and we refer to Milnor’s book(Chapter 4) for details. It follows that all fibres of φ are smooth submanifoldsof the sphere Sε of real codimension 1. In order to show that φ is actuallythe projection map of a C∞ fibre bundle one must prove that one has a localproduct structure around each fibre. This is achieved in [38] by constructinga vector field w on Sε \K satisfying:

i) the real part of the hermitian product 〈w(z), i grad log f(z)〉 is identicallyequal to 1; recall that this is the directional derivative of the argument of φin the direction of w(z).

ii) the absolute value of the corresponding imaginary part is less than 1:

|Re 〈w(z), grad log f(z)〉| < 1 .

Consider now the integral curves of this vector field, i.e., the solutions p(t)of the differential equation dz/dt = w(t). Set eiθ(z) = φ(z) as before. Sincethe directional derivative of θ(z) in the direction w(t) is identically equal to1 we have:

θ(p(t)) = t+ constant .

Therefore the path p(t) projects to a path which winds around the unit circlein the positive direction with unit velocity. In other words, these paths aretransversal to the fibres of φ and for each t they carry a point z ∈ φ−1(eito)into a point in φ−1(eito+t). If there is a real number to > 0 so that all thesepaths are defined for at least a time to then, being solutions of the abovedifferential equation, they will carry each fibre of φ diffeomorphically into all

22

the nearby fibres, proving that one has a local product structure and φ isthe projection of a locally trivial fibre bundle. Milnor proves this by showingthat condition (ii) above implies that all these paths are actually defined forall t ∈ , so we arrive to Theorem 2.1. 2

The previous formulation of Milnor’s theorem, besides of being the origi-nal formulation in [38], provides several geometric insights into the topologyof singularities, as for instance showing that the link K is a fibred knot (orlink) in the sphere Sε, a fact used by Milnor in his book to study the topol-ogy of the fibers and of the link itself. Furthermore, one has a “multi-link”structure (which is a richer structure, see [18], or section 4 below) determinedby the map f .

There is, however, another formulation of Milnor’s theorem, which is morefamous nowadays, and lends itself more easily to generalizations. This seemsto be the most appropriate view-point for algebraic geometry. We call thisthe Milnor-Le fibration theorem. For map-germs f as above, definedon

n+1, this result is essentially due to Milnor alone, except for a “little”

point that we explain below. Le completed this and extended the result toholomorphic maps defined on arbitrary complex spaces. Let us first state thetheorem in

n+1.

Theorem 3.7 Let Sε be a sufficiently small sphere in

n+1 centered at 0and choose δ > 0 small enough with respect to ε so that all the fibres f−1(t)with |t| ≤ δ meet Sε transversally. Let Cδ = ∂ δ be the circle in

of radius

δ and centered at 0, and set N(ε, δ) = f−1(Cδ) ∩ ε. Then:

f |N(ε,δ) : N(ε, δ) −→ Cδ∼= S

1 ,

is a C∞ fibre bundle, equivalent to the fibration in (3.5).

The manifold N(ε, δ) is usually called a Milnor tube for f .This result is due to Milnor ([37, 38]) when 0 ∈

n+1 is an isolated critical

point of f . Milnor also proved in [38, §5] that for arbitrary f , the fiberf−1(t) ∩ ε in (3.7) is diffeomorphic to the fiber in (3.5). The missing pointis that he did not prove that (3.7) is a fibre bundle (for f with non-isolatedsingularity). If f has an isolated critical point, this is an immediate extensionof Ehresmann’s fibration lemma, as noted by Milnor in [37]. However, forgeneral f one must prove first, for instance, that f has the so-called Thomaf -property. We recall the definition of this property in the simple situationenvisaged in this article.

23

Definition 3.8 Let (Xn+k, p) be a real analytic space-germ, and h : (X, p) →( n, 0) a real analytic map with an isolated critical value at 0. Then f hasthe Thom property if there exists a Whitney stratification of X such thatV = f−1(0) is union of strata, and for each sequence of points ym ⊂ X \Vthat converges to a point x ∈ V , such that the limit T of the tangent spacesTym

(f−1(f(ym))

)exists, one has that T contains the tangent space Tx(Xα),

where Xα is the stratum that contains α.

As mentioned before, Hironaka [24] proved that every holomorphic mapinto

has the Thom property. Using this, Le proved:

Theorem 3.9 Let (X, p) be a complex analytic germ in

N , and f : (X, p) →(

, 0) holomorphic. Let ε > 0 sufficiently small, so that KX = X ∩ S

2N−1ε is

the link of X and LV = V ∩ KX is the link of V = f−1(0). Choose a disc δ in

, ε >> δ > 0, around 0 such that all the fibers f−1(t), t ∈ δ \ 0,

meet KX transversally, and set N(ε, δ) = f−1(Cδ) ∩ ε, where Cδ = ∂ δ.Then:

f |N(ε,δ) : N(ε, δ) −→ Cδ∼= S

1 ,

is a continuous fibre bundle.

Furthermore, the tube in this fibration can be “inflated” (see explanationbelow), so that it takes the form of the classical Milnor fibration (3.5), nowdefined on KX \ LV .

We remark that the fact that one can choose δ so that all the fibersf−1(t), t ∈ δ \ 0, meet KX transversally follows easily from the factthat f has the Thom property, together with the first Thom-Mather isotopytheorem.

If X has an isolated singularity at p the proof of (3.9) is rather simple(and this was proved by H. Hamm [22]). The idea is that since 0 ∈

is an

isolated critical value, the map f in (3.9) is a submersion and we can lift avector field on the circle Cδ to a vector field on the tube N(ε, δ), transversalto the fibres of f . We may further choose this vector field to be a C∞ andintegrable, and also tangent to N(ε, δ) ∩ Sε (since the fibers are transversalto the sphere). Since the fibres are compact, we can assume that for eachfibre Fo there is a time to > 0 so that all solutions passing by Fo are definedfor at least time to. Then the corresponding local flow identifies Fo with allnearby fibres, which are thus diffeomorphic to Fo and one has a local productstructure, hence a fibre bundle.

24

The proof in general follows the same ideas, but it is technically moredifficult. The key point is noticing that also in this situation, the (proofof the) first Thom-Mather isotopy theorem gives that one can lift a smoothvector field on Cδ to a stratified, integrable vector field on the tube N(ε, δ)(smooth on each stratum) and transversal to the fibres of f . After that, theproof is exactly as in the isolated singularity case.

Once we have (3.9), in order to show that this fibration is equivalent toa fibration of the type of (3.5), one follows exactly the same steps followedby Milnor in [38, Ch. 5]. Up to this point, essentially everything worksin the real analytic category, but from now on the holomorphic structureis used thoroughly. The idea is to show that there exists a vector field on( ε∩X)\f−1(0) whose solutions move away from the origin being transversalto all the spheres around 0, transversal to all the tubes f−1(C ′

δ) and theargument of the complex number f(z) is constant along the path of eachsolution. This allows us to “inflate” the Milnor tube f−1(∂ δ) ∩ ε in (3.7)to become the complement of a neighbourhood T (ε, δ) of the link K in thesphere Sε, taking the fibres of (2.3) into the fibres of the map φ in Theorem2.1 (just as we did for the Pham-Brieskorn singularities at the beginning ofthis section). Then one must show that this fibration extends to T (ε, δ)\Kε.

Problem 2. Let (V, p) be a two-dimensional complex analytic non-isolatedsingularity in

3, say defined by a map-germ f with f(p) = 0, and equip V

with a Whitney stratification. Let r > 0 be sufficiently small, so that everysphere around p ∈

3 meets every Whitney stratum transversally, and let

K = V ∩ Sr(p) be the link of p. Now choose δ > 0 sufficiently small, sothat for every t with 0 < |t| < δ, the complex manifold f−1(t) meets Sr(p)transversally, and set ∂F = f−1(t)∩ Sr(p). What type of 3-manifold is ∂F ?,can we describe the topology of the link K from that of ∂F

We remark that the existence of r > 0 as above comes from the fact thatevery complex analytic map-germ

n →

has the Thom af -property, by a

theorem of Hironaka (improved by Parusinsky few years ago).Of course the same questions are valid in higher dimensions. Notice

F = f−1(t)∩ r(p) is the Milnor fibre of f . If p were an isolated singularity,then the link K is smooth and ∂F is diffeomorphic to K. In general, thereis a collapsing map F → V , defined by Le Dung Trang in [28] and others,which can be used to study the above problem.

Let us now say a few words about the topology of the fibers in Milnor’s

25

fibration. This will be explained in much more detail, and in a more generalsetting, in Le Dung Trang’s course next week.

First notice that the description of this fibration as in Theorem 3.9 im-plies that the fibers, Ft = f−1(t) ∩ r(p), are Stein spaces. Therefore thetheorem of Andreotti-Frankel [2] says that they have the homotpy type ofCW-complexes of real dimension n, where n is the complex dimension of Ft.This holds for all holomorphic map-germs, regardless of whether or not thecritical of f at 0 is isolated. This same statement was proved directly byMilnor using the real-valued function |f |, restricted to Ft. The first obser-vation for this is that Theorem 3.9 implies that Ft can be considered as acompact manifold with boundary the link K. Now, although |f | may not beitself a Morse function on Ft, its Morse index is well defined at each criticalpoint, and Milnor shows that the fact that f is holomorphic implies thatall its Morse indices are necessarily more than n. Then, one can alwaysapproximate |f | by a Morse function f , which agrees with |f | away from aneighborhood of its critical set on Ft. Then the Morse indices of f must bealso more than n, thus implying that Ft has the homotopy of a CW-complexof dimension n.

In the special case when f has an isolated critical point, the previousstatements can be made stronger, as proved by Milnor. In fact, using now thefibration as in Theorem 3.5, one has the the compact manifold with boundaryFt is embedded in Sr in such a way that it has the same homotopy type as itscomplement (since Sr \Ft is a fibre bundle over with fiber diffeomorphic toFt). Then, the Alexander duality theorem implies that the reduced homologyof Ft vanishes in dimensions less than n. With a little extra work one canshow that for n > 1 the fiber Ft is simply connected, which implies, togetherwith the previous statement, that if is (n−1)-connected. Milnor further showsthat the homology group Hn(Ft) (with integer coefficients) is free abelian,and the Hurewicz theorem implies that for n > 1 it is isomorphic to thehomotopy group πn(Ft). From this one gets:

Theorem 3.10 The fiber Ft has the homotopy type of a CW-complex ofmiddle dimension n. Furthermore, if f has an isolated critical point, then Ft

actually has the homotopy type of a bouquet of spheres of dimension n,

Ft '∨

µ

Sn,

the number µ of spheres in this wedge being some positive integer (unless fis regular at 0, then µ = 0).

26

For instance, for a Morse function z20 + · · · + z2

n one has µ = 1. Moregenerally, as proved by Pham in [] (and later by Milnor), for the polynomialmap f(z) = za0

0 + · · · + zan

n one has:

µ = (a0 − 1) · · · (an − 1) .

Definition 3.11 The above number µ is called the Milnor number of f .

This number is an important invariant of isolated hypersurface singulari-ties. By definition, it measures the rank of the middle homology of the Milnorfibre Ft, which has homology only in middle-dimension. This invariant hasbeen generalized in various ways for non-isolated singularities, giving riseto the so-called Le-numbers and classes, as well as to the so-called Milnorclasses. We refer to [36] for the former, and to [8] for the latter.

27

4 Additional geometric structures on surface

singularities

We now restrict the discussion the the case when V is a complex analyticsurface in some

N with an isolated singularity at 0, though much of what

we will say holds for higher dimensional complex analytic varieties.Thus V ∗ = V \ 0 is a complex 2-dimensional manifold and the link

K = V ∩ Sr is an oriented 3-manifold. We know from the previous sectionthat the diffeomorphism type of K does not depend on the choice of thesphere Sr used to define it. In fact, slightly more refined arguments, dueindependently to Durfee [17] and Le-Teissier [33] show that this statementcan be made much stronger: the diffeomorphism type of the link K dependsonly on the analytic structure of V at 0 and not on other choices, such asthe equations used to define the embedding of V in

N , the metric, etc.

This means that whatever invariant of 3-manifolds one has, is naturallyan invariant of surface singularities. This has been used for various authors,such as Laufer, Durfee and several others including myself, to obtain informa-tion about surface singularities (see for instance Chapter IV in [58] for moreon the subject). More recently, Nemethi and others have been getting re-markable results for surface singularities using some of the recent 3-manifoldsinvariants coming from gauge theory, such as the Seiberg-Witten invariants,Floer homology, etc.

In this section we look briefly at some finer structures one has on the man-ifolds V ∗ and K, that allow us to use important ideas and tools of other areasof geometry and topology to study surface singularities. These structures aresomehow the basic ground for the use of the recent new manifolds-invariants.

4.1 Contact structures

A contact structure on an odd dimensional manifold M is a field of tangenthyperplanes H, called contact hyperplanes, satisfying a certain maximal non-integrability condition, that we now explain. At each point of M the field Hof hyperplanes is defined locally by a (non-unique) 1-form α, whose kernel is,at each point x, the given hyperplane Hx. The 1-form α is called a contactform if the 2-form dα is non degenerate on each Hx. This means that ateach point x the alternate bilinear form dαx : Hx ×Hx → satisfies that ifdα(x, y) = 0 for all x ∈ V , then y = 0.

28

Notice that the classical frobenius integrability theorem says that a fieldof hyperplanes is integrable at a point x if the corresponding 1-form ω satisfiesω ∧ dω = 0. Thus a contact form is in some sense “as far as possible” frombeing integrable. The form that defines the hyperplanes is locally unique upto multiplication by a non-vanishing function, and this does not change thefact of being a contact form. Thus tis condition depends only on the choiceof hyperplanes.

The simplest example of a conatct structure is given on 2n+1 by equipingit with coordinates (q, p, y) = (q1, · · · , q2, p1, · · · , pn, y) and its usual contactform α = dy−p dq. An important theorem of Darboux acually says that thisexample is “universal”, in the sense that every contact form is equivalent tothis one, up to a smooth change of coordinates.

Now let V ⊂ N be an affine complex analytic variety of dimension

n > 1, with an isolated singularity at 0, and let K be its link. Then K is acodimension 1 oriented submanifold of V ∗ = V \0, and therefore its normalbundle ν(K) is trivial. At each x ∈ K, one has a real line νx(K); multiplyingeach vector in this line by the complex number i we obtain a real line iνx(K)in the tangent space TxK. If we endow V ∗ with the hermitian metric inheritedfrom that in

N , one can consider the orthogonal complement H§ of iνx(K)

in TxK. An important theorem of Varchenko says:

Theorem 4.1 The field of hyperplanes that one gets in this way satisfies themaximal non-integrability condition, and therefore defines a contact structureon K. Moreover, this contact structure is independent of all choices, up tocontact diffeomorphism.

We follow [13] and call this the canonical contact structure on K.

4.2 Spin and Spinc structures

We recall that given a smooth n-manifold M , one has its tangent bundleTM with structure group (Gl(n), ). This means one has a smooth atlasfor M , such that on each coordinate chart TM is trivial, and the transitionfunctions Ui ∩ Uj have natural liftings to (Ui ∩ Uj) × n, where n is beingregarded as the corresponding tangent space and the maps on these tangentfibers are linear isomorphisms, that is elements in (Gl(n), ). These clutchingfunctions on the charts (Ui ∩ Uj) × n allow us to reconstruct not only M ,but also its tangent bundle TM .

29

In other words, we have associated to TM a principal (Gl(n), )-bundleP (M ;Gl(n)) over M , whose fiber over each point x ∈ M is the group oflinear isomorphisms in the tangent space TxM . We can always equip Mwith a Riemannian metric. This means that on each tangent space we haveintroduced a non-degenerate bilinear form, that varies smoothly over M .Since (Gl(n), ) has the orthogonal group O(n) as a deformation retract,having a Riemannian metric on M allows us to reduce the structure group ofTM from (Gl(n), ) to O(n), that is, we can take all the clutching functionsof TM to be elements in O(n).

In other words, equipping TM with a Riemannian metric means thatwe have retracted the bundle P (M ;Gl(n)) over M to the principal bundleP (M ;O(n)), whose fiber over each point is the group O(n) of orthogonaltransformations in the tangent space. This group can also be regarded as thespace of orthonormal frames at the tangent space, and therefore P (M ;O(n))is often refered to as the bundle of orthonormal frames over M .

If we now suppose that the manifold M is orientable, then to actuallyequip M with an orientation means that we are choosing all the clutchingfunctions of TM to be linear maps with the same determinant, that is, el-ements in SO(n). This means we have reduced the structure group of TMfurther to SO(n). We thus have a corresponding principal SO(n)-bundleP (M ;SO(n)) over M .

We recall that SO(2) is the group of rotations of 2and therefore can beidentified with the circle S

1. For n ≥ 3 one has that the fundamental groupπ1(SO(n)) is cyclic of order 2. This means that its universal cover SO(n) is

a two-fold cover p : SO(n) → SO(n).

Definition 4.2 For n ≥ 3 the Spin(n) group is defined to be the universal

covering group SO(n) of SO(n).

For n = 2, Spin(2) can be defined to be S1 regarded as the double cover

of SO(2) ≡ S1 via the map z 7→ z2.

We remark that this geometric definition of the spin groups is certainlycorrect and useful for many purposes, but it hides away many importantadditional properties. In fact the Spin(n) group is naturally a subgroup ofthe group of units of the Clifford algebra of n and inherits from it its groupstructure. We refer to the literature for more on this topic.

For us, in this lecture it is enough to think of the spin groups as definedabove. In fact we are mostly interested in the cases n = 3, 4, and one knows

30

very well these groups: SO(3) is diffeomorphic to the real projective space P 3 and therefore

Spin(3) ∼= S3 ∼= SU(2) ,

where the 3-sphere S3 is being regarded as the group SU(2) of complex

matrices of the form, (z1 z2−z2 z1

),

with determinant 1.For n = 4 one knows that SO(4) ∼= (SO(3)× S

3) and therefore one has:

Spin(4) ∼= S3 × S

3 ∼= SU(2) × SU(2) .

Definition 4.3 A spin structure on an oriented manifold M means a liftingof its structure group from SO(n) to Spin(n).

In other words, one has a principal Spin(n)-bundle P (M ;Spin(n)) overM , together with a projection map P (M ;Spin(n)) → P (M ;SO(n)) whichon each fiber restricts to the double cover Spin(n) → SO(n).

Just as not every manifold admits an orientation, so too, not every mani-fold admits a spin structure. This can be easily said in terms of characteristicclasses (we refer to the literature for a proof of the following statements): Mcompact admits an orientation iff its first Stiefel-Whitney class w1(M) van-ishes, and it admits a spin structure iff one further has w2(M) = 0.

If M admits a spin structure, then it may admit several such structures,and any two of them are regarded as equivalent (or being lousy, as being”equal”) if the corresponding principal bundles are isomorphic. It can beproved that for M compact, the different spin structures on M , a manifoldwith w1(M) = w2(M) = 0, are classified by the cohomology group H1(M)with 2 coefficients.

Definition 4.4 A spin manifold means a (usually smooth, but this conceptcan be adapted to topological manifolds) manifold M , equipped with an ori-entation and a compatible spin structure.

Thus, for instance, every smooth manifold which is a 2-homotopy sphereadmits a unique spin structure.

Spin manifolds have remarkable geometric and topological properties, andwe refer for instance to the beautiful book of Lawson and Michelson for anaccount on the subject.

31

There are however very interesting manifolds which are not spin, as forinstance the complex projective space

P 2. Thus, we now introduce a larger

class of manifolds, the Spinc manifolds, which also have remarkable prop-erties and have the additional feature that every Spin manifold and everycomplex manifold is canonically Spinc.

These structures are the starting point for the theory of Seiberg-Witteninvariants, discussed in the courses of Ron Stern and A. Nemethi.

For this we must first define the Spinc groups.

Definition 4.5 Consider the subgroup SO(n) × SO(2) of SO(n + 2), andthe projection map,

p : Spin(n + 2) → SO(n+ 2) .

The group Spinc(n) is by definition p−1(SO(n) × SO(2)).

Definition 4.6 A Spinc structure on an oriented manifold M means thatwe have equipped M with an oriented 2-plane bundle D with structure groupSO(2) such that the principal SO(n)×SO(2)-bundle corresponding to TM×D lifts to a principal Spinc-bundle.

The bundle D is called the determinant bundle of the Spinc structure.Thus, for instance, if M has a spin structure, then we can take D to be

the trivial bundle M × 2, and one has an induced Spinc structure. That is,every Spin manifold is canonically Spinc, with trivial determinant bundle.

On the other hand, if M is a complex n-manifold, then TM is a complexbundle, then it is automatically oriented. In this case the structure groupof TM is GL(n,

) and by choosing a hermitian metric we can reduce it

further to U(n), which is a subgroup of SO(2n). Let K∗ =∧nTM be the

corresponding anti-canonical bundle of M , dual of the bundle of holomorphicn-forms on M . The complex bundle K∗ is 1-dimensional and so its structuregroup is U(1) ≡ SO(2). Then the structure group of TM ×K∗ is canonicallya subgroup of SO(2n)×SO(2) and it has a canonical lifting to Spinc. Henceevery complex manifold is canonicaly Spinc, with determinant bundle K∗ =∧

nTM .Now consider the case when M = V ∗ = V \ 0, where V is the above

complex surface with an isolated singularity. This is a complex manifold, andtherefore it has a canonical Spinc structure. Since the link K is a codimension1 oriented submanifold of V ∗, it follows that its normal bundle in V ∗ is trivial,

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and it has a canonical trivialization by the unit outwards normal vector field.Hence the canonical Spinc structure on V ∗ defines one on K.

That is,one has the following result, first due to Nemethi and Nicolaescuin [41], where they stduy the Seiberg-Witten invariant of the link K withthis Spinc structure.

Proposition 4.7 The link K of an isolated complex surface singularity hasa canonical Spinc structure.

4.3 The hypersurface case

Now assume the surface V is defined by a single equation f :

3 →

;V = f−1(0). Notice that on the regular points of each surface Vt = f−1(t)one has a canonical, nowhere-vanishing holomorphic 2-form ω obtained bycontracting the 3-form dz1 ∧ dz2 ∧ dz3 with respect to the vector field ∇f =( ∂f

∂z1

, ∂f∂z2

, ∂f∂z3

). In local coordinates ω is:

ω =dz1 ∧ dz2

∂f/∂z3

=dz2 ∧ dz3

∂f/∂z1

=dz3 ∧ dz1

∂f/∂z2

. (4.8)

It is easy to see that the 2-forms on the right hand side of this equation,which are defined where the denominator is not zero, coincide when any twoof them are defined and so they give a global 2-form on the regular part ofeach Vt, which coincides with the one given by contracting Ω with ∇f .

Assume for simplicity that 0 is the only critical point of f , and consideragain the complex manifold V ∗ = V \0. The structure group of its tangentbundle TV ∗ is GL(2,

), and we can always reduce it to U(2) by endowing

V ∗ with some Hermitian metric. If we try to reduce the structure group ofTV ∗ further to SU(2) we may run into problems, since this is not alwayspossible. The obstruction for doing so is the first Chern class of TV ∗ (c.f.IV.1 in [58]). To see this we notice that U(2), being the structure groupof TV ∗, acts naturally on the space of differential forms (of all degrees) onV ∗. In particular it acts on the forms of type (2, 0), i.e. on the sections

of the bundle associated to TV ∗ whose fibre at x ∈ V ∗ is∧(2,0) T ∗

xV∗; this

is isomorphic to the bundle of holomorphic 2-forms on V ∗, which is calledthe canonical bundle K of V ∗. The action of U(2) on

∧(2,0) T ∗xV

∗ is by thedeterminant, and SU(2) is precisely the subgroup of U(2) which acts triviallyon K. Hence we can reduce the structure group of TV ∗ from U(2) further to

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SU(2) if and only if the bundle K is trivial, and the specific reductions of thestructure group of TV ∗ to SU(2) correspond to the specific trivializationsof K. Now, the canonical bundle K is 1-dimensional, so it is classified (upto a differentiable isomorphism) by its Chern class, which is the negative ofthe first Chern class of V ∗. Hence c1(V

∗) vanishes iff K is trivial and thishappens iff we can reduce the structure group of TV ∗ to SU(2).

Now equip V ∗ with the above holomorphic 2-form ω. Then this formdefines a specific reduction to SU(2) of the structure group of TV ∗. SinceSU(2) is isomorphic to the group of unit quaternions Sp(1), this meansthat we have at each point of V ∗ multiplication of tangent vectors by thequaternions i, j, k.

Now choose, at each point x ∈ K, a normal vector field τ(x) of unitlength and pointing outwards. Multiplying τ(x) at each point of K by thequaternions i, j, k, we get a trivialization ρ of TK, the tangent bundle of thelink. This trivialization ρ of TK was called in [55] the canonical framing ofthe link K. This determines a canonical spin structure on K, and thereforedefines a Spinc structure too, which coincides with that of [41] explained inthe previous section.

We remark that the previous discussion generalizes immediately to allgerms of complex surfaces with an isolated complete intersection germ, andto some extent to all normal Gorenstein surface singularities (see [58]). Noticethat the above discussion also implies that the holomorphic 2-form ω is well-defined and never-vanishing on the Milnor fibers Ft of f . Hence it defines areduction to SU(2) of the structure group of TFt. Since SU(2) ≡ Spin(3)one also has a Spin structure on all of Ft. If one considers the fibre Ft asa compact manifold with boundary K, as in the previous section, then onegets that the spin structure on K extends to Ft.

One gets:

Theorem 4.9 Let (V, 0) be an isolated 2-dimensional complete intersectionsingularity. Let V ∗ = V \ 0 and let K be its link. Then:

i) The structure group of the tangent bundle TV ∗ has a canonical reductionto SU(2) ≡ Sp(1).

ii) This reduction to Sp(1) of the structure group of TV ∗ defines a canonicaltrivialization rho of the tangent bundle TK, determined by the three tangentvector fields iτ , jτ , kτ , where τ is the unit normal outwards vector field ofK in V ∗.

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iii) The trivialization rho of TK determines a canonical spin structure othe link, which in this case coincides with the canonical Spinc structure ofNemethi and Nicolaescu. With this spin structure, the link K is the spin-boundary of the Mlnor fibre Ft.

iv) The 2-plane field on K spanned by the vector fields jτ , kτ determines thecanonical contact structure on the link K, and the vector field iτ is (up toscaling) the Reeb vetor field of the contact structure.

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J. Seade, Instituto de Matematicas, Unidad Cuernavaca,Universidad Nacional Autonoma de Mexico,Av. Universidad s/n, Lomas de Chamilpa,62210 Cuernavaca, Morelos, A. P. 273-3, Mexico

jseade@matem.unam.mx

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