algebraic geometry: proceedings of the conference at berlin 9â€“15 march 1988

ALGEBRAIC GEOMETRY

ALGEBRAIC GEOMETRY

Proceedings of the Conference at Berlin 9-15 March 1988

Edited by

H.KDRKE Sektion Mathematik, Berlin, Germany

and

J. H. M. STEENBRINK Mathematical Institute, Nijmegen, The Netherlands

Reprinted from

COMPOSITIO MATHEMA TICA

Volume 76, Nos 1 & 2, 1990

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

ISBN-13:978-94-010-6793-5 e-ISBN-13:978-94-009-0685-3 001: 10.1007/978-94-009-0685-3

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COMPOSITIO MATHEMATICA

Volume 76, Nos 1 & 2, October 1990

Special Issue

ALGEBRAIC GEOMETRY

Proceedings of the Conference at Berlin, 9-15 March 1988

Preface vii

List of participants viii

F.A Bogomolov and AN. Landia: 2-Cocyc1es and Azumaya algebras under birational transformations of algebraic schemes 1

W. Decker: Monads and cohomology modules of rank 2 vector bundles 7

A Dimca: On the Milnor fibrations of weighted homogeneous polynomials 19

AH. Durfee and M. Saito: Mixed Hodge structures on the intersection cohomology oflinks 49

H. Esnault and E. Viehweg: Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields 69

J. Feustel: Eine Klassenzahlformel fUr singuUire Moduln der Picardschen Modulgruppen 87

J. Franke: Chow categories 101

M. Furushima: Complex analytic compactifications of (:3 163

H. Hauser and G. Milller: Analytic curves in power series rings 197

B. Hunt: A Siegel Modular 3-fold that is a Picard Modular 3-fold 203

J. Jurkiewicz: Linearizing some Z/21L actions on affine space 243

I. Luengo and G. Pfister: Normal forms and moduli spaces of curve singularities with semigroup <2p, 2q, 2pq + d> 247

U. Nagel: On Castelnuovo's regularity and Hilbert functions 265

D. Popescu and M. Roczen: Indecomposable Cohen-Macaulay modules and irreducible maps 277

M. Szurek and J.A Wisniewski: Fano bundles of rank 2 on surfaces 295

Compositio Mathematica 76: vii, 1990.

Preface

The Conference on Algebraic Geometry, held in Berlin 9-15 March 1988, was organised by the Sektion Mathematik of the Humboldt-Universitat. The organising committee consisted of H. Kurke, W. Kleinert, G. Pfister and M. Roczen. The Conference is one in a series organised by the Humboldt-Universitat at regular intervals of two or three years, with the purpose of providing a meeting place for mathematicians from eastern and western countries.

The present volume contains elaborations of part of the lectures presented at the Conference and some articles on related subjects. All papers were subject to the regular refereeing procedure of Compositio Mathematica, and H. Kurke acted as a guest editor of this journal.

The papers focus on actual themes in algebraic geometry and singularity theory, such as vector bundles, arithmetical algebraic geometry, intersection theory, moduli and Hodge theory.

We are grateful to all those who, by their hospitality, their presence at the Conference, their support or their written contributions, have made this Conference to a success.

The editors

Compositio Mathematica 76: viii, 1990.

List of participants

R. Achilles, Halle K. Altmann, Berlin Y. Alwadi, Berlin S. Aouira, Berlin A. Aramova, Sofia L. Badescu, Bucharest C. Banica, Bucharest F. Bardelli, Pavia G. Barthel, Konstanz K. Behnke, Hamburg A. Campillo, Valladolid J. Castellanos, Madrid J. Coanda, Bucharest W. Decker, Kaiserslautern F. Delgado, Valladolid B. Dgheim, Berlin A. Dimca, Bucharest K. Drechsler, Halle A. Durfee, Bonn D. Eisenbud, Waltham H. Esnault, Bonn J. EBer, Bonn J. Feustel, Berlin Th. Fiedler, Berlin T. Fimmel, Berlin H. Flenner, Gottingen J. Franke, Berlin M. Furushima, Bonn F. Gaeta, Madrid H. Grabe, Erfurt G.-M. Greuel, Kaiserslautern K. Haberland, lena Z. Hajto, Krakow H. Hauser, Innsbruck F. Herrlich, Bochum L. Hille, Berlin L. Hoa, Halle Th. HOfer, Bonn R. Holzapfel, Berlin B. Hunt, W. Lafayette D. Huybrecht, Berlin F. Ischebeck, M iinster J. Jurkiewicz, Warschau C. Kahn, Waltham G. Janelidze, Tbilisi

G. Kempf, Baltimore W. Kleinert, Berlin S. Kloth, Kothen B. KreuBler, Berlin H. Kurke, Berlin A. Landia, Tbilisi H. Lange, Erlangen O. Laudal, Oslo A. Lipkowski, Beograd V. Lomadse, Tbilisi N. Manolache, Bucharest F. Marko, Bratislava B. Martin, Berlin H. Meltzer, Berlin T. Muhammed, Halle G. MUller, Mainz W. MUller, Berlin U. Nagel, Halle N. Nakayama, Bonn Y. Namikawa, Bonn A. Nemethi, Bucharest V. Palamodov, Moscow A. Parisinski, Gdansk G. Pfister, Berlin D. Popescu, Bucharest P. Pragacz, Torun H. Reimann, Berlin M. Roczen, Berlin H. Roloff, Erfurt D. Rothe, Berlin P. Schenzel, Halle H. Schonemann, Berlin O. SchrOder, Halle T. Shiota, Princeton Th. Siebert, Berlin O. Song, Berlin J. Spies, Berlin J. Steenbrink, Nijmegen M. Szurek, Warsaw J. Tschinkel, Berlin N. Tu Cuong, Hanoi E. Viehweg, Bonn W. Vogel, Halle U. Vollmer, lena E. Zink, Berlin

Compositio Mathematica 76: 1-5, 1990. © 1990 Kluwer Academic Publishers.

2-Cocycles and Azumaya algebras under birational transformations of algebraic schemes

F.A. BOGOMOLOV1 & A.N. LANDIA 2

'Steklov Mathematical Institute of the Academy of Sciences of USSR, Vavilov Street 42, Moscow 117333, USSR; 2Mathematical Institute of the Academy of Sciences of Georgian SSR, Z. Rukhadze Street 1, Tbilisi 380093, USSR

Received 17 November 1988; accepted in revised form 23 November 1989

The basic question whether the injection Br(X) -+ H2(X, (91)tors is an isomorphism arose at the very definition of the Brauer group of an algebraic scheme X. Positive answers are known in the following cases: 1. the topological Brauer group Br(Xtop ) ~ H2(X, (9~p)tors ~ H3(X, Z)tors (J.-P.

Serre); in the etale (algebraic) case the isomorphism is proved for 2. smooth projective surfaces (A. Grothendieck); 3. abelian varieties; 4. the union of two affine schemes (R. Hoobler, O. Gabber). The first author has formulated a birational variant of the basic question, while considering the unramified Brauer group in [1]. The group Brv(K(X)) =

nBr(Av) S Br(K(X)) (intersection taken over all discrete valuation subrings Av of the rational function field K(X)) is isomorphic to H2(X, (9*), where X is a nonsingular projective model of X, i.e. a nonsingular projective variety birationally equivalent to X.

QUESTION. Given a cocycle class )I E H2(X, (9*), is it possible to find a nonsingular projective model X such that )I is represented by a [pln-bundle (i.e. by an Azumaya algebra) on X?

The case where X is a nonsingular projective model of V;'G, with G a )I-minimal group and Va faithful representation of G, was considered in [2]. O. Gabber in his letter to Bogomolov (12.1.1988) has given an affirmative answer to the question in the case of general algebraic spaces. In this paper we give a simple version of his proof for algebraic schemes.

Let X be a scheme, )I E H2(X, (9*), {V i} an affine cover of X. Then the restriction of)l to each Vi is represented by an Azumaya algebra Ai' If we would have isomorphisms Ai1u,nUj ~ AjlU,nup we could glue the sheaves {AJ and get an

2 F.A. Bogomolov and A.N. Landia

Azumaya algebra on X, representing y. But we have isomorphisms AilU,nUj ® End(Eij) ~ AjlU,nUj ® End(Eji) for certain vector bundles Eij, Eji on Ui n Uj .

THEOREM. Let X be a noetherian scheme, y E H2(X, (9;). There exists a proper birational morphism IX: X -+ X such that IX*(Y) is represented by an Azumaya algebra on X.

Proof. It is enough to consider X which are connected. Suppose that {Ui} is an affine open cover of X and that y is non-trivial on at least one Ui' We will construct an Azumaya algebra on a birational model of X by an inductive process which involves adjoining one by one proper preimages of the subsets Ui and, by an appropriate birational change of the scheme and Azumaya algebra obtained, extending the new algebra to the union. We start with some affine open subset U 0

and an Azumaya algebra Ao on it. Now suppose by induction that we already have an Azumaya algebra Ak on the

scheme X k , a Zariski-open subset of the scheme Xk , equipped with a proper birational map ak: Xk -+ X such that X k = akl(Uo u ... U Uk)' Let Uk+l intersect Uo U ... U Uk and Uk+l = a- 1(Uk+d. Suppose that on Uk+1> y is represented by the Azumaya algebra Ak+l' In the same vein as above we have an isomorphism

and we need to extend Ek,k+l to X k and Ek+l.k to Uk+l from their intersection. After this we will change Ak and a:'(Ak+ d by the other representatives Ak ® End(Ek,k+d, a:'(Ak+d ®End(Ek+l.k) ofthe same Brauer classes and glue these Azumaya algebras, hence the proof.

First, extend both sheaves E as coherent sheaves. This can be done by the following

LEMMA. Let X be a noetherian scheme, U £: X a Zariski-open subset, E a coherent sheaf on U. Then there exists a coherent sheaf E' on X such that E'lu ~ E. This is Ex. 11.5.15 in [4].

Note that we can assume that in our inductive process we add neighborhoods U k+ 1 of no more than one irreducible component (or an intersection of irreducible components) of X, different from those contained in X k• Thus we assume X k n Uk + 1 to be connected and the rank of E to be constant on X k n U k+ 1> hence E' will be locally generated by n elements, where n is the rank of E.

LEMMA (see [3], Lemma 3.5). Let X be a noetherian scheme, E a coherent sheaf on X, locally free outside a Zariski closed subset Z on X. Then there exists a coherent sheaf I of ideals onX such that the support of{9x/I is Z with thefollowing

2-Cocycles and Azumaya algebras 3

property. Let IX: X -+ X be the blowing up of X with center I, then the sheaf ri(E) := the quotient of IX*(E) by the subsheaf of sections with support in IX - l(Z), is locally free on X.

Proof. The proof consists of two parts. First: to reduce the number of local generators to get this number constant on the connected components of X (the minima are the values ofthe (local) rank function of E). Second, to force the kernel of the (local) presentations (!)'P -+ Elv -+ 0 to vanish for all neighborhoods from some cover {V}. Both parts are proved by indicating the suitable coherent sheaves of ideals and blowing up X with respect to these sheaves. Let (!)'P L Elv -0 be a local presentation of E. Then Ker(f) is generated by all relations L7'= 1 Ciai = 0 where {ai} stand for the free basis of (!)'P. The coherent sheaf of ideals in the first case is the sheaf defined locally as the ideal I v in (!)v generated by all Ci such that L7'= 1 Ciai E Ker(f) and in the second case as J x =

Ann(Ker(f)). As the number of generators is constant in the case we are interested in, we give the details only for the second part of the proof and refer to [3] for the first.

Let IX: X' -+ X be the blowing up of X with respect to J x and let ri(E) be as in the statement of the Lemma. Let

o -+ (Ker(J))lv' -+ (!)'P, L ri(E)lv' -+ 0

be the local presentation of ri(E). We have 1X-1(Ann(f)) 5; Ann(Ker(J)), Let P E Z', V' = Spec(A') an affine neighborhood of p in X' and let Li'!: 1 CiaiE Ker(J) I v' map to a nonzero element in Ker(J)p' Denote by y a generator of the invertible sheaf 1X-1(Ann(f)) on V" = Spec(A") 5; V' for suitable A". It is clear that there exists for given p and V" a finite sequence of open affine neighborhoods V1, ... , V; such that X'\Z' = V1, V" = V; and Vj n Vj+1 #- (j) for j = 1, .. " s - 1. So suppose V' n (X'\Z') #- (j) and q E V" n (X\Z). Then (c;)q = 0 for i = 1, ... , m and q E Spec(A~) hence ykCi = 0 for i = 1, ... , m for some k. Since y is not a zero divisor, we conclude that Ci = 0 for i = 1" .. ,m. Thus (maybe after considering a finite sequence of points q 1, ... , qs) we prove that (Ker(f))p is trivial for every p EX'. 0

In this way we glue the two sheaves Ak and Ak+ 1 and get an Azumaya algebra on X k U a k+ 1. As the scheme X is quasi-compact, we obtain an Azumaya algebra on X after a finite number of such steps.

Now we have to show that this process can be done in such a way that the class [A] of the Azumaya algebra A constructed in this way is equal to ri*(y). Again this goes by induction on k. We have X k+ 1 = U U V with U = rik-t\(UO u .. · u Uk) and V = rik-)-dUk+d. We have the exact sequence

4 F.A. Bogomolov and A.N. Landia

and by induction hypothesis, at+ 1 tV) - [Ak+ 1] maps to zero in H2(U, (9*) E9 H 2(V,(9*) so it comes from peHi(U () V, (9*). By blowing up Xk+i we may assume that p is represented by a line bundle which extends to U. Then p maps to zero in H2(Xk+1> (9*), hence at+dy) - [Ak+i] = o. 0

Note that we need not bother about the compatibility of isomorphisms, because at each step we choose a new isomorphism between the Azumaya algebra A on U i u ... U U j from the preceding step and Ak on Uk, modulo End(E), End(Ek)'

COROLLARY 1. Let G be afinite group, V afaithju[complex representation ofG. Then there exists a nonsingular projective model X of V/G such that Br(X) = H2(X, (9*).

Proof. The group H2(X, (9*) is a birational invariant of nons in gular projective varieties and is isomorphic to H2(G, Ql/l::) if X is a model of V/G (see [1]). It remains to recall that the group H2(G, Ql/l::) is finite. 0

COROLLARY 2. Let X be a noetherian scheme over C, Z a closed subscheme of X and y e HHx, (9*). Then there exists a proper morphism oc: X' --t X which is an isomorphism above X\Z and maps y to zero in H;-,(Z)(X', (9*).

Proof First, let's have IX(Y) map to zero in H2(X, (9*). To do this, desingularize X by X' --t X. Then in the following exact sequence (in etale cohomology), p will be injective:

Hi (X'\Z' ,(9*) --t Hl'(X', (9*) --t H 2(X', (9*) ~ H 2(X'\Z' , (9*)

i Hl(X, (9*)

The injectivity is due to the injectivity of H 2 (X', (9*) --t H 2 (K(X'), (9*) for a nonsingular irreducible scheme X'.

Now y comes from y' e Hi (X'\Z', (9*) = Pic(X'\Z'). It is obvious that Picard elements lift to Picard elements by the blowing ups from the theorem. Thus from the diagram

Hi (X" , (9*) --t Hi (X"\Z" , (9*) --t Hl'(X" , (9*)

i i Hi (X'\Z', (9*) --t Hl'(X', (9*)

we conclude that y becomes trivial on Z" by X" --t X' which extends y' to X". o

Now let us return to the problem of an isomorphism Br(X) --t H2(X, (9*) for

2-Cocycles and Azumaya algebras 5

nonsingular quasi-projective varieties. The theorem reduces the general problem to the following

QUESTION. Let X' be a blowing up of a nonsingular variety X along a smooth subvariety S and let A' be an Azumaya algebra on X'. Does there exist an Azumaya algebra A on X such that its inverse image on X' is equivalent to A'?

In case the restriction of A' to the pre-image of S is trivial, the question reduces to the one, whether a vector bundle on this preimage can be extended to X as a vector bundle. For example, if dim(X) = 2 then S is a point and its proper preimage is a iP>1 with self-intersection -1. Since the map Pic(X') -+ Pic(iP> 1 ) is surjective, any vector bundle on iP>1 can be extended to X'.

Therefore we obtain a simple proof of the basic theorem in the case dim(X) = 2 using the birational theorem.

In the case of dim(X) = 3 the same procedure reduces the basic problem to the analogous problem of extending vector bundles from iP>2 and ruled surfaces to a variety of dimension three.

References

[1] Bogomolov, F.A., Brauer group of quotients by linear representations. Izv. Akad. Nauk. USSR, Ser. Mat. 51 (1987) 485-516.

[2] Landia, A.N., Brauer group of projective models of quotients by finite groups. Dep. in GRUZNIITI 25.12.1987, no. 373-r87.

[3] Moishezon, B.G., An algebraic analog of compact complex spaces with sufficiently large field of meromorphic functions I. Izv. Akad. Nauk. USSR, Ser. Mat. 33 (1969) 174-238.

[4] Hartshorne, R., Algebraic Geometry. Graduate Texts in Math. 52, Springer Verlag, Berlin etc. 1977.

Compositio Mathematica 76: 7-17,1990. © 1990 Kluwer Academic Publishers.

Monads and cohomology modules of rank 2 vector bundles

WOLFRAM DECKER * Universitat Kaiserslautern, Fachbereich Mathematik, Erwin-Schrodinger-StrafJe, 6750 Kaiserslautern, FRG

Received 19 August 1988; accepted 20 July 1989

Introduction

Monads are a useful tool to construct and study rank 2 vector bundles on the complex projective space IP' n' n ~ 2 (compare [O-S-S]). Horrocks' technique of eliminating cohomology [Ho 2] represents a given rank 2 vector bundle lff as the cohomology of a monad

as follows. First eliminate the graded S = C[zo,"" zn]-module H1lff(*) = EE>mEzHl(lP'n' lff(m)) by the universal extension

° --+ lff --+ f2 --+ Lo --+ 0,

where

is given by a minimal system of generators ('" stands for sheafification). If n = 2 take this extension as a monad with d = 0. If n ~ 3 eliminate dually H n - Ilff( *) by the universal extension

(where c1 = c1 (lff) is the first Chern-class). Then notice, that the two extensions

* Partially supported by the DAAD.

8 Wolfram Decker

can be completed to the display

o 0

I I

I I

lifJ I

I I o 0

of a monad

for $.

To get a better understanding for f!4, qJ and", consider first the case n = 2,3. Then f!4 is a direct sum of line bundles by Horrocks' splitting criterion [Ho 1]. Taking cohomology we obtain a free presentation

with B = H°f!4(*). The crucial point is that this is minimal [Ra]. Moreover, if n = 3, then B is self-dual [Ra]: B V (c 1 ) ~ B. We will see below that up to isomorphism qJ is the dual map of "'.

Let us summarize and slightly generalize. Consider an arbitrary graded S-module N of finite length with minimal free resolution (mJ.r. for short)

If n = 2 then N ~ H1 $ (*) for some rank 2 vector bundle $ on IFD 2 iff rk L1 --;;;, rk Lo + 2 (compare [Ra]). In this case 8 is uniquely determined as kef 1X0 :

Monads and cohomology modules of rank 2 vector bundles 9

(This sequence is self-dual by Serre-duality [Ho 1, 5.2], since t&"V(c 1) ~ t&"). For n = 3 there is an analogous result. Answering Problem 10 of Hartshorne's

list [Ha] we prove:

PROPOSITION 1. N is the first cohomology module of some rank 2 vector bundle on iP' 3 ijf

(1) rkL1 = 2rkLo + 2 and (2) there exists an isomorphism <1>: LHcd --=-+ Ll for some C1 E 7L such that

0:0 0 <1>0 0:0' (cd = o.

In this case any satisfying (2) defines a monad

and t&" is a 2-bundle on iP'3 with H1t&"(*) ~ N (and C1 = c1(t&")) ijf(M(t&")) ~ (Mtl»for some <1>.

To complete the picture let us mention a result of Hartshorne and Rao (not yet published). If N ~ H1t&"(*) as above then LO'(c1) LLl is part of a minimal system of generators for ker 0:0 . In other words: There exists a splitting

inducing the monad

and the mJ.r.

resp. For n ;?; 4 there is essentially only one indecomposable 2-bundle known on

iP' n: The Horrocks-Mumford-bundle ~ on iP' 4 with Chern-classes C1 = -1, C2 = 4. We prove:

PROPOSITION 2. The mJ.r. of H2~(*) decomposes as

10 WoifTam Decker

with B = H O fJI( *), inducing the monad

and the minimal free presentation

The corresponding mJ.r. decomposes as

0--+ L5 --+ L4 --+ L3 tB H 2 (: ~J, L2 tB H 1 -(-*-P~-')-+l L1 ~ Lo--+

--+ H1 .1'(*) --+ 0

inducing the mJ.r.

(M(.1'» is the monad given in [H-M]. Using its display we can compute the above mJ.r.'s explicitly. Especially we reobtain the equations of the abelian surfaces in OJ> 4 ([Ma 1J, [Ma 2J).

Of course we may deduce from .1' some more bundles by pulling it back under finite morphisms n: OJ> 4 --+ OJ> 4' The above result also holds for the bundles n*.1' with (M(n* .1'» = n*(M(.1'».

There is some evidence (but so far no complete proof), that a splitting as in Proposition 2 occurs for every indecomposable 2-bundle on OJ> 4' This suggests a new construction principle for such bundles by constructing their H 2-module first.

Proof of Proposition 1

Let n = 3 and N be a graded S-module of finite length with mJ.r.

Suppose first that N ~ H1 cS'( *) for some 2-bundle cS' on OJ> 3 (with first Chern-class c1). As seen in the introduction, Horrocks' construction leads to a monad

for $. The dual sequence

is a monad for $V(c1 ) ~ $. The induced presentation of N has to be isomorphic to that one given by the mJ.r.:

Dualizing gives (2) since 0(0 0 oO(O'(c 1 )).

Conversely if N satisfies (2), we obtain a monad (Mil» by sheafification. (Since N = 0, 0(0 is a bundle epimorphism. Dually 0(0' (c 1 ) is a bundle monomorphism.) Let $ be the cohomology bundle of (MIl»' Then H 1$(*) ~ N. $ has rank 2, if N satisfies (1). 0

REMARK 1. (i) Let N ~ H l $(*) as above with induced splitting

as in the introduction. Recall that $ is stable iff HO(IP 3' $(m)) = 0 for m ::::; -c1 /2. Thus $ is stable iff L'z has no direct summand S(m) with m ~ cd2. Notice that this condition only depends on N.

(ii) If N satisfies (1) and has only one generator, then (2) is obviously equivalent to the symmetry condition L{(c1) ~ L l . Thus [Ra,3.1] is a special case of Proposition 1.

EXAMPLES. (i) The well-known Null correlation bundles are by definition the bundles corresponding to the S-module C. Consider the Koszul-presentation

The isomorphisms 4S ~ 4S with 0(0 0 0 O(o(c l ) = 0 are precisely the 4 x 4 skew symmetric matrices with nonzero determinant. Two such matrices give isomorphic bundles iff they differ by a scalar (use [O-S-S, II, Corollary 1 to 4.1.3]). The moduli space of Null correlation bundles is thus isomorphic to IP 5 \ G, where G is the Plucker embedded Grassmanian of lines in 1P 3 .

Unlike the case n = 2 the bundle is not uniquely determined by the module.

12 Wolfram Decker

(ii) The S-module

satisfies (1) and the symmetry condition L'{ ~ Lv i.e. the necessary conditions of [Ra]. But N does not satisfy (2).

Cohomology modules of the Horrocks-Mumford-bundle iF

We first recall the construction of iF [H-M]. Let

v = Map(Zs, C)

be the vector space of complex valued functions on Zs' Denote by

HeN c SL(S,C)

the Heisenberg group and its normalizer in SL(S, C) resp. Let Vo = V, VI' V2 , V3 and

be defined as in [H-M]. The V; are irreducible representations of Hand N of degree S. W is an irreducible representation of N /H of degree 2. It is unimodular, so it comes up with an invariant skew symmetric pairing.

Let IP 4 = IP(V) be the projective space of lines in V. The Koszul-complex on IP(V) is the exact sequence

(K)

obtained by exterior multiplication with the tautological sub bundle

@(-1)-+ V®@.

The exterior product provides (K) with a self-duality (with values in @(-I)®AS V ~ @(-1)).


This induces the natural pairings

and is compatible with the action of SL(5, IC). It can be extencp.d to (K) ® Wby tensoring with the invariant form, then being

compatible with the action of N. As in the proof of [H-M, Lemma 2.4] it follows, that (K) ® W decomposes as

0(-1)@VI (ari(O_I) ~) 0@V3

---+ EEl EEl ---+ ... , ... 0(-1)@VI@U 0@V3 @U

"" /

(A2T)( -3) @ W

/ "" ° 0

given by the splitting into irreducible N-modules. Moreover the induced

is the self-dual Horrocks-Mumford-monad, whose cohomology is $' (normalized such that c 1 $' = - 1).

To proof Proposition 2 consider the display

0 0

1 1 0-----+ (O( -1) ® V1 1% I $'-----+ 0

1 1 o -----+ (O( -1) ® V1 -+ (A 2 5"')( - 3) ® W -----+ f2 -----+ 0

1 1 (O® V3 (O® V3

1 1 0 0

It first follows that H2 $'( *) = W is a vector space, sitting in degree - 2 (compare [H-M]). Its m.f.r. is the Koszul-complex obtained from (K) ® Wby taking global

14 Wolfram Decker

sections. So it decomposes, inducing the presentation

and the Horrocks-Mumford-monad. But this is just

apply e.g. [O-S-S, II, Corollary 1 to 4.1.3] (notice that HOff = 0 implies HO~ = HO~*( -1) = 0 by construction of (M(ff))).

It remains to show that CXo is minimal and that the corresponding m.f.r. of H1ff(*) decomposes, inducing the m.f.r., say,

of F = HOff(*).

From the second row of the display we obtain the m.f.r. of Q = H02(*):

o

j S(-3)0 W

]1, S(-2)0 V0 W

j o o


The third column of the display gives rise to the commutative diagram

o

1

Y11 1P1

F 1 -=-~4 S( - 2) ® V ® w

1 1 0----+ F lQ l S ® V3 ----+ HI ff(*) ----+ 0

1 1 0 0

with exact columns and bottom row. The induced

is exact and it is minimal, iff O('b O(~, 0(3 have no entries in C\{O}. But since H O ff(1) = 0 [H-M], these maps have only entries in degrees ~ 1. D

REMARK 2 (i) Let us describe (M(ff)) more explicitly by choosing convenient bases of VI' V3 = Vr, Wand forgetting the N-module structure (compare the proof of [H-M, Lemma 2.5].)

Choose the basis eo, . .. , e4 of V = Map(Zs, q given by ei(j) = iiij and its dual basis zo, ... ,Z4 E V*.

16 Wolfram Decker

Define

A = (aij)0,;;i.;4 by O.;j.;l

Then wo, W1, given by Wiei) = aij is a basis of W. Identifying W ~ C2, the invariant form on W becomes the standard symplectic form Q = (_? b) on C2 •

We thus may rewrite (M(ff)) as

(M(ff)) 5(9( -1) ~ 2(A 2 .r)( - 3) ~ 5(9,

the matrices operating by exterior multiplication. (ii) From the explicit form of (M(ff)) we can compute 0:0 explicitly. Choose

a convenient basis of A 2V0 W = (V1 EB U). Then

15S( -1) ~ 5S -+ H 1.9F(*) -+ 0

is the matrix

0 Z3 0 0 Z2 0 0 Zl Z4 0 Zo 0 0 0 0

Z3 0 Z4 0 0 0 0 0 Z2 Zo 0 Zl 0 0 0

0:0 = 0 Z4 0 Zo 0 Zl 0 0 0 Z3 0 0 Z2 0 0

0 0 Zo 0 Zl Z4 Z2 0 0 0 0 0 0 Z3 0

Z2 0 0 Zl 0 0 Zo Z3 0 0 0 0 0 0 Z4

Resolving it (use e.g. [B-SJ), we obtain the mJ.r. of H 1.9F(*). Its shape is

o ~ 2S( - 8) ~ 20S( - 6) ~ 35S( - 5) EB 2S( - 3) ~ (15S( - 4) EB 4S( - 3)) EB lOS( - 2)

~ 15S( -1) ~ 5S ~ H 1.9F(*) ~ O.

(iii) Consider the induced mJ.r. of F and its dual

... ~ 35S( -5) ~ 15S( -4) EB 4S( -3) --~~ 15S(3) EB 4S(2) ~ 35S(4) ~ ... ~ /

F~FV(-1)

/ ~ o O.

r can be computed by resolving tyo (use again [B-SJ). We thus obtain explicit


bases for the spaces of sections H O $i'(m). Especially we get the equations of the zero-schemes of sections of $i'(3), including the abelian surfaces in IP 4'

(iv) Let n: IP 4 -+ IP 4 be a finite morphism and d4 its degree. Then n* $i' is a stable 2-bundle with Chern-classes c1 = -d, C2 = 4d2• Proposition 2 and the above remarks also hold for n* $i': Replace (K) by n*(K), (M($i')) by n*(M($i')) =

(M(n* $i')) and zo,'''' Z4 in !Xo by fo,'" ,J4' where fo,' .. ,J4 are the forms of degree d defining n.

Acknowledgement

I'd like to thank R. Hartshorne for helpful discussions.

References

[B-SJ [HaJ

[H-MJ

[Ho IJ

[Ho 2J

[Ma IJ

[Ma 2J

[O-S-SJ

[RaJ

D. Bayer and M. Stillman: Macaulay, a computer algebra system for algebraic geometry. R. Hartshorne: Algebraic vector bundles on projective spaces: A problem list. Topology 18 (1979) 117-228. G. Horrocks and D. Mumford: A rank 2 vector bundle on p 4 with 15,000 symmetries. Topology 12 (1973) 63-81. G. Horrocks: Vector bundles on the punctured spectrum of a local ring. Proc. London Math. Soc. 14 (1964) 689-713. G. Horrocks: Construction of bundles on P". In: Les equations de Yang-Mills, A. Douady, J.-L. Verdier, seminaire E.N.S. 1977-1978. Asterisque 71-72 (1980) 197-203. N. Manolache: Syzygies of abelian surfaces embedded in p4. J. reine angew. Math. 384 (1988) 180-191. N. Manolache: The equations of the abelian surfaces embedded in p4(C). Preprint, Bukarest, 1988. C. Okonek, M. Schneider and H. Spindler: Vector bundles on complex projective spaces. Boston, Basel, Stuttgart: Birkhiiuser 1980. P. Rao: A note on cohomology modules ofrank two bundles. J. Algebra 86, (1984) 23-34.


On the Milnor fibrations of weighted homogeneous polynomials

ALEXANDRU DIMCA Department of Mathematics, INCREST, Bd. Pacii 220, RO-79622 Bucharest, Romania; Current address: The Institute for Advanced Study, Princeton, NJ 08540, USA

Received 6 October 1988; accepted 20 July 1989

Let w = (wo, ... , wn) be a set of integer positive weights and denote by S the polynomial ring C[xo, ... , xn] graded by the conditions deg(xi) = Wi' For any graded object M we denote by M k the homogeneous component of M of degree k. LetfE SN be a weighted homogeneous polynomial of degree N.

The Milnor fibration of f is the locally trivial fibration f: cn + 1 \f - 1 (0) -+

C\{O}, with typical fiber F = f- 1(1) and geometric monodromy h: F -+ F, h(x) = (twoxo,' .. , tWnxn) for t = exp(2ni/N). Since hN = 1, it follows that the (complex) monodromy operator h*: H'(F) -+ H'(F) is diagonalizable and has eigenvalues in the group G = {t"; a = 0, ... ,N - 1} of the N -roots of unity.

We denote by H'(F)a the eigenspace corresponding to the eigenvalue t-a, for a = O, ... ,N-1.

When f has an isolated singularity at the origin, the only nontrivial cohomology group Hk(F) is for k = n and the dimensions dim Hn(F)a are known by the work of Brieskorn [2]. But as soon as f has a nonisolated singularity, it seems that even the Betti numbers bk(F) are known only in some special cases, see for instance [9], [14], [17], [22], [25].

The first main result of our paper is an explicit formula for the cohomology groups Hk(F) and for the eigenspaces Hk(F)a' Let Q' be the complex of global algebraic differential forms on Cn + 1, graded by the convention deg(udxi, /\ ... /\ dxik) = p + wi, + ... + Wik for UE Sp. We introduce a new differential on Q', namely Df(m)=dm-(Iml/N)df /\m, for mEQ~ with Iml=p the degree of m and d the usual exterior differential, similar to Dolgachev [8], p.61.

For a = 0, ... , N - 1 we denote by Q(a) the subcomplex in Q' given by EBs;;.OQ·-a+sN·

To a D rclosed form mE Qk+ 1 we can associate the element c;(m) = [i* A(m)] in the de Rham cohomology group Hk(F), where A is the contraction with the Euler vector field (as in [12], p. 467 in the homogeneous case and [8], p. 43 in the weighted homogeneous case) and i: F -+ Cn + 1 denotes the inclusion.

THEOREM A. The maps C;:Hk+1(Q',Df )-+Rk(F) and c;:Hk+1(Q(a),Df )-+ Rk(F)a are isomorphisms for any k ~ 0, a = 0, ... , N - 1, with R denoting reduced cohomology.

20 Alexandru Dimca

The proof of this Theorem depends on a comparison between spectral sequences naturally associated to the two sides of these equalities see (1.8).

Our second main theme is that these spectral sequences can be used to perform explicit computations and to derive interesting numerical formulas, in spite of the fact that the E 1 -term has infinitely many nonzero entries and that degeneration at the E2 -term happens only in special cases (see (3.10) and (3.11) below).

The eigenspaces H'(F)o are particularly interesting. If P = P(w) denotes the weighted projective space Proj(S), V the hypersurfacef = 0 in P and U = P\ V the complement, then there is a natural identification H'(F)o = H'(U). We conjecture an inclusion between the filtration on H'(F)o induced by the spectral sequence mentioned above and the (mixed) Hodgefiltration on H'(U), having a substantial consequence for explicit computations and extending to the singular case an important result of Griffiths [12], see (2.7.ii) below.

To prove the analogous result for these filtrations on the whole H'(F), we establish first some subtle properties of the Poincare residue operator

R: H,(cn + 1 \F) -+ H'-1(F) (see (1.6), (1.20), (1.21) and (2.6)) which may be useful in their own.

Note that the Betti numbers bk(V) are completely determined by bk(U) and hence one can get by our method at least upper bounds for all bk(V) as well as the exact value of the top interesting one (i.e. bn +m - 1(V) where m = dimf-1(0)sing) in a finite number of steps see (2.8).

Then we specialize to the case when f has a one-dimensional singular locus, a situation already studied (without the weighted homogeneity assumption) by N. Yomdin and, more recently and more completely, by D. Siersma, R. Pellikaan, D. van Straten, T. de Jong. We relate the spectral sequence (Er(f)o dr) to some new spectral sequences associated to the transversal singularities off, these being the intersections off -1 (0) with transversals to each irreducible component of f- 1(0)sing' We hope that these intricate local spectral sequences will play a fundamental role in understanding better even the isolated hypersurface singularities (see for instance the nice characterization (3.10') of weighted homogeneous singularities). Concerning the numerical invariants in this case, we get interesting and effective formulas for the Euler characteristics X(V) and X(F) extending in highly nontrivial way the known formulas for the homogeneous case (we conjecture them to hold in general and check them under certain assumptions on the transversal singularities off, see (3.19.ii)).

The last section is devoted to explicit computations with our spectral sequence. The first two of them are just simple illustrations of our technique, while the third offers a more subtle example, for which we know no other method to get even the Betti numbers for V. It is interesting to remark that if one wants to compute the Euler characteristic X(V) in this case using Theorem (3.1) in Szafraniec [26], then one is led to compute bases of Milnor algebras (and signatures of bilinear forms

On the Milnor fibrations of weighted homogeneous polynomials 21

defined on them) of a huge dimension (~67 ) and this is an impossible task even· for a computer!

A more theoretical application (improving a result of Scherk [20]) is given in the end, the key point in the proof being again an explicit computation with the spectral sequence.

A basic open problem is to decide whether the spectral sequence (E,(f), d,) or its local analog (E,(g, 0), d,) degenerates always in a finite number of steps and, in the affirmative case, to determine a bound for this number in terms of other invariants off or g.

1o Some spectral sequences

In this section we shall use many notations and results from Dolgachev [8] without explicit reference.

Let A: Ok --+ Ok -1 denote the contraction with the Euler vector field "£WjXjo/OXj. For k ~ 1 we put Cik = ker(A: Ok --+ Ok-I) = im(A: ok+ 1 --+ Ok) and let Oi denote the associated sheaf on P. One has also the twisted sheaves Oi(s), for any SE 7L.

Let i: U --+ P denote the inclusion and put Ot(s) = i*Oi(s). The Milnor fiber F is an affine smooth variety and according to Grothendieck

[13] one has HO(F) = HO(r(F, OF )). Let p: F --+ U denote the canonical projection and note that

N-l

p*OF = EB 0iT( -a). (1.1) a=O

If we let A~ = qu, 0iT( - a)) and AO = EB: ~ 6- A~, then we clearly have

HO(F) = HO(A), HO(F)a = HO(A~). (1.2)

There is a natural increasing filtration F. on A~, related to the order of the pole a form in A~ has along V, namely

F.A~ = 0 for s < 0 and F.A~ = {W/!";WE O~N-a} for s ~ 0 similar to [12]. (1.3)

But for obvious technical reasons it is more convenient to consider the decreasing filtration.

(1.4)

22 Alexandru Dimca

The filtration Ps is compatible with d, exhaustive (i.e. A~ = UPS A~) and bounded above (Fn+ 1 A' = 0). Here d denotes the differential of the complex A~ which is induced by the exterior differential d in nj, via (1.1) and which is given explicitly by the formula

d(OJ/!') = df (OJ)·f-s-l where df(OJ) = fdOJ - (I OJ liN) df /\ OJ. (1.5)

By the general theory of spectral sequences e.g. [16], p. 44 we get the next geometric spectral sequence.

(1.6) PROPOSITION. There is an E 1 -spectral sequence (Er(f)a, dr) with

and converging to the cohomology eigenspace H'(F)a'

Moreover one can sum these spectral sequences for a = 0, ... , N - 1 and get a spectral sequence (Er(f), dr) converging to H·(F). And (Er(!)o, dr) and (Er(!), dr) are in fact spectral sequences of algebras converging to their limits as algebras. Note that H'(F)o ~ H'(U), either using the fact that U = FIG, G acting on F via the geometric monodromy or the fact that nu is a resolution of C [24].

We pass now to the construction of some purely algebraic spectral sequences. Let (Ba,d',d") be the double complex B~·t = n::':~l, d' = d and d"(OJ) = -I OJ II N df /\ OJ for a homogeneous differential form OJ. Note that the associated total complex B~, with B~ = EBs+t=kB~·t,D = d' + d" is precisely the complex (n~-1, Df )·

Similarly B' = EB B~ = (n' - 1 ,D f). Consider the decreasing filtration FP on B~ given by FPB~ = EBs;.pB~·k-S and

similarly on B'. Using the contraction operator i\, we define the next complex morphisms, compatible with the filtrations:

5: B~ -+ A~ and 5: B' -+ A',

5(OJ) = i\(OJ}f-t for OJE B~·t.

Note that B' and A' are in fact differential graded algebras, but 5 is not compatible with the products.

(1.7) PROPOSITION. There is an E 1 -spectral sequence (' Er(f)a' dr) with

and converging to the cohomology H·(B~). The operator 5 induces a morphism (j,: (' Er(f)a, dr) -+ (Er(!)a .dr) of spectral sequences.


Moreover one can sum these spectral sequences' E,(f)a and get a spectral sequence (' E,(f), d,) converging to H'(B') and a morphism (' E,(f), d,) --+ (E,(f), d,). The proof of these facts is standard e.g. [16], p. 49. Let E,(f)o (resp. E,(f)) denote the reduced spectral sequence associated to E,(f)o (resp. E,(f)) which is obtained by replacing the term at the origin E?'O = E~o = C by zero. For a -1= 0,

we put E,(f)a = E,(f)a' We clearly have natural morphisms '5,:' E,(f)a --+ E,(f)a' '5,: ' E,(f) --+ E if)

induced by ~,. We can state now a basic result.

(1.8) THEOREM. The morphisms '5, are isomorphismsfor r;;?; 1 and they induce isomorphisms H'(Ba) ~ jj'(F)a and H'(B) ~ jj'(F).

Proof. Since Fn + 1 B' = Fn + 1 A' = 0, the filtrations F are strongly convergent [16], p. 50 and hence it is enough to show that '51 is an isomorphism. The vertical columns in ' E 1 (f) correspond to certain homogeneous components in the Koszul complex K'.

(1.9)

of the partial derivatives.!; = (of)/(ox i ), i = 0, ... , n in S. To describe the vertical columns in E 1 (f) is more subtle. Note that f K' is a subcomplex in K' and let K' denote the quotient complex K' /fIC. There is a map Ll: K' --+ K'-l induced by A which is a complex morphism and hence j(' = ker Ll is a subcomplex in K'.

Let Li denote the composition K' --+ K' ~ j(' - 1 •

Then the vertical lines in E1 (f) correspond to certain homogeneous components in the cohomology groups H'(j('). The morphism 51 corresponds to Li*: H'(K') --+ H'(j('-l) and a well-defined inverse for Li* is given by the map

V: H'(j('-l) --+ H'(K'), V[A(w)] = [df /\ A(w)!(Nf)]. (1.10)

To check this, use that df /\ w = 0 implies 0 = A(df /\ w) = Nfw - df /\ A(w).

(1.11) EXAMPLE. Assume thatf has an isolated singularity at the origin. Then fo,'" ,J,. form a regular sequence in S and we get' Ett(f)a = 0 for s + t -1= nand

where Q(f) = S/(fo, ... , J,.), w = Wo + ... + wn • Moreover, the Poincare series for Q(f) (see for instance [7], p. 109) implies that Q(f)k = 0 for k > (n + 1)N - 2w. Hence in this case all our spectral sequences are finite and degenerate at the E1 -term (the degeneracy of the component a = 0 being equivalent to Griffiths' Theorem 4.3 in [12]). Note that one can have , E 11 ,n + 1 (f)a -1= O. In general, one has the next result about the size of the spectral sequence' E,(f).

24 Alexandru Dimca

(1.12) Proposition. 'E:·t(f) = 0 for any r ~ 1 and s + t < n - m, where m = dimf-1(0)sing'

Proof The result follows using the description of' Ei·t (f) in terms of the Koszul complex and Greuel generalized version of the de Rham-Lemma, see [11], (1.7).

(1.13) Corollary. jjk(F) = 0 for k < n - m.

This result is implied also by [15], but (1.12) will be used below in (2.8) in a crucial way.

Now we show that our complexes can be used to describe very explicitly the Poincare residue isomorphism R: Hk+ l(C n + 1 \F) -+ jjk(F) and the SebastianiThom isomorphism.

When X is a smooth complex manifold and D is a smooth closed hypersurface in X there is a Gysin exact sequence

(1.14)

where i* is induced by the inclusion i: X\D -+ X and R is the Poincare residue, see for instance [24], Section 8.

Let Cj denote the complex 0' with the differential D f introduced above (up to a shift Cj = B'!) and note that

or:: Cj -+ r(C n + l\F, 0Cn+'\F) (1.15)

or:(w) = w - (df /\ A(w))jN(f - 1)

is a morphism of differential graded algebras (i.e. dOr:(w) = Or:D f(w) and Or:(Wl /\ wz) = or:(w1) /\ Or:(wz)).

Using the definition of the Poincare residue as in [12], p. 290 it follows that

ROr:(w) = (-INN[)(w). (1.16)

Since R is an isomorphism by using (1.14) in the case X = (;"+ 1, D = F and [) is an isomorphism by Theorem A, it follows that Or::H'(Cf)-+H,((;"+l\F) is an isomorphism too.

To discuss the Sebastiani-Thorn isomorphism (see for instance [17]), we introduce a new complex associated to f, namely C f which is the complex 0' with the differential i5 fW = dw - df /\ w.

Define 8: Cf -+ Cf to be the C-linear map which on a homogeneous form w with k = Iwl acts by the formula 8(w) = A(k)'w, where A(k) = 1 for k ::::; Nand A(k) = (k - N)··· (k - tN)' N- t for tN < k ::::; (t + I)N, t ~ 1.

Then it is obvious that 8 induces a complex isomorphism between the

corresponding reduced complexes. In particular we get isomorphisms 0: }f(CI ) --+ Ht(CI ) for any t ~ 1.

Let w' = (w~, . .. ,w~.) be a new set of weights and f' E C[Yo,' .. ,Yn'] be a homogeneous polynomial of degree N with respect to these weights. Then it is easy to check that

(1.17)

and that there is no such result for C I+ r. Using the isomorphisms 0 and Theorem A we get the Sebastiani-Thom

isomorphism

Hk(F") = EB HS(F) ® Ht(F') (1.18) s+t=k-l

where F', F" denote the Milnor fibers off' and f + f' respectively. Keeping trace of the homogeneous components in (1.17) we get

with c = 0, ... ,N - 1 and H·(F')N = H·(F')o' When f' = y~, Example (1.11) shows that ii·(F')o = 0 and ii·(F')c = <15(y~-c-l dyo), a onedimensional vector space for c = 1, ... , N - 1. It follows that dim Hk(F")o =

dimiik - 1(F)*o where iiS(F)*o = (f)c=l,N-1HS(F)c This equality of dimensions is related to the next geometric setting. Let

H: Yo = 0 denote the hyperplane at infinity in the compactification pew, 1) of cn + 1, let V" c P(w,l) be the hypersurface given by f(x) - y~ = 0 and set V" = P(w,I)\V". Since H n V" = V, V"\H = Cn + 1 \F, the Gysin sequence (1.14) applied to X = V", D = H n V" gives

(As a matter off act V" may be singular and then to apply (1.14) one has to do as follows. Let q: pn+ 1 --+ pew, 1) be the covering map induced by

(xo :'" :xn :Y)f-+(x1)o: ... : x;:'" :y)

and let G be the corresponding group of covering transformations. If we set G = q-l(V"),H = q-l(H), then there is a Gysin sequence associated

to X = G, D = H n G. And the <i-invariant part of this exact sequence is precisely the exact sequence which we have written above).

26 Alexandru Dimca

Note that

dim Hk(Cn+ l\F) = dim 11k- 1 (F) = dim11k - 1 (F),.0 + dim11k - 1 (F)0

= dim Hk(U") + dim 11k-(U).

It follows that the first and the last map in the above exact sequence are trivial. Note also that the geometric monodromy h acts on Cn + 1 \F and hence it makes sense to define H s(Cn + 1\F),.0 as above.

It will be clear from what follows that the image ofi* is precisely Hk(Cn + 1 \F),.o and hence we can write the next diagram of isomorphisms:

Hk(F")o ~ Hk(U") ~ Hk((Cft+1\F),.o ~ Hk- 1(F),.0

'I ~ l' 1"'( C f d, "-- j.'( C f)"

Hk + 1 (C f+ f')o ..... ( --------'-"'-----

(1.19)

Here 1/1 is defined in a natural way: if WE Hk( C f)c (i.e. W is a sum W 1 + ... + wp of homogeneous forms such that Iw;I == -c modulo N) then I/I(w) = W /\ Yo- 1 dyo

The formula (1.16) tells us that the triangle in the diagram (1.19) is commutative up to a constant. The big rectangle in the diagram is commutative in a similar way by the next result.

(1.20) LEMMA. Ri*b(}1/1 = -l/Nb(}. Proof We have to show that both sides of this equality yield the same result

when applied to an element W = w 1 + ... + wp E Hk(Cf)c as above. Since these computations are rather tedious, we treat here only the case p = 2 and let the reader check that the general case is completely similar.

So let W = W 1 + w2 with q = tN - c = I w 1 I and q + N = I w21 (when IW21 -lw1 1 > N the forms W 1 and w2 are themselves cycles in Hk(Cf)c and the proof is easier!).

The condition D fW = 0 is equaivalent to

(i) df /\ w2 = 0 (ii) df /\ w 1 = dW2

(iii) 0 = dw1 .


It is easy to see that

To compute the residue of this element we proceed as follows. First we apply A to the equality (i) and get

(iv) w 2/(f - 1) + w 2 = df /\ A(W2)/(N(f - 1)).

Next we can divide this equality by (f - 1)' and get

(v) w 2/(f - 1y+l + w 2/(f - 1)' = -d(A(w2)/Ns(f - 1)S) + dA(w2)/Ns(f -)'.

If we apply A to (ii), we get

This should be put in (v), one should apply once more this trick getting a term containing dA(w 1 ) and then replace this by qWl as follows by applying A to (iii).

Let As = wds(f - 1)' + w 2/(f - 1y+ 1 and note that As is a closed form on C"+ 1 \F for any s ~ 1. The above computation implies that the associated cohomology classes satisfy [As] = ((q - N(s - 1))/Ns)[As- 1 ] and hence

This ends the proof of (1.20) in this case.

(1.21) REMARK. There is a nice geometric consequence of the existence of the diagram (1.19). One can think of the weighted projective space P(w, 1) as a compactification of C"+ 1 \F such that the complement P(w, 1)\(C" + 1 \F) consists of two irreducible components, namely V" and H. Using the isomorphism ct, it follows that any cohomology class in H"(Cn + 1 \F) can be represented by a closed differential form on C" + 1 \F having a pole of order 1 along V" and a pole (possibly of a higher order) along H.

On the other hand, the isomorphism i* (jel/! shows that any class in H'(C" + 1 \F)",o can be represented by a closed differential form on C"+ 1 \F having a pole on V" and no poles at all along H. It can be shown similarly that any class in H'(C" + 1 \F)o can be represented by a form having a pole of order 1 along Hand a pole along V". It would be nice to have a more geometric understanding of this phenomenon.

In conclusion, the natural isomorphism Hk(F) = Hk(F)o EB Hk(F)",o =

Hk(U) EB Hk+ l(U") shows that it is enough to concentrate on the cohomology groups H'(U) and this is what we do in the next two sections.

28 Alexandru Dimca

2, The relation with the Hodge filtration

Let us consider the decreasing filtration F" on H'(U) defined by the filtration F" on Ai>, namely

F"H'(U) = im{H'(F"Ai» = H'(U)}. (2.1)

On the other hand there is on H'(U) the decreasing Hodge filtration F~ introduced by Deligne [5].

(2.2) THEOREM. One has FSH'(U):::> F~+ 1 H'(U) for any sand FO H'(U) = F1H'(U)FCJtH'(U) = H'(U).

Proof. Let p: pn --+ P be the projection presenting P as the quotient ofpn under the group G(w), the product of cyclic groups of orders Wi.

Then J = p*(f) = f(x'O°, .. . , x:n) is a homogeneous polynomial of degree N and let U be the complement of the hypersurface J = 0 in pn.

Since H'(U) can be identified to the fixed part in H'(U) under the group G(w) and since the monomorphism p*: H'(U) --+ H'(U) is clearly compatible with the filtrations F S and F~, it is enough to prove (2.2) for U.

To simplify the notation, we assume that w = (1, ... ,1) from the beginning. Then U is smooth and it is easier to describe the construction of the Hodge filtration [24].

Let p: X --+ pn be a proper modification with X smooth, D = p-l(V) a divisor with normal crossings in X and (J = X\D isomorphic to U via p.

From this point on it is more suitable to work with hoi om orphic differential forms on our algebraic varieties. Ifnu is this holomorphic sheaves complex, anu the algebraic version of it and i: U --+ pn is the inclusion, then one has inclusions i*(anu) c: ni>n(*V) c: i*nu, where ni>n(*V) denotes the sheaves of merom orphic differential forms on pn with polar singularities along V. By Grothendieck [13], the inclusion i*enu) c: ni>n(* V) induces isomorphisms at the hypercohomology groups. And the same is true for the inclusions nx(log D) c: nx(*D) c: j*n'a where j: (J --+ X is the inclusion, nx(*D) is defined similarly to npn(*V) and nx(log D) is the complex of hoi om orphic differential forms with logarithmic poles along D[24].

Recall that there is a trivial filtration (J;;o on any complex K', by defining (J ;;o.K' to be the subcomplex of K' obtained by replacing the first s terms in K' by O. The Hodge filtration is given by

(2.3)

via the identifications

H'(nx(logD)) = H(j*niJ) = H'(niJ) = H'((J) = H'(U).

The filtration F' on the complex Ai> is related to a filtration F' on the complex Qp'(* V) defined in the following way: Fsn~.(* V) is the sheaf of meromorphic j-forms on pn having poles of order at most j - s along V for j ~ sand F'Q~n(* V) = 0 for j < s.

Note that F'Qtn(* V) !::::: Qtn«(j - s)N) for j ~ s. We get next a filtration on the complex Qx(*D) !::::: p*(Qpn(* V» by defining FSQx(*D) = p*(F'npn(* V».

At stalks level, a germ WE n~(*D)x belongs to F'n~(*D)x if and only if p*(u)i- s, W E nL, where u = 0 is a local equation for V around the point y p(x).lf Vb" ., Vn are local coordinates on X around x such that VI ••. Vk = 0 is a local equation for D, then p*(u) vanishes on D and hence p*(u) = V ~' ••• v:kw for some germ WE @x,x and integers aj ~ 1.

Using the definitiions, it follows that Q~(log D) c FSQ~(*D) forj > s andj > O. And n~(logD) = Q~ c F'n~(*D) for s:::::; O. We can state this as follows.

(2.4) LEMMA. (i) u;;.s+lnx(logD c F'nx(*D)for s > 0; (ii) nx(log D) c FOnx(*D).

We can hence write the next commutative diagram

1 lp H'(Qx(log D» ------,-~I H'(Qx(*D» +f- H'(Qpn(* V».

Now H'(Qpn(*V» = H,(uQiT) = H'(Ai»:;: H'(U). To compute H'(F'Qpn(*V» we use the E2-spectral sequence E~,q = HP(Hq(pn, K'» converging to H'(K'), where K = F'npn(* V) and Bott's vanishing theorem [8].

It follows that E~'o = HP(F' Ai», E~s = Hs(pn, n~n) and E~,q = 0 in the other cases. The spectral sequence degenerates at E2 since one can represent the generator of E~s by a a-harmonic form y and hence ily = O. On the other hand P(y) = 0, since ')I belongs to the kernel of the map H2s(pn) 4 H2s(U). In fact this map is zero for s> O. To see this, it is enough to show that i*(c) = 0, where c = CI (@(1» is the first Chern class of the line bundle @(1) (in cohomology with complex coefficients!). But Ni*(c) = 0, since it corresponds to the Chern class of @(N)lu and this line bundle has a section (induced by f) without any zeros.

It follows that im(p) = F' H'(U) and this gives the first part in (2.2). The similar diagram associated to the inclusion (2.4. ii) gives FO H'(U) =

F~H'(U) = H'(U). To see that F~ = Fl we relate the mixed Hodge structure on H'(U) to the

mixed Hodge structure on H'(V) Consider the exact sequence in cohomology with compact supports of the pair (pn, V)

(2.5)

30 Alexandru Dimca

This is an exact sequence of MHS (mixed Hodge structures) and it gives an isomorphism of MHS H~+ l(U) ~ HMV), the primitive cohomology of V [10]. Poincare duality gives a natural identification (U is a Q-homology manifold):

Since H;n(u) ~ H2\pn) ~ q - n), we get the following relations among mixed Hodge numbers

This gives hO,q(HS(U)) = 0 for any q and s, which shows that FIJrHO(U) = FhHO(U), ending the proof of (2.5).

(2.6) REMARK. In spite ofthe fact that FSHO(U) = F'}t 1 HO(U) for any s in many cases (e.g. when V is a quasi-smooth hypersurface or when V is a nodal curve in p2), this equality does not hold in general. A simple example is the next: take V: x[xy(x + y) + Z3] = 0 the union of a smooth cubic curve in p2 with an inflexional tangent. Then it is easy to show that in this case dim Fl H2(U) =

2 > dim FiI H2(U) = 1. There is a similar inclusion F'HO(F) => F~+ 1 HO(F) among the analogous

filtrations on the cohomology of the Milnor fiber F. The proof of this fact can be reduced to (2.2) as follows. The geometric monodromy h is analgebraic map and hence h* preserves both filtrations F' and F~ on HO(F). If we define F'HO(F)a =

FSHO(F) n HO(F)a it follows that F'HO(F) = tBaF'HO(F)a. And one has a similar result for the Hodge filtration F~. In particular, it is enough to prove

(i) F' HO(F)o => F~+ 1 HO(F)o, and (ii) F'HO(F)#o => F~+ 1 HO(F)#o

where F'HO(F)#o = F'HO(F) n HO(F) # ° = tBa#oFSHO(F)a and similarly for F H'

Now (i) is clearly implied by (2.2), since the isomorphism HO(U) ~HO(F)o c

HO(F) is clearly compatible with both filtrations. To get (ii) from (2.2) we use the diagram (1.19) and the next two facts. The Poincare residue map R is a morphism of MHS of type (-1, -1) and

hence

Using the definition of the filtrations F' and (1.20) it follows that

Note also that the filtration po on HO(F) is very close to the filtrations considered by Scherk and Steenbrink in the isolated singularity case in [21].

(2.7) COROLLARY. (i) E".;;(f)o = for s < 0 and E"';;(f)a = 0 for s < -1 and a=l ... ,N-1.

(ii) Any element in Hk(U) can be represented by a differential k-form with a pole

along V of order at most k.

We note that (ii) can be regarded as an extension of Griffith's Theorem 4.2 in [12]. On the side of numerical computations of Betti numbers we get the following important consequence. Recall that m = dim f - 1(0)sing'

(2.8) THEOREM. Let bJ(V) = dim Hb(V) denote the primitive Betti numbers of

V. Then

(i) bJ(V) = 0 for j < n - 1 or j > n - 1 + m; (ii) For kE [0, m] and r > lone has

n-k-1 b~_1+k(V) = bn-k(U) ~ L dimE~,n-k-S(f)o .

• =0

When k = m and r ~ n - m the above inequality is an equality. Proof Use (1.6), (1.7), (1.8), (1.12) and (2.7). There is also an analog of (2.8) for dim H i(F)a but we leave the details for the

reader.

30 The case of a one-dimensional singular locus

We assume in this section that f has a one-dimensional singular locus, namely

f- 1(0)sing = {ZE cn+1;df(x) = O} = {O} u U C*ai i=1,p

for some points ai E cn + \ one in each irreducible component of f -1(0)sing' If Hi is a small transversal to the orbit C*ai at the point a;, then the isolated

hypersurface singularity (y;, ail = (Hi n f -1(0), ail is called the transversal singularity of f along the brach C*ai of the singular locus.

The weighted homogeneity of f easily implies that the isomorphism class (%-equivalence) of the singularity (y;, ail does not depend on the choice of ai (in the orbit C*ai) or of Hi'

In this section we get a better understanding of the sequence (Er(f)o, dr) by relating it to some spectral sequences associated to the transversal singularities (Y;, a;) for i = 1, ... , p.

32 Alexandru Dimca

First we describe the construction of these new (local) spectral sequences. Let g: (C"' 0) --+ (IC, 0) be an analytic function germ and let (Y, 0) = (g-l(O), 0) be

the hypersurface singularity defined by g. Let Q~,o denote the localization of the stalk at the origin of the hoI om orphic de Rham complex Qen with respect to the multiplicative system {gS; s ~ O}.

Choose e > 0 small enough such that Y has a conic structure in the closed ball Be = {YE ICn; lyl ~ e} [4]. Let Se = aBe and K = Se (l Y be the link of the singularity (Y, 0). Then Thm. 2 in [13] implies the following.

(3.1) PROPOSITION. H'(Se \K) ~ H'(Q~,o).

One can construct a filtration Ps on Q~,o in analogy to (1.4), namely

PSQ~,o = {w/gi-s;WEQbn,o} for j ~ s and PSQ~,o = 0 for j < S.

(3.2) PROPOSITION. There is an E1 -spectral sequence of algebras (Er(g, 0), dr) with

and converging to H'(Se\K) as an algebra.

Assume from now on that (Y, 0) is an isolated singularity and let L' = (Qc',o dg) denote the Koszul complex of the partial derivatives of g. In our case these derivatives form a regular sequence and hence Hi(L') = 0 for j < nand Hn(L') = M(g), the Milnor algebra of the singularity (Y, 0), see for instance [7], p. 90. Let r denote the quotient complex L'/gL' If g: M(g)--+M(g) denotes the multiplication by g, it follows that Hi(I') = 0 for j < n - 1, W- 1(I') =

ker(g) and W(I') = coker(g) = T(g), the Tjurina algebra of (Y, 0), see [7], p. 90. There is the next analog of (1.8), computing E1 (g, 0) in terms of H'(I').

(3.3) LEMMA. The nonzero terms in E 1 (g, 0) are the following.

(i) E'I'°(g, O) = Q~n,o for s E [0, n] (ii) E'I'l(g,O)=Q~ for sE[0,n-3], there is an exact sequence O--+Q~-2~

E~-2,1(g,0)~ker(g)--+O and Er- 1,1(g,0} = Qc.,o/gQc.,o, where

Qr = (Qen,o)/(gQen,o + dg /\ Qen,o).

(iii) E1- t- 1,t(g,0) = ker(g),m-t,t(g,O) = T(g) for t ~ 2.

Proof To get the more subtle point (ii), one uses the well-defined maps

u: Q~ --+ E'I'l(g, 0), u(a) = [(dg /\ a)/g]

v: E~,l(g,O) --+ H S + 2 (L'), v[fJ/g] = [(dg /\ fJ)/g]

and note that im(v) c ker(g) for s = n - 2.


(3.4) COROLLARY. The only (possibly) nonzero terms in Ez(g,O) are E~'o =

E~·i = IC and E~-l-t.t,E~-t.t for t ~ 1.

Proof Use the exactness of the de Rham complexes [11]:

o --+ IC --+ n2n.o --+ ... --+ n~n.o --+ 0

0--+ IC --+ n~ --+ ... --+ n~-i.

We can also describe the differentials

An (n - 1) form rx induces an element in ker(g) if dg /\ rx = gf3 and then

(3.5)

(3.6) EXAMPLE. Assume that (Y,O) is a weighted homogeneous singularity of type (Wi'"'' W n ; N), i.e. (Y,O) is defined in suitable coordinates by a weighted homogenous polynomial 9 of degree N with respect to the weights w.

Then M(g) = T(g) = ker(g) and they are all graded IC-algebras. Let rx = Li=l.n(-l)i+iwiXidxi /\ ... /\ dXi /\ ... /\ dXn and note that

dg /\ rx = N' gWn' with wn = dX i /\ ... /\ dxn. It follows that the class of rx generates ker(g) For a monomial XD = X~' ... x:n of degree IXDI = ai Wi + ... + anwn one has by (3.5)

with W = Wi + ... + wn •

It follows that ker di ~ coker di ~ M(g)tN-w' Hence the Ez-term Ez(g, 0) has finitely many nonzero entries and the spectral sequence Er(g, 0) degenerates at Ez (compare to (1.11)).

The next result gives a large class of singularities having the E3 -term of the spectral sequence Er(g,O) with finitely many nonzero entries. The reader should have no difficulty in checking that this class contains in particular the next more familiar classes of singularities:

(i) all the non weighted homogeneous ~-unimodal singularities, see for instance [0], p. 184 for a complete list;

(ii) all the semi weighted homogenous singularities (see [7], p. 115 for a definition) of the form 9 = go + g' with go weighted homogeneous of type (Wi>" ., W n ; N), g' weighted homogeneous of type (Wi"'" Wn ; N') and such that

N' ~ (n + 1)Nj2 - Wi - ... - Wn •

34 Alexandru Dimca

To state the result, note that there is a linear map d~ : ker(g) --. T(g) defined by taking t = 0 in the formula (3.5).

(3.7) PROPOSITION. Assume that the singularity Y: g = 0 satisfies the condi

tion:

(i) g2 = 0 in M(g) and d~ I (g) = 0 (resp. (ii) /leg) - reg) = 1).

Then E~-l-t.t = ker d~ = (g) for t » 0 (resp. dim ker dt1 = 1 for t » 0 and the lines

ker d~ in ker(g) converge to the line C-g when t --. CIJ) and the E3 - term E3(g, 0) has finitely many nonzero entries.

Proof (i) Let K c M(g) be a vector subspace which is a complement of the ideal Ker (g) c M(g).

Then multiplication by g induces a vector space isomorphism K --""'--+ gK = (g). For t large enough, it is clear using (3.5) that kerdt1 = (g) and that the canonical projection M(g) --. T(g) induces an isomorphism K ~ coker di.

Via these isomorphisms we may regard d~ as an endomorphism of K for t » O. Next di(ag) =0 implies that we may write ag2wn =dg /\ (X and the (n-l)-form

(X satisfies doc = dg A P + ),gwn for some (n - I)-form P and function germ),. But then we have

This shows that the endomorphism d~ has a matrix of the form - t' Id + A + B(t - 1) -1 for A, B some constant matrices. It follows that for t » 0 this matrix is invertible and this clearly ends the proof. The proof in case (ii) is similar.

Now we come back to our global setting and assume first that we are in the homogenous case, i.e Wo = ... = Wn = 1. Let Z denote the singular locus of V.

Consider the restriction morphism

(3.8)

and the associated morphisms

A moment thought shows that Grpp is a quasi-isomorphism for s < O. A computation using an E2-spectral sequence shows that

H'(Grp(ni>n(* V))) = H'(Gr}Ao).

Assume from now on that Z is a finite set, namely Z = {a 1 , ••• , ap }. Note that

the singularity (V, ai) is precisely the transversal singularity of f along the line C*" a i as defined in the beginning of this section.

Choose the coordinates on pn such that H: Xo = 0 is transversal to V and Z c pn\H c:::: cn. We denote again by ai the corresponding points in Cn and let g(y) = f(1, y).

Then Qim( * V)lz = EB j= 1,pQ~,ai' this identification being compatible with the F filtrations. Thus we get

H"(Gr}(Qim(* V)lz)) = EB H"(Gr}(Q~,aj))' j= 1,p

We can restate these considerations in the next form.

(3.9) THEOREM. The restriction map p induces a morphism p,: E.(f)o --+

EB j= 1,pEr (g, a) of spectral sequences such that at the E1 -level Pl,t is an isomorphism

for s < O.

(3.10) COROLLARY. For a projective hypersurface V: f = 0 with isolated

singularities the next statements are equivalent

(i) all the singularities of V are weighted homogeneous; (ii) E~t(f)o = 0 for s < 0;

(iii) E~t(f)o #0 for finitely many pairs (s, t).

Proof Using (3.6) and (3.9) we get (i) => (ii). The implication (ii) => (iii) is obvious. To prove (iii) => (i) we compute the Euler Poincare characteristic X(U) in two ways. First we use the fact that U = pn\ V and the well-known formula for x(V) given in (3.12) below and get

x(U) = x(U 0) + (_1)n-1 L .u(V, ai )

i= 1,p

where U 0 is the complement of a smooth hypersurface Vo in pn.

Next using (1.8) and a standard property of spectral sequences we get

where the sum is finite by our assumption. Choose m > n such that E~t(f)o = 0 for t > m. Then

x(U) -1 = (_1)n-1 L (dimE~-l-t,t(f)o - dimE~-t,t(f)o) t=l,m

=(_1)"-1 L (dimE1-1-t,t(f)0 -dimE1-t,t(f)0) t=l,m

36 Alexandru Dimca

with

qJ = L dimE~-t,t-l(f)o - dimE~-t,t(f)o)· t=l,m

By (3.3.iii) and (3.9) it follows that

dimE~-l-m,m(f)o = L '["(V,a i )

i=l,p

where '["(V, ai ) = dim T(g, ai ) = dim ker(g, aJ are the corresponding Tjurina numbers. On the other hand, using the connection of E 1 (f) with the Koszul complex, it is easy to see that the sum qJ does not depend on f Since one can compute X(U 0) in the same way, it follows that

x(U) = X(U 0) + (-It- 1 L '["(V, ai )·

i=l,p

Comparing the two formulas for X(U) we get JL(V, ai ) = '["(V, ai ) for any i = 1, ... , p and hence by K. Saito's Theorem (see for instance [7], p. 119 for a discussion) all the singularities (V, ai ) are weighted homogeneous.

Since for any isolated hypersurface singularity (Y,O) there is a projective hypersurface V havingjust one singular point a1 and such that (V, a1 ) ~ (Y, 0), see for instance [2], we get the next result using (3.6), (3.9) and (3.10).

(3.10') COROLLARY. For an isolated hypersurface singularity (Y, 0) defined by g = 0 in (Cn, 0), the next statements are equivalent:

(i') (Y,O) is a weighted homogeneous singularity; (ii') the spectral sequence Er(g, O) degenerates at E2 ;

(iii') E~t(g, 0) #- 0 for finitely many pairs (s, t).

We conjecture in analogy with (3.10') that the statements in (3.10) are equivalent to the next stronger version of (ii):

(iv) the spectral sequence Er(f)o degenerates at E2 •

(3.11) REMARK. Let f be a homogenous polynomial such that V has an isolated singularity of the type considered in (3.7). Then E.(f)o surely does not degenerate at E2 • Note thatf: (Cn + 1 ,0) --+ (C, 0) is concentrated in the terminology of [25], p. 206 and our spectral sequence Er(f)o is a subobject in the huge spectral sequence considered in [25], p. 209. Hence in this case that spectral sequence does not degenerate at E2 and this gives a negative answer to the question at the top of p. 209 in [25].

By Theorem (2.8) the interesting Betti numbers for V in the isolated singularities case are just bn - 1 (V), bn(V) and we can get bn(V) from En - 1 (f)o.


But one has a simple formula for the Euler-Poincare characteristic in this case [6]:

x(V) = x(Vo) + (-1t L /l(V, aJ (3.12) i= l,p

where Vo denotes a smooth hypersurface in pn of degree Nand /l(V, aJ =

dim M(g, ai ) are the corresponding Milnor numbers. In this way we get bn-I(V) knowing bn(V). We remark that there is a formula

for X(F) similarto (3.12) and which appears in the special case n = 2 as Theorem 6. A in [9].

(3.13) PROPOSITION. X(F) = 1 + (-1)n[(N - 1)n+ I - N L /l(V, aJ]. i=l,p

Proof If F denotes the closure of Fin pn+ I, one has X(F) = X(F)\X(V). One then use (3.12) and the remark that the singularities of F are just the N-fold suspensions of the singularities of V and hence

/l(F,(ai:O)) = (N - 1)fl(V,aJ,

(3.14) REMARK. An important invariant ofthe singularity f is the zetafunction Z(h) of the monodromy operator h. Explicitly one has

where /\ (hk) denotes the Lefschetz number of the map hk. Using the second expression above for Z(h) it follows that for any homogeneous polynomial f one has

When V has only isolated singularities, this formula may be used to compute dim Hn(F)a for a = 1, ... , N - 1 assuming that we know dim Hn - I (F)a via computations with the spectral sequence ErU) as in the remark after (2.8).

Next we describe briefly the additionalfacts necessary in order to treat the case when f has arbitrary weights w = (wo,' .. , wn).

First we have to include a group action in the local setting. Let G c U(n) be a finite group and consider the induced action on en. Then the ball B. and the sphere S. are G-invariant subsets. Assume that Y:g = 0 is a reduced hypesurface singularity which is also G-invariant (i.e. y E Y, Y E G => y(y) E Y for a representative Y of (Y, 0) in B.).

38 Alexandru Dimca

There is an associated action of G on Q{;n,o given by y. W = (y - 1 )* W And there is character Xy: G -+ IC* such that y. g = Xy(y)g for any y E G. In this situation we call (Y, 0) a G-singularity. Note that this setting is larger than in Wall [27] where one takes Xy = 1, but coincides (in the case of G cyclic) to the hyperquotient singularity notion of M. Reid [19].

Let (n~?o, d) be the subcomplex in (n~,o, d) consisting of the fixed elements under the obvious action of G. If K· is any complex of IC-vector spaces with G-actions compatible with the differentials, then there is a natural isomorphism H·(K· G) = H·(K·)G which says that taking cohomology commutes with taking the fixed parts under G. Moreover in Proposition (3.1) both cohomology groups have natural G-actions and the isomorphism considered there is compatible with these actions. It follows that

(3.15)

Next, using again the above commutativity, we get an E1 -spectral sequence (Er(g,O)G,dr) consisting of the fixed parts of the spectral sequence described in (3.2) and converging to H·((Se\K)/G).

Assume now that (Y, 0) is an isolated singularity and note that G acts on the complex E considered above. Since the G-action commutes with the differentials in E up-to multiplicative constants, it follows that there is an induced action on the cohomology H·(E). And one has exactly as in Wall [27] an isomorphism of G-vector spaces

with wn + 1 = dxo A ... A dxn • Let Xo be the character of the action of G on ICWn +1' If W is any G-vector space and x: G -+ IC* is a character we set

WX = {WE W;y·w = X(y)w for all yE G}.

With this notation, note that

/ t r,..G·f d I if n·X' W g E ug,O 1 an on y WE Ucn~o'

Combining these remarks we get the next analog of (3.3.iii):

(3.16)

for all t ~ 2, where ker(g) and T(g) have the obviously induced G-actions. We consider now the global setting. Let a E e+ 1 \ {O} be a point in the singular

locus f - 1 (O)sing' Let G a be the isotropy subgroup of a with respect to the C* -action on cn + 1 given by

Then Ga is the finite cyclic group of the unity roots of order

ka = g.c.d.{w j ; the component aj of a is nonzero}.

Take H to be a transversal to the orbit C*.a at the point a which is Ga-invariant. For instance, we may assume that ao =f. 0 and then take H: Xo - ao = O. We identify the germs (cn, 0) and (H, a) via the isomorphism ({J given by (Yl' ... , Yn) f-+

(ao'Yl"'" Yn)' Then the transversal singularity (Y, a) = (H (l f- 1(0), a) is in an obvious way a Ga-singularity and moreover

Xy = N, Xo = Wo + ... + Wn = W

under the identification of the (multiplicative) group of the characters of Ga with the (additive) group ZlkaZ (the character t f-+ tm corresponds to the class of m modulo kaZ, denoted again by m!).

Note that the germ (P, a) (resp. (V, a)) can be identified to (HIGa, a) (resp. (YIGa, a)) and hence the latter is a hyperquotient singularity in the sense of Reid [19]. It follows that Op,a ~ Oc~~a and Op(* V)a ~ O~~~a where ga(Y) = f(aa, Yl"" ,Yn) is a local equation for (Y,a), compare with [24], Section 5.

Let Z c V be the finite set corresponding to the singular locus f- 1 (0)sing' Then we have, (with exactly the same proof) the next analog of Theorem (3.9):

(3.17) THEOREM. The restriction map p: Op( * V) --+ Op( * V) Iz induces a morphism Pr: Er(f)a --+ EBaEZEr(ga,a)Ga of spectral sequences such that at the E 1 -level pyt is an isomorphism for s < O.

As an application we derive now new formulas for the Euler characteristics xCV) and X(F) similar to (3.12), (3.13). Our result should be compared to the more explicit formulas of Siersma [22] (obtained in the very special case when f - 1 (O)sing is a complete intersection and all the transversal singularities are of type A 1) and, on the other hand, to the very general formulas of Yom din [28] (which involve some numerical invariants defined topologically and hence difficult to compute in general concrete cases).

Consider the Poincare series

pet) = «(1 - tN - wo ) ... (1 - tN - Wn ))/«l - tWO) ... (1 - tWn )) = I Ck(W, N)tk

k;.O

40 Alexandru Dimca

associated to the weighted homogenity type (w, N). Define next the virtual Euler characteristics of order m of V and F by the formulas:

Xm(V(w,N)) = n + (_1)n-1 L C.N-w(W,N) s=l,m

Xm(F (w, N)) = 1 + ( _1)n L c.(w, N) (3.18) .=1,mN-w

where w = Wo + ... + wn •

Note that if there is a weighted homogeneous polynomial of type (w, N) having an isolated singularity at the origin and if V' (resp. F') denotes the corresponding hypesurface in P (resp. Milnor fiber) then

Xm(V(w, N)) = X(V') for m ~ n

(resp. Xm(F(w,N)) = X(F') for m ~ n + 1). To see this you may find useful to read first the proof of (3.19. ii) below!

(3.19) PROPOSITION. (i) Assume that a polynomial I' as above exists. Then

X(V) = X(V') + (_1)n L dimM(ga)-W aeZ

X(F) = X(F') + (_1)n+1 L L dimM(ga)-N-w+ j. aeZ j= 1,N

(ii) Assume that any transversal singularity ga = 0 for a E Z is either weighted homogeneous or satisfies the assumptions in (3.7). Then

X(V) = Xm(V(w, N)) + (_1)n L dim M(gatN- w aeZ

X(F) = Xm(F(w, N)) + (_1)n+1 L L dimM(ga)(m-1)N-w+j aeZ j= 1,N

for all m large enough. When all the singularities ga are weighted homogeneous, it is enough to take m ~ n + 1.

Proof On a formal level, note that the formulas in (i) are a special case of the formulas in (ii), obtained by taking m divisible by all ka = 1 Gal, a E Z. The proof of (i) is purely topological and independent of our previous results. Let a, H, ... , be as above. We may takef' close enough to f such that for all aE Z the intersection Fa = B. n (I' 0 <p)-1(0) can be identified to the Milnor fiber of the singularity (Y, a). Note also that Fa is Ga-invariant. Let B'(a) be the image ofthe small ball B.

On the Milnor jibrations of weighted homogeneous polynomials 41

under the natural projection (Cn , 0) --+ (P, a). Then there is a homeomorphism V\B' ~ V'\:B' where B' = UaEZ B'(a). Moreover B'(a) (") V is contractible, while B'(a) (") V' can be identified to Fa/Ga and hence has middle Betti number

as in [27]. Then a Mayer-Vietories argument gives the result for X(V). The result for X(F) then follows from that for X(V) as in the proof of (3.13).

To prove (ii) we use basically the same argument as in the proof of (3.10) (if necessary starting the computations with the E3 -term of the spectral sequence Er(f)O) together with (3.17) and (3.16). First we express the sum <p from the proof of (3.10) in terms of the Poincare series P(t). Following Siersma [22], we define a new grading on Q' by setting for a homogeneous p-form WE QP:

degw = (n - p + l)N - w + Iwl

where Iwl denote the degree of w as defined in our introduction. Then multiplication by df becomes a map of degree 0 and one has

P(t) = L (-1t+ l-k p(Qk)(t) k=O,n+ 1

where p(Qk) is the Poincare series of Qk with respect to this new grading [22]. Then it is obvious that

<p = - L CsN-w(W, N). s= 1,m

To treat the case of transversal singularities covered by (3.7) one has to use the next isomorphisms of vector spaces, which are clear by the proof of (3.7):

E~-l-m,m(g,O)G + E1- 2 - m,m+l(g,0)G = ker(grN-W + (g)(m+l)N-W

=ker(grN- W + K mN - w = (ker(g) + K)mN-w = M(g)mN-w.

The case of singularities in (3.7.ii) can be treated similarly.

(3.20) EXAMPLE. The polynomial f = X6 65 + xoxP + xox~ + x2x1 has degree N = 265 with respect to the weights W = (1, 24, 33, 58). The singular set Z consists of one point, namely a = (0, -1,1,0) with transversal singularity ga of type A3 • The corresponding isotropy group Ga is 7L/37L and acts on M(ga) such that dim M(ga)i = 1 for any i. It is konwn that the Poincare series P(t) is a polynomial in this case, in spite of the fact that there is no isolated singularity f'

42 Alexandru Dimca

of this homogeneity type (w, N), see [0], p. 201. It follows that

Xm(F(w,N)) = 1 - P(l) = -66515 for m» 0 and

X(F) = Xm(F(w, N)) + 265 = -66250.

Using a computer to determine the coefficients of P(t), one gets X(V) = 254.

(3.21) REMARKS. (i) We conjecture that the formulas (3.19.ii) hold for any transversal singularities.

(ii) If all the transversal singularities (Y, a) for a E Z have links which are Q-homology spheres, then the hypersurface V is a Q-homology manifold and hence satisfies the Poincare duality over Q. In this case bn(V) = bipn -1) and the remaining interesting Betti number bn - 1 (V) can be determined from X(V) once this Euler characteristic is known.

For concrete computations it is useful to use the following general remark. Assume that fl' ... ' fn is a regular sequence in S(this can be always achieved by a linear change of coordinates in the homogeneous case!). Then the Koszul complex K· (1.9) is quasi-isomorphic to the complex

(3.22)

where Ql (f) = S/(fl'· .. , f,,) and fo denotes multiplication by fo. An indication of the dimensions of Hn+ 1 (K.)k ~ Q(f)k-n-l and Hn(K·)k ~ ker(fo)k-n can be obtained from the exact sequence

(3.23)

since the Poincare series of Ql (f) is known.

4. Explicit computations

(4.1) EXAMPLE (Computation of Hl(U)).

Let f = H' ... ftk be the decomposition of f in distinct irreducible factors. Then it is known that b1(U) = bgn - 2 (V) = k - 1 and it is easy to check that the closed forms

Wi = (dJ;)/(J;) - (NdN(df)/(f)

where Ni = deg(J;), i = 1, ... , k generate Hl(U) with only one relation: ~aiwi = 0 Compare to (2.7.ii).

(4.2) EXAMPLE (with isolated singularities for V).

Let f = xyz(x + y + z), n = 2. Then V consists of 4 lines in general position in p 2 and its topology is simple to describe. However, the dimensions of the eigenspaces H'(F)a are more subtle invariants.

First we compute explicit bases for the homogeneous components of Q(f):

Q(f)o = (l>,Q(f)1 = <x,y,Z>,Q(f)2 = <X2,y2,z2,xy,yz,zx>

Q(f)3 = <x3,l,z3,x2y,y2z,z2x,xyz> and

Q(fh = <x\ yk, z\ xk-Iy, yk-Iz, zk-I X> for k ~ 4.

Then we look for the elements in H2(K') and define:

wxy = x(x + 2y + z)dy /\ dz + y(2x + y + z)dx /\ dz

and w yz , Wzx by cyclic symmetry. Then df /\ wxy = df /\ Wyz = df /\ Wzx = 0 and these three forms give a basis

for H2(K')4' The six forms xWxy , yWxy. YW yz , zWyz , zWzx , xWzx generate H2(K')s with one

relation among them (their sum is trivial). And the six forms xkWXy , ykWXy , . .. , form a basis for H2(K')k+4 for any k ~ 2. It is now easy to compute dl: H2(K')k --+ H3(K')k and the nontrivial kernels

and cokernels are listed below together with E~'°(f)o:

E~'°(f)o = E~,2(f)2 =E~,2(f)3 = E~,I(f)1 = C,

E~,I(f)O = m,I(f)o = e.

The computations also show that the spectral sequence degenerates at E2 and hence we get the complete results. One can restate them by saying that the monodromy operator h* acts trivially on HO(F) = C, HI (F) = C3 and its action on H2(F) = C6 has characteristic polynomial (t - 1?(t + 1)(t2 + 1).

(4.3) EXAMPLE (with nonisolated singularities for V).

An irreducible cubic surface in p 3 with nonisolated singularities is projectively equivalent to one of the next normal forms [3]

(i) a cone on the nodal cubic curve; (ii) a cone on the cuspidal cubic curve;

(iii) S': x 2 z + y2 t = 0; (iv) S: x 2z + y3 + xyt = O.

The topology of the surfaces (i)-(iii) can be described easier e.g. using [18], so

44 Alexandru Dimca

that we concentrate on the last case: f = x2z + y3 + xyt. The homogeneous components of Q(f) are given by

where (z, t)k denotes the vector space of all homogeneous polynomials in z, t of degree k. Hence dim Q(f)k = k + 4 for k ~ 2. Consider now the differential forms:

W1 = xdx 1\ dy 1\ dz + ydx 1\ dy 1\ dt,

W2 = xdx 1\ dz 1\ dt + tdx 1\ dy 1\ dt - 3ydx 1\ dy 1\ dz,

W3 = tdx 1\ dy 1\ dz - 2zdx 1\ dy 1\ dt + xdy 1\ dz 1\ dt,

Then some tedious computations show that:

H3(KO)4 = (W 1,W2,W3),

H3(KO)s = (Z,t)lW2 + (Z,t)lW3 + (XW2,YW2,YW3),

H3(KO)k+4 = (Z,t)kW2 + (Z,t)kW3 + (Zk-1 X,Zk-l y,Zk-2 y2)W3,

for k ~ 2. This last vector space has dimension 2k + 5. And similarly one gets

H2(K")k+4 = (z, t)kW with

w=(6yz-t2)dx 1\ dy-xtdx 1\ dz-ytdx 1\ dt-3xydy 1\ dz-3y2dy 1\ dt.

After these complicated formulas it comes as a surprise that the spectral sequence Er(f) degenerates at E2 and the only nonzero terms are E~O(f)o =

E~2(f)1 = E~2(fh = c. It follows that HO(S) ~ Ho(P2) and hence S has the same rational homotopy type

as p2, according to Berceanu [1], who has proved that a projective complete intersection (with arbitrary singularities) is an intrinsically formal space.

Concerning the Milnor fiber one has HO(F) = C with trivial action of h*, H2(F) = C2 with the characteristic polynomial of h* equal to t2 + t + 1 and Hl(F) = H3(F) = O.

Our next result is an improvement of Corollary (3.11) in Scherk [20] (to see the connexion between these two results have a look at the exact sequences (1.3) in [20]!).

Let (Y, 0) be an isolated hypersurface singularity given by g = 0 in cn. We define the Jl-constant determinacy order of(Y, 0) (denote by Jl-det(Y,O)) to be the smallest integer s > 0 such that the family l = g + th (t E [0, 1]) is Jl-constant for any hE(Yl,"" Yn)" with small enough coefficients. Note that Jl-det(Y,O) can be

easily computed for large classes of singularities (e.g. weighted homogeneous or Newton nondegenerate singularities) and is always less or equal to the strongly %-determinancy order O(g) see [7], p. 75. In Scherk's notation, one has.

(4.4) PROPOSITION. Let VC pn be a hypersurface having just one singular point a and such that N = deg(V) > fl-det(V, a).

Then bn- 1(V) = bn- 1(V0) - fl(V,a) and bn(V) = bn(Vo), where Vo is a smooth hypersurface in pn with deg(Vo) = N.

Proof Choose the coordinates on pn such that a = (1 :0: .. -:0) and H: Xo = Ois transversal to V. If f = ° is an equation for V, then we set g(y) = f( 1, y l' ... , Y n) = g2(y) + ... + gN(y), with gk a homogeneous polynomial of degree k. Using the assumptions, we can find a continuous family

gt(y) = gi(y) + ... + g~(y) for t E [0, 1]

with the properties:

(i) gO = g, g~ = gk for k < N - 1; (ii) For any t > 0, the hypersurfaces in pn-1

Wl: gl = 0 for i = N - 1, N

are smooth and intersect transversally; (iii) gt is a fl-constant family; (iv) The projective hypersurfaces Vt with the affine equations gt = 0 have no

singularities except a. According to [6], the cohomology of Vt is determined by a lattice morphism

q/: L~ c. L! --+ [I, = L! /Rad L!

where L~ (resp. L!) is the Milnor lattice of the singularity l = 0 (resp. g~ = 0). When t varies, these Milnor lattices are constant and hence the morphism cpt has to be constant too.

Hence H"(V) = H"(V1) and so we can assume from the beginning that gN -1, gN satisfy the condition (ii).

Let cp: L1 C:L--+L be the lattice morphism in this case. We have to show that i(L 1 ) n Rad L = 0, where i is the embedding of Milnor lattices arising from the small deformation gr(y) = g(r" y)"r- N (r» 0) of the singularity gN = 0, see [6], proof of (1.2).

But we may think of gr as being a even smaller deformation (of order r- 2 ) of the

46 Alexandru Dimca

germ g' = gN - 1 • r - 1 + 9 N, which is a small deformation of 9 N. If I; denotes the

Milnor lattice of the singularity g' = 0, then the inclusion i above factorizes as

Ll ~ I; ~L and hence it is enough to show that

(v) j(I;) (\ Rad L = O.

Now jis related to the cohomology of the hyper surface V' c pn with the affine

equation gN - 1 + gN = O. Note that V' has just one singular point too, namely a.

By a Il-constant argument as above, we can assume that

gk(Y) = Y~ ... + Y~ for k = N - 1, N.

Next (v) is equivalent to Ho(V') = 0 and we show this using the spectral

sequence Er{f') for f' = XOgN-l(Xb"" xn) + gN(X 1, ... , xn). It is enough to

show that d 1 is injective. And this follows easily using the fact that a base for

Hn(K) is given by the forms x~o ... x~n·w with ai < N - 2 for i = 1, ... , nand

W = Wi 1\ ... 1\ W n, where the 1-forms

are the obvious solution of the equation

df = L xf- 2 Wk'

k= l,n

Compare to [22J, [25J, but note that here the transversal type is not A 1 for N > 3.

Note added in proof

The proof of Theorem (2.2) above contains an error on p. 11 lines 7 and 8. It is

possible to repair this in some special cases, e.g. when all the singularities of V

are isolated and weighted homogeneous. A more general result implying

Theorem (2.2) has been proved by the author and P. Deligne (to whom I am very

grateful for pointing out the above mentioned error!). For details, see our

preprint "Hodge and order of the pole filtrations for singular hypersurfaces".

References

O. V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable Maps, vol. I,

Monographs in Math. 82, Boston - Basel - Stuttgart, Birkhauser 1985.

1. B. Berceanu, Formal algebraic varieties, unpublished manuscript.

2. E. Brieskorn, Die Monodromie der Isolierten Singularitaten von Hyperflachen. Manuscr. Math.

2 (1970) 103-161.

On the Milnor jibrations of weighted homogeneous polynomials 47

3. J.W. Bruce and C.T.C.Wall, On the classification of cubic surfaces, J. London Math. Soc. (2) 19, (1979) 245-256.

4. D. Burghelea and A. Verona, Local homological properties of analytic sets. Manuscr. Math. 7, (1972) 55-66.

5. P. Deligne, Theorie de Hodge II, III. Publ. Math. IHES 40, 5-58 (1971) and 44, (1974) 5 -77. 6. A. Dimca, On the homology and cohomology of complete intersections with isolated

singularities. Compositio Math. 58, (1986) 321-339. 7. A. Dimca, Topics on Real and Complex Singularities, Braunschweig-Wiesbaden: Vieweg 1987. 8. I. Dolgachev, Weighted projective varieties. In: Carrell, J.B. (ed) Group Actions and Vector

Fields, Proceedings 1981. (Lect. Notes Math., vol. 956, pp. 34-71) Berlin Heidelberg New York: Springer 1982.

9. H. Esnault, Fibre de Milnor d'un cone sur une courbe plane singuliere, Invent. Math. 68, (1982) 477-496.

10. A. Fujiki, Duality of mixed Hodge structures of algebraic varieties, Publ. RIMS Kyoto Univ. 16, (1980), 635-667.

11. G.-M. Greuel, Der Gauss-Manin-Zusammenhang isolierter Singularitaten von vollstandigen Durchschnitten. Math. Ann. 214, (1975) 235-266.

12. P. Griffiths, On the periods of certain rational integrals I, II, Ann. Math. 90, (1969) 460-541. 13. A. Grothendieck, On the de Rhan cohomology of algebraic varieties, Pub/. Math. IHRES 29,

(1966) 351-358. 14. H. Hamm, Ein Beispiel zur Berechnung der Picard-Lefschetz-Monodromie fur nichtisolierte

Hyperfiachensingularitaten Mat. Ann. 214, (1975) 221-234. 15. M. Kato, Y. Matsumoto, On the connectivity of the Milnor fiber of a holomorphic function at

a critical point. In: Manifolds, Proceedings Tokyo 1973, pp. 131-136, Univ. ofTokyo Press 1975. 16. J. McClearly, User's Guide to Spectral Sequences, Publish or Perish: 1985. 17. M. Oka, On the homotopy types of hypersurfaces defined by weighted homogeneous

polynomials. Topology 12, (1973) 19-32. 18. M. Oka, On the cohomology structure of projective varieties. In: Manifolds, Proceedings Tokyo

1973, pp. 137-143, Univ. of Tokyo Press 1975. 19. M. Reid, Young person's guide to canonical singularities, Proc. AMS summer Institute Bowdoin

1985, Proc. Symp. Pure Math. 46, AMS 1987. 20. J. Scherk, On the monodromy theorem for isolated hypersurface singularities, Invent. math. 58,

(1980) 289-301. 21. J. Scherk, J.H.M. Steen brink, On the mixed Hodge structure on the cohomology of the Milnor

fiber. Math. Ann. 271, (1985) 641-665. 22. D. Siersma, Quasihomogeneous singularities with transversal type A .. Contemporary Mat

hematics 90, Amer. Math. Soc. (1989) 261-294. 23. J.H.M. Steenbrink, Intersection form for quasihomogeneous singularities. Compositio Math. 34,

(1977) 211-223. 24. J.H.M. Steen brink, Mixed Hodge Structures and Singularities (book to appear). 25. D. van Straten, On the Betti numbers of the Milnor fiber of a certain class of hypersurface

singularities. In: Greuel, G.-M., Trautmann, G. (eds) Singularities, Representations of Algebras and Vector Bundles, Proceedings, Lambrecht 1985 (Lect. Notes Math. Vol. 1273, pp. 203-220) Berlin Heidelberg New York: Springer 1987.

26. Z. Szafraniec, On the Euler characteristic of complex algebraic varieties, Math. Ann. 280, (1988) 177-183.

27. C.T.C. Wall, A note on symmetry of singularities, Bull. London Math. Soc. 12, 169-175 (1980). 28. LN. Yomdin, Complex varieties with I-dimensional singular locus, Siberian Math. J. 15, (1974)

1061-1082.


Mixed Hodge structures on the intersection cohomology of links

ALAN H. DURFEE 1* & MORIHIKO SAIT02** 1 Mount Holyoke College, South Hadley, MA USA-01075; 2 RIMS Kyoto University, Kyoto 606 Japan

Received 4 November 1988; accepted in revised form 16 March 1990

Keywords: Mixed Hodge structures, links of singularities, intersection homology, mixed Hodge modules, semipurity, topology of algebraic varieties.

Abstract. The theory of mixed Hodge modules is applied to obtain results about the mixed Hodge structure on the intersection cohomology of a link of a subvariety in a complex algebraic variety. The main result, whose proof uses the purity of the intersection complex in terms of mixed Hodge modules, is a generalization of the semipurity theorem obtained by Gabber in the l-adic case. An application is made to the local topology of complex varieties.

Introduction

Let X be a complex algebraic variety, assumed irreducible and of dimension n, and let Z be a closed subvariety. This paper studies the mixed Hodge structure on the intersection cohomology ofthe link of Z in X, derives a semipurity result, and deduces some topological consequences. The mixed Hodge structure is obtained using the theory of mixed Hodge modules developed by the second author.

Although the concept of 'link' of Z in X is intuitively obvious, its precise meaning is unclear. In this paper, we will define it as the nearby level set of a suitable nonnegative real valued distance function which vanishes exactly on Z. If Z is compact, a reasonable concept of link results if the distance function is assumed to be real analytic. Another stronger, concept results if N is a neighborhood of Z in X and if there is a proper continuous retraction map r from aN to Z such that the closure of N is the total space of the mapping cylinder of r; the distance function is then the projection of the mapping cylinder to [0, 1]. For most of this paper, however, we will use a third, weaker type oflink and distance function which combines topological and homological properties. In fact, the homological notions of link alone are enough for most of our results. These homological notions are functors which can be canonically defined in the derived category and fit well with the theory of mixed Hodge modules which we

* Partially supported by NSF grant DMS-8701328, the Max-Planck-Institut fur Mathematik and the Universita di Pisa. ** Partially supported by the Max-Planck-Institut fUr Mathematik and the Institut des Hautes Etudes Scientifiques.

50 Alan H. Durfee and Morihiko Saito

will be using. The material on the various definitions of link is in the beginning of Section 2.

Whichever definition is used, a link L is an oriented topological pseudomanifold of (real) dimension 2n - 1 with odd-dimensional strata. In particular, its Goresky-MacPherson middle-perversity intersection homology is defined. If L is a rational homology manifold, then intersection homology is ordinary homology. Furthermore, N\Z is a rational homology manifold if and only if L is.

This paper applies the theory of mixed Hodge modules to put a mixed Hodge structure on the intersection cohomology with rational coefficients of a link L. Various elementary properties of these groups are derived. The main result is as follows:

THEOREM 4.1. If dim X = nand dimZ =:::;; d, then IH~(L) has weights =:::;;kfor k < n - d.

Duality then immediately shows that I Hk(L) has weights > k for k ~ n + d. The notation here is as follows: The intersection cohomology group IH~(L) is (topologically) the homology group of geometric (2n - 1 - k)-dimensional intersection chains with compact support, and IHk(L) is the similar group with closed support.

For varieties over finite fields and Z a point this result was proved by Gabber [Ga]. The result of Gabber is equivalent to the local purity of the intersection complex by definition and self duality; it implies the purity of intersection cohomology by De1igne's stability theorem for pure complexes under direct images for proper morphisms [De2]. This local purity also follows from the existence of the weight filtration on mixed perverse sheaves, since intersection complexes are simple [BDD 4.3.1].

For varieties over the complex numbers, Z a point and X\Z smooth, the above result for mixed Hodge structures on ordinary cohomology was deduced by several people [Stl, EI] using the characteristic 0 decomposition theorem. Later Steenbrink and Navarro found a more elementary proof using Hodge theory [St2, Na]. We show that Theorem 4.1 follows naturally from the second author's theory of mixed Hodge modules combined with the theory of gluing tstructures from [BBD]. The proof is in the spirit of the second proof of local purity in the l-adic case. (See the end of 1.4.)

This theorem is then used to show that certain products in the intersection homology of a link must vanish (Theorems 5.1 and 5.2). This result is a generalization of [DH], which treated the case where Z is a point and X\Z is smooth. For example, Theorem 5.2 implies that the five-torus Sl x ... X Sl is not a link of a compact curve Z in a three-fold X, and Theorem 5.1 implies that certain pseudomanifolds L cannot be links of points in a complex variety. The only previous result is this area is, we believe, one of Sullivan: If L is a link ora

Mixed Hodge structures 51

compact subvariety, then the Euler characteristic of L vanishes [Su]. Sullivan's proof is entirely topological and holds for any compact oriented pseudomanifold L with only odd-dimensional strata. The above examples are independent of this result, and hence provide new restrictions on the topology of complex algebraic varieties.

Although the results and proof of this paper are given in terms of mixed Hodge modules, they can actually be read in three settings:

The derived category of sheaves on X: On a first reading, this paper can be understood in the derived category D(X) of sheaves of vector spaces over a field on X, together with the derived functorsj*,f!, ... (Following recent convention, we omit the R or L when referring to derived functors.) We use no more properties than those summarized in [GM2 Sect. 1]. Various properties of the intersection homology of links are stated and proved in this language. Of course no conclusions can be drawn about weights (Sect. 4) or the resulting corollaries on the topological structure of varieties (Sect. 5).

The category of mixed Hodge modules on X: The additional material we need about mixed Hodge modules is basically the same as the formalities of [BBD]. This material is summarized in the first section of this paper.

The category of mixed l-adic sheaves on a variety in characteristic p: Lastly, this paper can be read in the setting of [BBD], with the conclusions of Sect. 4 about weights for varieties in characteristic p. Of course, the concept of link as topological space makes no sense here, so the isomorphisms of 2.11 should be taken as the definition of cohomology, homology and intersection cohomology of 'link' in this case. However the applications of Sect. 5 to the topology of varieties can still be obtained by the methods of reduction modulo p as in [BBD Sect. 6].

Throughout the paper references are given for each of these three settings. The first author wishes to thank the Max-Planck-Institut fUr Mathematik,

Bonn, the Universita di Pisa, and the Katholieke Universiteit, Nijmegen for their cordial hospitality during his sabbatical year when a first draft of this paper was written. The second author would like to thank the Max-Planck-Institut fUr Mathematik and the Institut des Hautes Etudes Scientifiques.

1. Background material

General references for the following material on intersection homology and derived functors are [GM2, Bo, GM3 §1, Iv]. All groups will be assumed to have the rational numbers as coefficients, unless otherwise indicated. Following the convention of [BBD], we will use the same symbol for a functor and its right or left derived functor.


1.1. Suppose that W is a topological pseudomanifold of dimension m with strata of even co dimension. For example, W can be a complex algebraic variety or a link of a subvariety. The following groups (all with rational coefficients) can be associated to W:

Hk(W), resp. H~(W): The kth cohomology group, resp. cohomology group with compact supports.

Hk(W), resp. HfM(W): The kth homology group, resp. Borel-Moore homology group (homology with closed supports).

IHdW), resp. IHfM(W): The kth intersection homology group with middle perversity k-dimensional chains with compact support, resp. closed support [GM2, GM3 1.2; Bo l]. Since W has even codimensional strata, the middle perversity is well defined.

IHk(W), resp. IH~(W): The kth intersection cohomology group, defined (topologically) as IH!~k(W), resp. cohomology with compact supports, defined as IHm-k(W).

1.2. We also have the following:

aw: W--+ pt

Qw = (aw)*Q = the constant sheaf on W

Dw = (aw)!Q = the dualizing sheaf on W

lCW' = the intersection complex on W [GM2 2.1]

These complexes have the following properties:

Hk(W) = H\fV, Qw) and H~(W) = H~OV, Qw)

Hk(W) = Hc-k(W, Dw) and HfM(W) = H-k(W, Dw)

IHk(W) = IH~-k(W) = Hc-k(W, lC~P) and

IHfM(W) = IHm-k(w) =:= H-k(W, lCW')

1.3. Let X be a complex algebraic variety (a reduced separated scheme of finite type over C), assumed irreducible and of complex dimension n. A general reference for the following material is [BBD].

Let

as objects of D~(X) as in [BBD]. Ifj: U --+ X is the inclusion of a smooth dense Zariski-open set, then


We also have

D~(X) = The derived category whose objects are bounded complexes of sheaves of Q-modules with constructible cohomology.

Perv(X) = The full subcategory of perverse sheaves over Q.

1.4. General references for the theory of mixed Hodge modules are [Sal, Sa2]. Many of their formal properties are similar to those of mixed perverse sheaves [BBD 5.1].

We have

MHM(X) = the abelian category of mixed Hodge modules on X

rat: MHM(X)-+ Perv(X), an additive, exact, faithful functor which assigns the underlying perverse sheaf over Q.

Functors f*, /I, f*, f', D, ®, lEI on DbMHM(X) compatible with the corresponding derived functors on D~(X) and with the corresponding perverse functors on Perv(X) via

rat) DbPerv(X) real) D~(X)

,~j~ rat

---+) Perv(X)

where 'real' is an equivalence of categories [Be, BBD 3.1.10]. See also [BBD 1.3.6, 1.3.17(i), 3.1.14].

Adjoint relations for f*, f* and fi, 1', the natural morphismfi -+ f* and the usual relations D2 = id, Df* = fiD and Df* = f'D.

The fact that the category MHM(pt) is the category of polarizable mixed Hodge structures (over Q).

Let QH E MHM(pt) be the mixed Hodge structure of weight 0 and rational structure rat(QH) = Q. In this language the cohomology of X has a mixed Hodge structure since we can write

H'(X) = H'«ax)*(ax)*QH)

H;(X) = H'«ax),(ax)*QH)

and so forth. These mixed Hodge structures coincide with those of Deligne

54 Alan H. Durfee and M orihiko Saito

[Del] at least if X is globally embeddable into a smooth variety (for example, if X is quasiprojective). We let

Q~ = (ax)*QH

D~ = (ax)'QH

IC~ = j,*Qff[n]

note that

DIC~ = IC~(n)

where (n) denotes the Tate twist, which can be defined by ~QH(n). As in the category of mixed complexes of l-adic sheaves, there is a weight

filtration in MHM(X) and the notions of 'weight ~ k', etc., in DbMHM(X) such that [BBD 5.1.8, 5.1.14, 5.3.2; Sal 4.5]:

In MHM(pt) these are the usual notions of mixed Hodge theory.

fi, f* preserve weight ~k.

f*, J' preserve weight ~ k.

j,* preserves weight = k.

In particular, this implies the (local) purity of the intersection complex IC~, since Qff[n] is pure and IC~ = j,*(Qff[n]) for j as in 1.3. This is the same argument as in the l-adic case, which uses stability by direct images and subquotients. Note that local purity can also be proved using only the existence of the weight filtration, since the weight filtration of IC~ must be trivial by the simplicity of ICx and the faithfulness and exactness of the forgetful functor rat.

In the l-adic case, the existence of the weight filtration is proved [BBD 5.3.5] after showing the purity of intersection complexes with twisted coefficients [BBD 5.3.2]; in fact, this existence is not used in the definition of 'weight ~ k', etc., nor in the proof of its stability by the functors as above. However, in the mixed Hodge module case, the existence of the weight filtration is more or less assumed from the beginning, since the Hodge filtration and the rational structure are not together strong enough to determine the weight filtration uniquely. The latter fact is of course even true for mixed Hodge structures.

2. Links

We define a link L of a subvariety Z in a variety X to be the level set of a suitable


distance function d whose zero locus is Z. We put three kinds of conditions on this distance function, and consequently get three different kinds of links: a 'weak topological link', which will be used for the rest of this paper, an 'analytic link', which can be defined if Z is compact, and a 'topological link', which is the strongest kind of link. These links are stratified topological pseudomanifolds [GM2 1.1]. We expect that an analytic link is a topological link. We also expect that an analytic link is unique in some sense as stratified topological pseudomanifold.

Since we cannot expect these links to be in general unique, we also introduce functors which describe the cohomology, homology and intersection cohomology of a link intrinsically. These are analogous with the vanishing cycle functors [DE3]. The 'local link cohomology functor' assigns a complex of sheaves on Z to a complex of sheaves on X\Z. We will only apply this functor to the constant sheaf, the dualizing complex and the intersection complex on X\Z. The 'global link cohomology functors' are obtained by composing the local link cohomology functor with the direct image (with and without compact supports) of the map of Z to a point. These link cohomology functors are intrinsically defined. Furthermore, they are naturally related to mixed theories (Hodge or Zadic). The connection between the topological types of link and the cohomological types of link is that a topological link determines a functor which is canonically isomorphic to the local link cohomology functor, and that a weak topological link correspondingly determines a functor canonically isomorphic to the global link cohomology functor.

2.1. Let X be an irreducible complex algebraic variety of dimension n, and Z a closed subvariety. Let

i: Z c.. X,

j: U = X\Z c.. X.

In each of the following three definitions, there will be an open neighborhood N of Z in X and a distance function

d: N ~ [0, I}

with Z = d- 1(0}. Let N* be defined as N\Z with stratification induced by a complex analytic Whitney stratification of X.

The link L will be of (real) dimension 2n - 1, have odd-dimensional strata, and have an orientation induced from that of X. We will have

for ° < e suitably small. Let

k: L,-+ U

be the inclusion whose image is d- 1(e). Also, N* will be a rational homology manifold if and only if Lis.

2.2. A stratified topological pseudomanifold L is a weak topological link if there exist an open neighborhood N of Z in X and a continuous distance function d: N --+ [0, 1) such that Z = d- 1(0) and such that the following conditions are satisfied:

(i) There is a stratified homeomorphism a: N* ~ L x (0, 1) such that d is identified with the second projection.

(ii) For any F in D~(N*) such that HiF is locally constant on each stratum of N*,

the natural morphisms afd*j*F)o --+ H"(Z, i*j*F) are isomorphisms.

Note that condition (ii) is always satisfied if d is proper.

2.3. Suppose Z is compact. A stratified topological pseudomanifold L is an analytic link of Z in X, if there exists a nonnegative real analytic distance function d: N --+ [0,1) where N is an open neighborhood of Z in X, such that Z = d- 1(0) and L is isomorphic to d- 1(e) as stratified space (up to a refinement of stratification) for ° < E « 1. (The stratification of d- 1(e) is induced by a complex Whitney stratification of X compatible with Z).

We may assume that d is proper over its image [0, <5) by taking two relatively compact open neighborhoods U 1, U 2 of Z in N such that (j 1 C U 2, and

restricting d to U 1 \d- 1d(U 2 \ U d. When X is quasiprojective, analytic links can be shown to exist by arguments

similar to [Du].

2.4. PROPOSITION. An analytic link is a weak topological link. Proof Condition (ii) is satisfied by the above remark. By the curve selection

lemma we may assume that the restriction of d to the strata disjoint with Z is smooth. Thus, by the Thom isotopy theorem, the restriction of the Whitney stratification to a- 1(e) is well-defined and N\Z is isomorphic to d -1(e) x (0, <5) as stratified space (See [Du]). Thus condition (i) is checked. 0

2.5. A stratified topological pseudomanifold L is a topological link of Z in X, if there exist a continuous proper surjective map (the retraction of L onto Z)

r: L --+ Z

and a homeomorphism

a: N --+ N

M ixed Hodge structures 57

of an open neighborhood N of Z in X with an open neighborhood N of Z in the open mapping cylinder CyIO(r) of r which is the identity on Z and induces a stratification preserving homeomorphism N\Z ~ N\Z. Here CyIO(r) is defined as ZIlL x [0, l)/r(x) '" (x, 0), and the stratifications of N\Z and N\Z in L x (0, 1) are induced by a complex analytic Whitney stratification of X (defined on a neighborhood of Z) compatible with Z and the stratification of L, respectively.

This definition is inspired by [De3]. In this case we define the distance function d from Z to be the composition of IX with the natural projection CyIO(r) -+ [0, 1) c R. We define the retraction r of N onto Z to be the composition of IX with the natural morphism Cyqr) -+ Z.

In the definition of topological link, we may assume that N = CyIO(r) and that aN is identified with L by replacing d with (d/g) orand N with the subset 1X- 1[ZIl{(x,a)ELx(0, 1): a < g(r(x))}J, where 9 is a continuous function on Z whose value is in (0, 1] and goes to zero appropriately at infinity. Then r = rk where k is as in 2.l.

If X\Z is smooth, the existence of topological links can be shown by reducing to the case where X is smooth and Z is a divisor with normal crossings. The existence of topological links in general will be treated elsewhere.

2.6. PROPOSITION. A topological link is a weak topological link. Proof We may assume N = CyIO(r) as above. Then the condition (i) is

satisfied. For (ii) we have

(ffd*j*F)o = lim H·(d- 1(0, 15), F) ~

and

H·(Z, i*j*F) = lim H·(V\Z, F) ~

where V runs over open neighborhoods of Z in X. Since every V contains an open neighborhood ~ corresponding by IX to Zll {(x, a) E L x (0, 1): a < g(r(x))} for some continuous function g, and H·(~\Z, F) is independent of 9 by the hypothesis on F, we get the assertion. D

When Z is compact we expect that an analytic link is a topological link. The argument is similar to the proof of the Thorn isotopy theorem, but also the limit of integral curves must be controlled using a good partition of unity. This problem will be treated elsewhere.

2.7. Assume that the stratifications of the objects in D~(U) and D~(Z) are algebraic. In the notation of 2.1, the local link cohomology Junctor Az of Z in X is defined for F E D~(U) by


as an object of D~(Z). We also let

Note that the canonical isomorphism of 3.1 implies that

and that there also is the duality isomorphism

We will take F to be Qu, Du and leu in the applications of this paper.

2.8. With the above notation, the global link cohomology functors of Z in X are defined by

for F as above, where (az)*: Z -4 pt. Note that

2.9. PROPOSITION. If L is a topological link of Z in X and if F is in D~(U) and Hi F is locally constant on each stratum of the stratification of X in 2.5, then in

Db(Z) there is a natural functorial isomorphism

Proof Let rf be the restriction of r to N*. We have natural morphisms of r~F to i*j*F and r *k* F, and since r is proper it is easy to check that they are quasiisomorphisms. 0

2.10. PROPOSITION. If L is a weak topological link of Z in X, if F is in D~(U) and if HiF is locally constant on each stratum of the stratification of X in 2.2(i), then there are natural functorial isomorphisms in Db(pt)

Proof The natural morphism k!Qu ® k* F -4 kiF gives a natural isomorphism k* F [ - 1] = k! F. So by duality it is enough to show the first isomorphism. There is a commutative diagram


Z N N* ( k L ) (

j az

i'

jd j'

jd k'

j a,

0 ) [0, 1) ( (0, 1) ( e

By 2.2(ii) and the commutativity of the diagram we have

By 2.2(i), H·d~F is constant on (0,1), and the natural morphisms

are quasi-isomorphisms. o If L is a topological link, the isomorphisms of 2.10 coincide with the direct

images of the isomorphisms of 2.9.

2.11. PROPOSITION. If L is a weak topological link, then

(i) Hk(L) = Hk«az)*AzQu) (ii) Hk(L) = H-k«az)!AzD u)

(iii) IHn+k(L) = Hk«az)*AzICu) = Hk+ l«az)*AzICu)

and the same for H~(L), HfM(L) and IH~+k(L) with (az)* and (az)! exchanged. Proof This follows immediately from 2.10. For the isomorphism (i), let

F = Qu. For (ii), let F = Du and use k!Du = DL [Bo V10.11]. For (iii), let F = Iq? and use k!IC~p = IC~P [GM2 5.4.1] and IC~P = ICu[n]. The proofs in the other cases are similar. 0

3. Mixed Hodge structures

In this section we derive various elementary properties for the mixed Hodge structure on the cohomology, homology and intersection homology of a link. Throughout the section, the notation is that of 2.1.

By the general theory of mixed Hodge modules and making the obvious modifications (replacing Qu by Q~ and so forth), the right hand side of the six isomorphisms in 2.11 have mixed Hodge structures. We define the mixed Hodge structures on the cohomology, homology and intersection cohomology groups associated to a weak topological link by these isomorphisms. Various maps and operations on the right-hand groups are compatible with the corresponding operations on a weak topological link by [GM II]. Note that the results of this


section and the following sections are actually proved for the groups on righthand of 2.11. We define the mixed Hodge structures on the intersection homology groups by setting

IHk(L) = IH?n-l-k(L)(n),

IH:M(L) = IH2n - 1 - k(L)(n)

If U is smooth, these mixed Hodge structures have already been found by the methods of Deligne [EI, Stl, Na]. The mixed Hodge structures here agree with these, because K:= qOi' -+ 8*05. EB Og(1og D)) together with natural Hodge and weight filtrations is isomorphic to the filtered de Rham complex of the underlying filtered complex of ~-Modules of C(hQ~ -+ ]*Qm in a bifiltered derived category, where n: eX, D) -+ (X, Z) is an embedded resolution with I: U = X\Z -+ X, and 8: D. -+ X the semisimplicial space defined by the intersections of the irreducible components of D. In fact (K, F) is a filtered differential complex with weight filtration W as in [Sa3 2.5.8], and it can be checked that the filtered ~-Modules associated to K 1 := C(0i' -+ 8*05. )[ -1] and K 2 := Og(log D) (together with the weight filtration) are canonically isomorphic to the underlying filtered ~-modules of j~Q~ and J:.Q~ respectively by the calculations as in [Sa2 §3]. Moreover the morphism Kl -+ K2 induced by the natural morphism Oi' -+ Og(logD) corresponds to the natural morphism ;;Q~ -+ j:Q~ by the inverse of the de Rham functor [Sa3 §2], which is the identity on U.

3.1. LEMMA. In the notation of 2.1:

(i) For FE DbMHM(U) there are isomorphisms in DbMHM(X)

(ii) For GEDbMHM(X) there are isomorphisms in DbMHM(Z)

Proof (i) For G E DbMHM(X) there is a distinguished triangle [Sal 4.4.1, BBD 1.4.34, GM2 1.11]

Letting G = j*F gives


which gives the first isomorphism of (i). The dual of this triangle with DF replacing F is

which shows the second isomorphism of (i). (ii) The dual of the first triangle in the proof of (i) is

Taking i* gives

which shows the first isomorphism of (ii). The second follows by duality as above. D

3.2. PROPOSITION. If L is a weak topological link of Z in X, the natural morphisms

(and similarly with compact supports) are morphisms of mixed Hodge structures. These morphisms are isomorphisms if X\Z is a rational homology manifold in a neighborhood of Z.

Proof There is a morphism in DbMHM(U) [Sa 4.5.8; GM2 5.1]

QZ[n] -+ ICZ·

The dual of this is

which combines with the first morphism to give

QZ[n] -+ ICZ -+ DZ( -n)[ -n].

Taking i*j* and Hk-n(z, -) gives

so we get the first assertion. The statement with compact supports is similar. The


last assertion is well-known, and follows immediately from the definition of intersection complex [GM2]. 0

3.3. PROPOSITION. If L is a weak topological link of Z in X, the duality

isomorphism [Bo p. 149J

is an isomorphism of mixed Hodge structures. Proof In DbMHM(pt)

where the second isomorphism is by 3.1(i). Taking cohomology, we get the proposition. 0

3.4. PROPOSITION. If L is a weak topological link of Z in X, the natural

intersection pairings [GM2 5.2J

HP(L) ® Hq(L) ~ Hp+q(L)

HP(L) ® IHq(L) ~ IHp+q(L)

HP(L) ® H:M(L) ~ H:~p(L)

IHP(L) ® IHq(L) ~ H~~-l- p-iL)( - n)

(and similarly with compact supports) are morphisms of mixed Hodge structures.

Proof Let ~: X ~ X x X be the diagonal embedding. We have the natural morphisms

Q~ ® F:= ~*(Q~ ~F) ~ F

for F = Q~, IC~ and D~, and

In fact the first is clear by Q~[gj = pr~ and the functoriality of pull-backs. The second is equivalent to


by the adjoint relation. For a nonsingular Zariski-open dense subset V of X we have

using a stratification and the support condition ofICx. Moreover they are equal to 0 for i> 0 and Q( -n) for i = 0, because ICy ® ICy = Qy[2n] and V is connected since X is irreducible. So the morphism is obtained by

By the adjoint relation for j*j*, we have the morphisms

and

because ~ * 0 (i x i)* = i* 0 ~ * (same for the others). Then the assertion follows from

and the same for (ax)!, and the canonical morphism

Note ~! = ~* because ~ is a closed embedding.

In the l-adic case this result follows by an argument similar to [GM 5.2]. In general we have a canonical isomorphism

Hom(F ® G, D~) ~ Hom(F, DG)

o

(i.e. Hom(Q~, Hom(F, G)) ~ Hom(F, G) by duality) for F, GEDbMHM(X), and we can use it to obtain the morphism IC~ ® IC~ --+ D~( - n).

If S is the family of supports in U defined by S = {C c U: C is closed in X}, then H~(U, F) = Hk(X,j!F); see [Go]. If X is compact, then H~(U, F) = H~(U,F).

3.5. PROPOSITION. If L is a weak topological link of Z in X, thefollowing are exact sequences of mixed Hodge structures:

(i) ... --+ H~(U) --+ Hk(U) --+ Hk(L)--+ ... (ii) ... --+IH~(U) --+ IHk(U) --+ IHk(L)--+···

(iii) ... --+ H~(X) --+ Hk(Z) --+ Hk(L)--+ ...

Proof By 3.1(i) for FE DbMHM(U) there is a distinguished triangle

The long exact sequence of hypercohomology on X with F = QZ gives (i), and with F = ICZ and 3.1(i) gives (ii).

By 3.1(ii) there is for G E DbMHM(X) a distinguished triangle

--+ i'G --+ i*G --+ i*j,J*G.

The long exact sequence of hypercohomology on Z with G = Q~ gives (iii). D

4. Semipurity

This section contains the main result of this paper.

4.1. THEOREM. Let L be a weak topological link of Z in X.If dim X = nand dimZ ~ d, then the intersection cohomology IH~(L) has weights ~k for k < n - d.

For the case Z a point, see also [Sa2 1.18].

Proof We use the notation of 3.4. By [BBD 1.4.23(ii)]

where 't"~ -1 is from [BBD 1.4.13]. Now ICx = j!*ICu, so the above together with the distinguished triangle of [BBD 1.4.13] with X = j*ICu gives a triangle

Taking i* gives

Since rat: MHM(X) -+ Perv(X) is faithful and rat 0 't = P't 0 rat, we have

Thus there is a long exact sequence

The middle term above is lH!(L). The left term has weight ~k, since IC~ has weight n, which implies i*ICx has weight ~ n. The right term is zero for k - n < -d = -dimZ, since Hi(P't>_li*j*ICu) = 0 for i < -d as a consequence of the support condition for a perverse sheaf. 0

4.2. COROLLARY. The group lHk(L) has weights >kfor k ~ n + d.

This follows immediately from 4.1 and 3.3.

5. Applications

In this section the results of the previous sections are applied to the topology of a complex variety. We use again the notation of 2.1. Let L be a weak topological link of Z in X. Let

be the morphism from 3.2. The following theorem is concerned with the composite

lH~(L) x lH~(L) :::H2n-l-p-iL)( -n) ~ H~~-l-P-iL)( -n)

where the first morphism is the cup product from 3.4, and the second morphism is the obvious one.

5.1. THEOREM. Let L be a weak topological link of Z in X, and assume that dim X = nand dimZ ~ d.lf IXEIH~(L), {3ElH~(L), and U(IX U {3) is the image of v, with p, q < n - d and p + q ~ n + d, then U(IX U {3) = O.

Proof Weight IX ~ P and weight {3 ~ q by 4.1, so weight U(IX U {3) ~ p + q by 3.4. Since weight (im v) > p + q by 4.2 and 3.2, this implies U(IX U {3) = O. 0

Let

be the natural map.

5.2. THEOREM. Let X, Z and L be as in 5.1. Suppose X\Z is a rational

homology manifold in a neighborhood of Z. If k 1, ... , km < n - d and

k = kl + ... + km ~ n + d, then the composition

is the zero morphism.

The proof is similar to the proof of 5.1. Note that since H;n -1(L) has weight 2n by 3.3, the above theorem is true

without the composition with w when k = 2n - 1. Also note that if Z is compact, both u and ware isomorphisms.

5.3. EXAMPLES. Let

(i) If n ~ 2 and d = 0, then L = (T 2n - 1 with two odd-dimensional submanifolds identified) cannot be a weak topological link of Z in X: In Theorem 5.1 take p = q = n - 1. The map IW- 1(L) ® IHn-1(L) ~ H 1(L)( -n) is nonzero and factors through IH1(L)(-n)~H1(L)(-n) as is seen by using IH(L) ~ H(T2n-1) [GM1 4.2] and the diagram

(ii) If Z is compact and n - d > 1, then T 2n - 1 cannot be a weak topological link of Z in X: In Theorem 5.2 take ki = 1.

Note that the above examples L are compact oriented topological pseudomanifolds with odd-dimensional strata, so that they are not excluded from being links by [Su].

Bibliography

[Be]

[BBD] [Bo]

A. Beilinson: On the derived category of perverse sheaves, in: K-Theory, arithmetic and geometry, Lecture Notes in Math. 1289, Springer-Verlag, Berlin, 1987,27-41. A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers, Asterisque 100 (1982). A. Borel et aI., Intersection cohomology. Birkhiiuser Boston 1984.

[Del] [De2] [De3]

[Du] [DH]

[El] [Ga]

[GMl]

[GM2] [GM3]

[Go] [IV] [Na]

[Sal] [Sa2]

[Sa3] [Stl]

[St2] [Su]


P. De1igne: Theorie de Hodge III. Pub!. Math. IRES 44 (1974) 5-78. P. Deligne: La conjecture de Weil II. Publ. Math. IRES 52 (1980) 137-252. P. Deligne: Leformalisme des cycles evanescents. In SGA7 XIII and XIV, Lecture Notes in Math. 340, Springer-Verlag, Berlin, 1973, pp. 82-115 and 116-164. A. Durfee: Neighborhoods of algebraic sets. Trans. Amer. Math. Soc. 276 (1983) 517-530. A. Durfee and R. Hain: Mixed Hodge structures on the homotopy of links. Math. Ann. 280 (1988) 69-83. F. Elzein: Mixed Hodge structures. Proc. Symp. Pure Math. 40 (1983) 345-352. O. Gabber: Purite de la cohomologie de MacPherson-Goresky, redige par P. Deligne. (IHES preprint Feb. 1981). M. Goresky and R. MacPherson: Intersection homology theory. Topology 19 (1980) 135-162. M. Goresky and R. MacPherson: Intersection homology II. Inv. Math. 72 (1983) 77-129. M. Goresky and R. MacPherson: Morse theory and intersection homology theory. Asterisque 101-2 (1982) 135-192. R. Godement: Thtiorie des faisceaux. Hermann, Paris 1958. B. Iverson, Cohomology of sheaves. Springer, Berlin, 1986. V. Navarro-Aznar: Sur la theorie de Hodge des varietes algebriques a singularites isolees. Asterisque 130 (1985) 272-307. M. Saito: Mixed Hodge modules. Pub!. RIMS, Kyoto Univ. 26 (1990), 221-333. M. Saito: Introduction to mixed Hodge modules. Preprint RIMS-605 (1987), to appear in Asterisque. M. Saito: Modules de Hodge polarisables. Publ. RIMS, Kyoto Univ. 24 (1988),849-995. J. Steen brink: Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math. 40 (1983) 513-536. J. Steenbrink: Notes on mixed Hodge theory and singularities. To appear, Asterisque. D. Sullivan: Combinatorial invariants of analytic spaces, In: Proceedings of Liverpool Singularities Symposium I, Lecture Notes in Math. 192, Springer-Verlag, Berlin, 1971, 165-168.

Effective bounds for semipositive sheaves and for the height of points on curves over complex function fields *

HEL1~;NE ESNAULT1 & ECKART VIEHWEG2

lMax-Planck-Institutfur Mathematik, Gotifried-Claren-Str. 26, D-5300 Bonn 3, FRG; 2FB6, Mathematik, Universitiit-GH-Essen, Universitiitsstr 3, D-4300 Essen 1, FRG

Received September 1988; accepted 20 July 1989

In this note we prove an effective version of the positivity theorems for certain direct image sheaves for fibre spaces over curves and apply it to obtain bounds for the height of points on curves of genus 9 ~ 2 over complex function fields. Similar positivity theorems over higher dimensional basis and their applications to moduli spaces [13J were presented by the second author at the conference on algebraic geometry, Humboldt Universitiit zu Berlin, 1988.

Let X be a complex projective surface, Y be a curve and f: X --+ Y be a surjective, non isotrivial morphism with connected general fibre F. In 1963 y. Manin [6J showed that the number of C(Y) rational points of F is finite if the genus 9 of F is larger than 1. A C(Y) rational point p E F gives rise to a section 0': Y --+ X of f. If one assumes that the fibres of f do not contain exceptional curves the height of p with respect to WF is h(p) = h(O'(Y)) = deg(O'*wx/y).

It is well known that Manin's theorem "the Mordell conjecture over function fields" can be proved by bounding h(O'(Y)) from above for semistable morphisms f. The main result of this note is:

THEOREM 1. Assume that f: X --+ Y is relatively minimal. Let q be the genus of Y, 9 ~ 2 the genus ofF and s the number of singular fibres off. Then for all sections 0' of f one has

h(O'(Y)) < 2· (2g - 1)2. (2q - 2 + 2s).

If moreover f is semis table, then

h(O'(Y)) < 2'(2g - If'(2q - 2 + s).

In fact, if f is not semistable, a closer look to the semistable reduction of f gives a slightly better bound (see Corollary 4.10).

*Supported by "Heisenberg Programm", DFG.

70 Helene Esnault and Eckart Viehweg

Effective bounds for the height were first given by A. Parshin [8] and, in arbitrary characteristics, by L. Szpiro [9]. In [15] A. Parshin announced that, using H. Grauert's proof [4], it is possible to bound h(u(Y» by a polynomial of degree 13 in g. As S. Lang and Y. Miyaoka pointed out, one can use the Miyaoka-Yau inequality to get a bound, linear in g.

The proof of Theorem 1 presented in this paper is given in two steps: First we show, that the sheaf f * wi,y(u(Y» can not have an invertible quotient sheaf oflow degree, if h(u (Y» is large. Then, in Section 4, we use the Kodaira-Spencer map to show that this sheaf always has a quotient of degree 2q - 2 + 2s. The existence of global one forms is only used in this second step. Of course, it would be quite interesting to get along without using Ok at all. May be, combining methods from P. Vojta's proof of Manin's theorem in [14] and from this paper, this could be done. In fact, the methods used in the first part are overlapping with those used in [1] to prove Dyson's lemma in several variables. Hence the relation between [14] and this paper might be quite close.

The experts will see immediately that the second step in our proof is not too different from the arguments used by Y. Manin [6], H. Grauert [4] and L. Szpiro [9]. The "Parshin-construction" used in [8] and [9], however, is replaced by the effective bounds on the "positivity of certain direct image sheaves". This part (see 2.4 for the exact statement) is presented in the first two sections of this article. Without having any other application, we took Theorem 1 as a pretext allowing us to work out for fibre spaces over curves an effective version of the results of [11]. The reader not familiar with the notations used there should have a look to S. Mori's survey article [7].

In Section 3 we just evaluate the constants obtained for general fibre spaces in the special case of families of curves and we verify the assumptions made in 2.4 in this special case.

The motivation to write this note grew out of discussions with A. Parshin during his and our stay at the Max-Planck-Institute for Mathematics in Bonn. The details were worked out during our stay at the TATA-Institute in Bombay.

1. The lower degree of direct images of sheaves

Let Y be a nonsingular compact curve defined over C and :F be a coherent locally free sheaf on Y.

DEFINITION 1.1. (a) The lower degree of:F is defined as

Id(:F) = min{deg(.;V); .;V invertible quotient sheaf of :F}.

Effective bounds 71

(b) The stable lower degree of ff' is

sld(ff') = inf f~~r: ~); r: Y' --+ Ya finite map of non singular curves}.

If ff' = 0 we put ld(ff') = sld(ff') = 00. I

(c) ff' is called semi-positive if sld(ff') ~ 0 (Fujita, [3]).

1.2. Some properties

(a) If ff is an invertible sheaf of degree d, then

ld(ff' ® ff) = ld(ff') + d and

sld(ff' ® ff) = sld(ff') + d.

(b) If p: Y" --+ Y is a non singular covering then

sld(p* ff') = deg(p)' sld( ff')

(c) ff' is ample if and only if sld(ff') > o. (d) The following three conditions are equivalent:

(i) sld(ff') ~ o. (ii) ff' is weakly positive over Y (see [11]). (iii) If JIf is an ample invertible sheaf on Y then for all Y > 0 the sheaf

SY(ff') ® JIf is ample. (e) For all y > 0 Id(®Y(ff')) ~ Id(SYff') ~ Y'ld(ff') and

sid (®Y(ff')) ~ sld(SYff') ~ y·sld(ff').

Proof. (a) and (e) follow directly from the definition. (c) If ff' is ample then SY(ff') ® JIf -1 will be ample for some Y» 0 and sld(ff') ~ l/y· deg(JIf). If sld(ff') ~ 8 > 0 then (0[1'(1) on I!Jl = 1!Jl(ff') satisfies the Seshadri criterium for ampleness (see for example: R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math. 156, springer 1970, or [3] §2). The proof of (d) is similar. In [11] Section 1 and [13] Section 3 the reader can find some generalizations for higher dimensional Y.

In (b) it is obvious that sld(p*ff') ~ deg(p)·sld(ff'). On the other hand, if r: Y' --+ Y is another nonsingular covering we can find r': Y'" --+ Y dominating both, rand p. One as

ld(r'*ff') Id(r*ff')'deg(Y'" --+ Y') ld(r*ff')'deg(p) sld(p*ff') ~ deg(Y'" --+ ylI) ~ deg(Y'" --+ ylI) = deg (r) .


(1.3) In our estimations of the stable lower degree we will frequently use vanishing theorems for integral parts of iIJ divisors. Let g: Z -+ X be a morphism of complex projective manifolds. For a normal crossing divisor D = I.viDi and e E ~ we write [e' D] = I.[e· va' Di where [e' va is the integral part of e' Vi. Recall that an invertible sheaf 2 on Z is called numerically effective if deg(2ld ~ 0 for all curves C in Z. We will say that 2 is g-numerically effective if deg(2ld ~ 0 for all curves C in Z with dim(g( C)) = O. K(2) denotes the Iitaka dimension of 2 (see [7] for example) and wZ/x = Wz ® g*wx 1 the difference of the canonical sheaves.

LEMMA 1.4. (a) Assume that there exists an effective normal crossing divisor D on Z such that 2N( - D) is g-numerically effective. If for a general fibre G of g

K(2( - [DIN]) ® {DG)=dim G then for i > 0, Rig*(wz/x ® 2( - [DIN])) =0. (b) Assume that g is birational.

Ifr is an effective divisor on X, D = g*r a normal crossing divisor and N > 0, then Rig*wz/x ® (Dz(-[DIN]) = o for i > o. If moreover r is a normal crossing

divisor then g*wz/x ® (Dz( - [DIN]) = (Dx( - [r/N]). Proof. (a) The assumptions imply that 2N( - D) ® g* ytN will be numerically

effective and K(2( - [DIN]) ® g* yt) = dim Z for all "very very" ample invertible sheaves yt on X. As in [10], 2.3, the vanishing theorem due to Kawamata [5] and the second author ([10] and [2], 2.13) implies (a) by using the Leray spectral sequence. (b) Is shown in [10], 2.3.

(1.5) Let in the sequel Y be a nonsingular compact curve, X a projective manifold of dimension n andf: X -+ Ya surjective morphism. The general fibre of f will be denoted by F. Let 2 be an invertible sheaf on X.

All estimates of sld(f *(2 ® wx/y)) will follow from the following corollary of Fujita's positivity theorem [3] (see [11] 5.1):

LEMMA 1.6. Let D be an effective normal crossing divisor on X and N > 0 such

that 2N = (Dx(D). Then f*(wx/y ® 2( - [DIN])) is semi-positive. Proof. The case 2 = {Dx is Fujita's original theorem. It may be easily obtained

by Hodge theory on cyclic covers ([12], 6 and 8). The general case follows from this one, applied to the cyclic cover given by

2N = (Dx(D) (as in [2], 2.7 or [11], 2.2).

Recall the following notation: If U c X is open, we call 2 very ample with

respect to U if HO(X, 2) ®c (Dx -+ 2 is surjective over U and the natural map U -+ IP(HO(X, 2)) is an embedding. Correspondingly we call 2 ample with

respect to U if for some a>02D is very ample with respect to U (see [13],1.16).

We will call 2 numerically effective with respect to U if there exists a birational morphism g: X' -+ X, isomorphic over U, and an invertible numerically effective sheaf 2' and an inclusion 2' -+ g* 2, isomorphic over U.

Effective bounds 73

Obviously, if !l' is ample with respect to U it is numerically effective with respect to U. Moreover, if yt' is ample with respect to U and !l' numerically effective with respect to U then yt' ® !l'a is ample with respect to U, for all a ~ O.

COROLLARY 1.7. Let D be a normal crossing divisor on X and N > 0, such that !l'N( - D) is numerically effective with respect to a neighbourhood of F and K(!l'N( -D)) = n. Then f*(wx/y ®!l'( - [DIN])) is semi-positive.

Proof. By 1.4(b) the statement is compatible with blowing ups. As K(!l'N( - D)) = n, we may assume (for example as in [2],2.12) that there exists an effective divisor r on X such tha D + r has normal crossings and such that !l'N( - D - r) is ample. Then for M ~ 0 !l'N.M( - M· D - r) will be ample with respect to a neighbourhood of F. Replacing N, D and r by some common multiple we can find a divisor H, smooth on F such that (blowing up a little bit more) D' = H + M· D + r has normal crossings and !l'N'M = (!}x(D'). For M big enough

[~JI = [H + M·D + rJI = [DJI M·N F M·N F N F

and therefore 1.7 follows from 1.6.

DEFINITION 1.8. Let Z be a manifold vN be an invertible sheaf and r be an effective divisor.

(a) Let .: Z' ---t Z be a blowing up such that r' = .*r is a normal crossing divisor. We define

(b) e(vN) = max{e(r); r zero divisor of SE HO(Z, vNn. By 1.4(b) the definition of e(r) is independent of the blowing up choosen. In

Section 2 we will give upper bounds for e(r). 1.8.1. Especially one obtains e(vN) < co. For e ~ e(r) and. as above one has .*wz'/z( -[r'/e]) = (!}z. If r = ~Viri is

a normal crossing divisor, then e(r) = max{v;} + 1.

COROLLARY 1.9. Assume that !l' is numerically effective with respect to some neighbourhood ofF and K(!l') = n. Assume moreover thatfor some invertible sheaf yt' on Y of degree h and some N > 0 one has an inclusion qJ: f* yt' ---t !l'N. Then


Remark. Especially, if h > 0 the sheaf f*(2 ® wX/Y) will be ample by 1.2(c) and 1.8.1. A similar result for higher dimensional Y can be found in [11] 5.4.

Proof. Let M = max{N,e(2N IF)} (1.8.1). We can choose a cover 11: Y' ~ Yof degree M such that the fibres of f over the ramification locus are non singular. Then X' = X x Y Y' is non singular and

By 1.2(b)itsenough to show 1.9 for X' ~ Y'. Hence, by abuse of notations we may assume that M divides h. Let JV E PicO(y). One has for example by Seshadri's criterion 1((2 ® f* JV) = 1((2). Using 1.2(a) we may replace 2 by 2 ® f* JV and Jf by Jf ® JVN. Therefore we may assume that Jf = (9y(h· p) for some point p E Y. Let r be the zero divisor of q> and let r: X' ~ X be a blowing up such that T*(r + h· f -l(p)) becomes a normal crossing divisor. Let f' = for, r' = r*r and 2' = r* 2. Since M ~ e(qF) the inclusion

is surjective at the general point of Y. This implies

One has 2IM(_rl_h·f'-1(p))=2'M-N and by 1.6 (if M=N) or 1.7 (if

M > N) the sheaf

f~(21(- [~J -~.f'-l(p)) ®WX'/y)

= f~( 2'( -[~J) ® wX'/y ) ® (9y( - !. p) is semipositive. From 1.2, a we obtain sld(f*(2 ® wx/y)) ~ hiM.

2. Bounds for e(2) and the main theorem

(2.1) Consider a complex projective manifold Vand an effective divisor r on V. We write./lt = (9v(r) and choose a blowing up r: V' ~ V such that r*r = r ' has

Effective bounds 75

normal crossings. Let us write ~(e) = coker(r*wy ,( - [r'/e]) -+ wy ). By 1.4(b)~(e) is independent of the blowing up choosen.

LEMMA 2.2 Let H be a smooth prime divisor of V which is not a component of r. Then Supp(~(e)) n H = (j) for e ~ e(r!H)'

Proof We may assume by l.4(b) that r' intersects the proper transform H' of H transversally. Then [r'/e]lw = [r'lw/el One has a commutative diagram

By the vanishing Theorem 1.4(b), a is surjective. If e ~ e(r!H)' f3H is surjective. Then f3 has to be surjective in a neighbourhood of H.

PROPOSITION 2.3. Let Zj, i = 1, ... , r, be projective manifolds, Yfj be a very ample invertible sheaf on Zj and m, d > 0 such that C1 (Yf;)dim(Z,) ~ dim. Let V be the r10ld product Z 1 X ... X Zr and .11 = ®~ = 1 pr{ Yf'('. Then e(.I1) ~ d + 1.

Proof. Consider first the case r = 1. Set Z = Zl, Yf = Yf1 • We prove 2.3 by induction on dim Z. If Z is a curve r is an effective divisor on it of degree ~ d. Therefore [rid + 1] = O. Assume dim Z ~ 2. Choose, r, r as in 2.1 and H a smooth hyperplane section with Yf = (!J(H). Then c1(YfIH )dimZ-l ~ dim. If H is not a component of r, then by induction and 2.2 Supp ~(d + 1) does not meet H. As we may find such a H containing any given point, we obtain ~(d + 1) = O.

We proceed by induction on r. We assume that 2.3 holds for T = Zl X ... X

Zr-1 and f£ = ®~: i pr{ Yf'('. If Zr is a point, then 2.3 holds by induction. Assume that Zr is a curve. Choose

rand r as in 2.1. Take a point pE Z" and define D = T x p ~ T. Let v be the maximal integer such that v 0 D ~ r. As deg Yf~ ~ d, one has 0 ~ v ~ d, We may assume that the proper transform D' of D in V' meets !1' = r' - v 0 r* D transversally. From the inequality

A' ~ r' - (d + l)o(r*D - D') - voD' = A' - (d + 1 - v)o(r*D - D')

one obtains -[A'id + 1] + D' ~ - [r'ld + 1] + r*D. The multiplicity of D' is


one on both sides of the inequality. One has thereby a commutative diagram

( [ A' ] ) IX ([ A'ID' J) fJD ,*wv' - d+1 +D' ----+"WD' - d+1 ----+WD

1 II

'* wv{ - [d : 1 J) ® (D(D) L wy(D) -------+1 WD

By the vanishing Theorem 1.4(b) rx is surjective. As (DD(r - v'D) = (DD(r) ~ 2, fJD is an isomorphism by induction. Therefore Supp C6'(d + 1) does not meet D. Moving p, we obtain 2.3 for dim Zr = 1. Assume dim Zr ~ 2. Choose F a general hyperplane section in Zr with (D(F) = Yl'r. As c 1 (Yl'rIF )dimF ~ dim, we have e(A'IH) ~ d + 1 by induction for H = T x F, and 2.2 implies that for all r with A' = (Dv(r), supp C6' (d + 1) does not meet H. As we may find such a F such that H = T x F is not a component of r and contains any given point, we obtain 2.3.

The main result ofthis note is the following theorem, which for 2 = w1il is an effective version of a special case of [11], 6.2.

THEOREM 2.4. Let Y be a nonsingular compact curve; X be a projective manifold of dimension nand f: X ---? Y be a surjective morphism. Let 2 be an invertible sheaf on X with K(2) = dim X. Assume thatfor some N > 0 the sheaf 2NIF is very ample on the general fibre F off and that 2 is numerically effective with respect to some neighbourhood of F. Write d = C1 (21F )n-1. Then for all m > 0

Proof. Let us start with the case m = 1:

Let r = rank(f*2N), x r the r-fold product X Xy X··· Xy X and j': xr ---? Y the induced map. If JV is any locally free sheaf on X we obtain by flat base change f~( ®~ = 1 pr'{ JV) = ®r f*%. f is a flat Gorenstein morphism and WXjY the same as the dualizing sheaf of f. Therefore WxrjY = ®r=l priwxjY (see [11], 3.5, for similar constructions). Let u: x(r) ---? xr be a desingularization, isomorphic on the general fibre F x '" x F, and f(r) = j'ou. For A' = U*(0r=lpri 2) we have

inclusions ®ri = 1 pr'{ 2N ---? U*A'N and

u * (A' ® Wx(r)jY) = (~ pri 2) ® Yl'Q?n«)X'(u * (DX('j , wX'/Y) ---?

---? C~ pri 2) ® WxrjY = ~ pr'{(2 ® WXjy)·

Effective bounds 77

The induced inclusions

r r

® f*!l'N --+ f~)vI(N and f~)(vI( ® rox(rl/Y) --+ ® f*(!l' ® rox/y)

are both isomorphisms at the general point of Y. Especially one has

r

sId ® f*(!l' ® rox/y) ~ sldf~)(vI( ® rox(rl/Y).

One has ,,(vi() = dim x(r) and vl(N is very ample on the general fibre F x ... x F of pr). We have a natural inclusion

r

det(f*!l'N) --+ ® f*!l'N --+ f~)vI(N

Up to now we did not use that!l'N is very ample and the last inequality holds for all exponents. Especially replacing N by m· N we find that (using 1.2(e»

On the other hand we have shown in 2.3 that

COROLLARY 2.S. Under the assumptions of 2.4 let !l' be even numerically

effective on x. Then

Proof. If!l' is numerically effective, the dimension of the higher cohomology groups of !l'm·N is bounded from above by a polynomial of degree n - 1 in m.


Since the Leray spectral sequence gives an inclusion

the same holds true for h1(Y,f*ym.N).

The Riemann-Roch-Theorem for vector-bundles on Y and for invertible sheaves on X implies that

Then, using in the same way the Riemann-Roch on F and taking the limit over m we get 2.5 from 2.4.

REMARK 2.6. Especially for those, mostly interested in the case that f is a family of curves, it might look quite complicated that the proof of 2.5 and 2.4 forced us to consider higher dimensional fibre spaces. In fact, if one is just interested in 2.5 this is not necessary and we sketch in the sequel a proof which is avoiding the products in 2.3 and 2.4: If p is a point on Y the Riemann-Roch theorem shows that hO(Y,f*yN.mQ!)(l)y(-h·p)) is larger than or equal to deg(f*yN.m) - rank(f*yN.m)·(h + q-l). Therefore, whenever we have

we will find an inclusion of lDy(+h'p) in f*yN.m.

Applying 1.9 and 2.3 (for r = 1) we obtain the same inequality as in 2.4, except that we have to add a - q on the right hand side. Since in the proof of 2.5 we were taking the limit over m anyway, this is enough to obtain 2.5.

3. Examples and applications

The first application of 2.4 is not really needed in the proof of theorem one and it is just added for historical reasons.

THEOREM 3.1. Let f: X --+ Y be a surjective morphism with general fibre F,

where X is a projective manifold of dimension nand Ya non singular curve, and let v > 1. Assume that for N > 0 OJ~ is very ample and that f is non isotrivial. Then,

for all multiples m of v - 1

Effective bounds 79

Moreover for v > 1,f*wx/y is ample, (if it is not trivia0. Proof. Results due to J. Kollar and the second author show that K(Wx/y) =

dim(X) (see [7] or [13], §1(c) for example). In fact, one first shows that deg(f*w~il~') > 0 for m » 0 and then one uses methods similar to those used in the proof of 2.4 to show that f * wilY is ample for J1 » O. Then wx/y will be ample with respect to a neighbourhood of F and the inequality follows then from 2.4 for !f? = wxil. By 1.4(a) f*wx/y will be ample whenever it is not trivial.

EXAMPLE 3.2. Assume that X is a surface, and moreover that f is a non isotrivial family of curves of genus g ~ 2.

(a) For N > 1 onehasrankf*w~/y = (2N - l)·(g - 1). Applying 3.lfor v = 2, one obtains

(b) Let!: X ~ Y be the relative minimal model of f. By definition the fibres of J do not contain any (-1)-curves and wX/Y is J-numerically effective. On the other hand, if B is a curve on X which dominates Y, then l*l*wg/y ~ Wg/y ~ Wg/ylB is non trivial. Since the sheaf on the left hand side is the pullback of a semi positive sheaf, deg(wg/yIB) ~ O. Therefore Wg/y is numerically effective.

From 2.5 we obtain that sld(f*wi/y)·(2g - 2)2 ~ !Cl(wg/yf.

(c) One has Cl (wg/y)2 > 0 as K(Wg/y) = 2 and since Wg/y is numerically effective (see [10], §3). If one does not want to use this non trivial fact, one can get along with Cl (wg/y)2 ~ 0 if one replaces all strict inequalities in the sequel by" 2:". The weak inequality follows directly from (b).

(3.3) From now on f: X ~ Y will denote a non isotrivial family of curves of genus g ~ 2 and 0": Y ~ X a section. Let C = O"(Y) and let!: X ~ Y be the relative minimal model. The image C of C in X intersects the fibres of 1 in smooth points. Therefore we may assume that all fibres of f are normal crossing divisors, and that C does not meet any exceptional divisor contained in the fibres. Of course, h(C) = cdwg/y)·C is the same as cdwx/y)·C under this assumption.

LEMMA 3.4. For N > 1 one has

deg(f*wx/y(ct) = deg(f*w~/y) + !N(N - 1)· h(C).

rank(f*wx/y(ct) = N·(2g - 1) - (g - 1).

Proof. We may assume here that f is relatively minimal, i.e. X = X. Then wx/y as well as wx/y(C) arefnumerically effective (see 1.3). Then by 1.4(a) we have for 0 :( J1 < N exact sequences

Since wx/y( C)lc = (!h the sheaf on the right hand side is

Adding up we obtain

N-1

deg(f*wx/y(ct) - deg(f*w~/y) = L (N - 11- 1)·h(C). fl=O

The second equality is trivial.

COROLLARY 3.5. Under the assumptions made in 3.3 we have for N ~ 2

sld(f*wi/y(C))'(N'(2g - 1) + 1)'(N'(2g - 1) - (g - 1))

N N(N - 1) ~ deg(f *wx/y) + 2 . h(C)

and

Proof Since w1'/yis numerically effective, the same holds for W1'/y(C). Moreover,ifh(C) #- 0,c1(W1'/y(C))2 = C1(W1'/y)2 + h(C) > ° and hence K(W1'/y(C)) = 2. The first inequality follows from 3.4 and 2.4 applied to !I! = W1'/y(C) and the second one from 2.5 applied to !I! = W1'/y( C).

REMARK. Since the arguments used in 3.4 also show that wX/YIc is a quotient of f*w}/y(C) we can state as well h(C) ~ sld(f*w}/y(C)) and

COROLLARY 3.6. Using the notations and assumptions made in 3.3

4. Effective bounds for the height

We want to finish the proof of Theorem 1. (4.1) Let f: X -+ Y be a family of curves. Let S = {y E Y; f -1 (y) singular} and

D = f*(S). We assume that D is a normal crossing divisor (i.e. an effective divisor, locally in the analytic topology with nonsingular components meeting transversally). Recall that f is called semistable when D is a reduced divisor.

Let ni(D> = nl<Dred > be the sheaf of differential forms with logarithmic poles along D. The natural inclusion f*n~<s>-+nl<D> splits locally. In fact, f is locally given by t = xa • yP, where x and yare parameters on X and

Effective bounds 81

t a parameter on Y. Then dt/t = Ct" dx/x + f3 0 dy/y is part of a local bases of n1(D). The quotient sheaf, denoted by nl/y<D), is therefore invertible. Comparing the highest wedge products one finds

As C = a(Y) meets D only in points which are smooth on D, the cokernel nl/y<D + C) off*n}<S) --+ nl<D + C) will be invertible as well and one has nl/y<D + C) = nl/y<D) ® @x(C).

LEMMA 4.2. Iff is non isotrivial, there is a nonzero map

Moreover, if C is a section y factors through

Proof of Theorem 1. If f is semis table then nl/y<D) = wx/y. 4.2 implies that sld(f*(wi/y(C)) ::::; 2q - 2 + s. In general one has an inclusion

and using 4.2 and 1.2(a) one obtains sld(f*wi/y(C))::::; 20q - 2 + 20s. In both cases 3.5 gives (2 0 g - 1)-2 oh(C)::::; 20sld(f*wi/y(C)).

The construction of y': (4.3) We have a commutative diagram of exact sequences:

o o

1 1

0--+ f*n}<S) --+ n1(D + C) --+ nl/y<D + C) --+ 0

I j

o o

(4.4) Applying Rf* to the diagram 4.3 we obtain

®n}<s)

15 is the Kodaira-Spencer class and, since f is not isotrivial, both 15 and 15' are non zero. If we tensorize 4.3 with wX/y and apply Rf* again we get

(4.5)

f*(wx/y ® nl/y<D») --,---Y ---+) R1f*wx/y ®Q}<S)

j j=

Since the left vertical arrow is injective 1" cannot be zero if l' is non zero.

LEMMA 4.6. There is a commutative diagram

where m is multiplication and U the cup product. Proof One has a natural mapf*f*wx/y ---+ wX/y and taking the tensor product

with the first row of 4.3 we get a commutative diagram

Applying Rf* we obtain the diagram in 4.6.

Proof of 4.2. It is enough to show that l' is non zero. Let U be some open subvariety of Y such that n~<S) is generated by a differential form rx. Since 15 is non zero we may choose s E f*nl/y<D)(U) with J(s) #- O. For some

Effective bounds 83

o #- AER1f*(I)x(U) we can write b(s) = A ® IX. Since U induces a perfect pairing

there is some J.1. E f*wx/y(U) with J.1. U A #- O. Then J.1. U A ® IX #- 0 and by 4.6 this is the same as y(m(J.1. ® s)).

(4.7) Of course, 4.2 also implies that for non isotrivial families

If f is semistable and minimal this together with 3.2 shows (see [9] for better bounds):

COROLLARY 4.8. Iff is semis table, relatively minimal and non isotrivial and if q is the genus of Y and s the number of degenerate fibres of f, then for N > 1

(2q - 2 + s)(N ·(2g - 2) + 1)(2N - 1)(g - 1) ~ deg(f*w~/y)

and

If f is not semistable let r: Y' ---+ Ybe a non singular Galois cover such that the ramification index of s E Y is divisible by the multiplicities of f - 1(S). Let b: X' ---+

X Xy Y' be the normalization, f' = pr2 ob: X' ---+ Y' and r = pr1 ob: X' ---+ X. If the fibres of f are normal crossing divisors, then X' has at most rational Gorenstein singularities. Especially WX' is invertible and, if r': X" ---+ X' is a desingularization, then r~wX" = WX" Moreover X' is non singular in a neighbourhood of a section C. In fact, under the assumptions made, we may choose X" such that X" ---+ Y' is semis table.

LEMMA 4.9. Let 5£ be an invertible sheaf on X and 5£' = r* 5£. Then there is an inclusion f~(5£' ® wx'/y') ---+ r* f*(5£ ® wx/y) isomorphic in the general point ofY'.

Proof As in the proof of 2.4 one obtains by duality theory an inclusion

4.9 follows by flat base change.

COROLLARY 4.10. If f: X ---+ Y is not semis tab Ie, then the bound for h(O"(Y)) given in Theorem 1 can be improved to

h(O"(Y)) < 2·(2g -1f·(2q - 2 + s) + s.

Sketch of proof If f: X ---+ Y is any morphism, not necessarily relatively

minimal, and C=a(Y), then h(C):::;c1(wX /Y)·c. Let r: Y'--+Y be a nonsingular Galois cover with Galois-group G, such that X x y Y' --+ Y' is birational to a semis table family of curves cp: T --+ Y', relatively minimal over Y'. Let B be the image of the section of cp lifting a. The action of G on Y' extends to an action of G on T, such that X is birational to T/G.

CLAIM 4.11. There exists a sequence of at most s· ord( G) blowing ups of points 1]: X" --+ T such that G acts on X" and such that the quotient X" /G is non singular in some neighbourhood of C"/G, where C" is the proper transform of B.

Using the notations from 4.11 one has Cl(WX"/Y')·C":::; s·deg(r) + h(B). We choose X to be a desingularisation of X"/G, isomorphic to X"/G near C"/G, such that f: X --+ Y has fibres with normal crossings. As above, let 1': X' --+ Y' be a minimal desingularization of X x Y Y' --+ Y', r': X' --+ X the induced map, C' = r,-l(C) and D' = r'-l(D). One has r'*nhy,<D' + C') = nhy,<D' + C') and, since I' has a semistable model, both f~wx'/y'(C') and

are independent of the model chosen for 1'. Especially both sheaves coincide. Using 3.5, 4.9, 1.2(b) and 4.2 one obtains

t(2g _1)-2. h(C') = t(2g - 1)-2. h(B) :::; sld(f~whdc))

:::; sld(f*wx/y ® nl/y<D + C»)· deg(r)

:::; (2q - 2 + s)· deg(r).

On the other hand, near C the sheaves nl/y<D) and wX/y coincide and, using the notations introduced above, one has

Proof of 4.11. The question is local in Y and we may replace G by the ramification group of some p E Y'. Hence we assume Y to be a small disk and G = <a) to be cyclic of order N. Let Q = B n cp-l(P)E T. We can find local coordinates x and y near Q, such that x is the pullback of a coordinate on Y', and such that the zero set of y is B. Moreover we can assume that a(x) = e· x and a(y) = ell., where e is a primitive Nth root of unit and 0 :::; Ji, < N. Blowing up Q we obtain T' and local coordinates x' and y' near Q' = cp' -l(p) n B' with x' = x and y' = y/x. Therefore a(x') = e· x' and a(y') = ell' • y' for Ji,' = Ji, - 1. After at most N blowing ups we may assume that Ji,' = O. Then, however, the quotient is non singular.

Effective bounds 85

COROLLARY 4.12. Assume that f is not isotrivial. Let q be the genus of Yand s be the number of degenerate fibres of f. Then one has

4.12 implies the well known fact that a non isotrivial family of curves of genus g ;::: 2 over Y must have at least three degenerate fibres, if Y = IP 1, and at least one degenerate fibre, if Y is an elliptic curve.

(4.13) If f: X -+ Y is a family of higher dimensional canonically polarized manifolds with degenerate fibres one can consider the iterated Kodaira-Spencer map f*D.'Xil<D) C(D.}<S»)"-10R"-lf*@x and

With 3.1 and the same arguments we used above one obtains

PROPOSITION 4.14. Assume that £5"-1 is non trivial then (2q - 2 + s) > O.

However, we do not know any reasonable criterion implying that £5" -1 #- O.

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2. H. Esnault and E. Viehweg, Logarithmic De Rham complexes and vanishing theorems. Invent. Math. 86 (1986) 161-194.

3. T. Fujita, On Kiihler fibre spaces over curves. J. Math. Soc. Japan 30 (1978) 779-794. 4. H. Grauert, Mordell's Vermutung iiber rationale Punkte auf algebraischen Kurven und

Funktionenkorper. Publ. Math. IHES 25 (1965) 131-149. 5. Y. Kawamata, A generalization of Kodaira-Ramanujam's vanishing theorem. Math. Ann.

261 (1982) 57-71. 6. Yu. I. Manin, Rational points on an algebraic curve over function fields. Trans. Arner. Math.

Soc. 50 (1966) 189-234. 7. S. Mori, Classification of higher-dimensional varieties. Algebraic Geometry. Bowdoin 1985.

Proc. of Syrnp. in Pure Math. 46 (1987) 269-331. 8. A.N. Parshin, Algebraic curves over function fields I. Math. USSR Izv. 2 (1968) 1145-1170. 9. L. Szpiro, Seminaire sur les pinceaux de courbes de genre au moins deux. Asterisque 86 (1981).

10. E. Viehweg, Vanishing theorems. J. Reine Angew. Math. 335 (1982) 1-8. 11. E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre

spaces. Adv. Stud. Pure Math. 1 (1983) 329-353 North-Holland. 12. E. Viehweg, Vanishing theorems and positivity in algebraic fibre spaces. Proc. Intern. Congr.

Math., Berkeley 1986, 682-687. 13. E. Viehweg, Weak positivity and the stability of certain Hilbert points. Invent. Math. 96 (1989)

639-667. 14. P. Vojta, Mordell's conjecture over function fields. Preprint 1988. 15. A.N. Parshin, Algebraic curves over function fields. Soviet Math. Dokl. 9 (1968) 1419-1422.

Eine Klassenzahlformel fUr singuHire Moduln der Picardschen Modulgruppen

JAN FEUSTEL Akademie der Wissenschaften der DDR, Karl- Weierstraf3-Institut fur Mathematik, Mohrenstraf3e 39, DDR-I086 Berlin

Received 3 December 1988; accepted in revised form 6 February 1990

Es ist altbekannt, dap die elliptische Invariante J(.) den Korper der Modulfunktionen auf der oberen Halbebene bzgl. SI2(Z) erzeugt und auf den Punkten • E H, die imaginar-quadratische Zahlkorper erzeugen (und damit Fixpunkte von Elementen aus Gli(Z) sind), algebraische Werte annimmt. Durch Zuordnung dieser Punkte • mit Q(.) = K zu Idealen in Ordnungen des imaginar-quadratischen Zahlk6rpers K erhalten wir mit der 'Klassengleichung' eine algebraische Gleichung fUr J(.) mit rationalen Koeffizienten. In diesem Zusammenhang sei erinnert, dap J(.) zusammen mit der Weierstrapschen ftFunktion der zugehorigen elliptischen Kurve E t aIle Abelschen Erweiterungen von Q(t) erzeugt ('Kroneckers Jugendtraum').

Hecke verallgemeinerte in seiner Dissertation [3] diese Theorie auf die Wirkung der Hilbertschen Modulgruppe Sliok) (k reeIl-quadratischer Zahlk6rper) auf das Produkt zweier oberer Halbebenen (siehe Kapitel III).

Fur die komplexe Kugel B und die Picardsche Modulgruppe U((2, 1), OK) = r K (wobei K ein imaginar-quadratischer Zahlkorper ist) lagen bis vor kurzer Zeit keine einschlagigen Ergebnisse vor. In seiner Arbeit [4] bestimmte Holzapfel erstmals den Ring der automorphen Formen bzgl. r K und

einiger Untergruppen fUr K = Q(j=3). Der Ring dieser automorphen Formen von r K wird erzeugt durch Formen G;

vom Gewicht i (i = 2, 3, 4) (exakter: der Ring der automorphen Formen von r K

bzgl. eines Twistes mit einem Charakter). Dabei sind die G; die elementarsymmetrischen Funktionen Ll ~jl < ... <j;~4 Xii ... Xi; fUr gewisse automorphe Formen einer Kongruenzuntergruppe von r K (genauer flir

r K(j=3) = {YEr K I dety = 1, y == Idmodj=3}), die sich nach den Ergebnissen von Shiga [10] und dem Autor [1] als Summen von Theta-Nullwerten auf Jt' 3 (obere Siegelsche Halbebene) mit einer automorphen Einlagerung

'7: B ~ Jt' 3 und explizit bekannter Charakteristik darstellen lassen (exakt: Xl = CPt + cP~ + cP~, Xj = Xl - 4cp]-1 fUr 2 ~ j ~ 4, CPj Theta-Nullwert).

88 Jan Feustel

Shiga zeigt nun auf analytischem Weg, da(3 fUr a11e r, fUr die ein

o l Y' = pZ [1 1 -d 0

o l. p "age in K/Q}

-d existiert, so da(3 r isolierter Fixpunkt von I' ist, die Modulfunktionen

b - G~ und b _ G~ 1 - G~ z - G4

von r K algebraische Werte annehmen [11]. Holzapfel komplettiert das Resultat fUr a11e singularen Moduln z, d.h. {z E B 131' E U«2, 1)m K): z isolierter Fixpunkt von y}; der Beweis arbeitet algebraisch-geometrisch und ist 'unkonstruktiv', d.h. liefert keinerlei Aussagen, in welchem Zahlk6rper die Werte der Modulfunktionen liegen.

Das stellt naturlich erst einmal folgendes Programm auf: Zuerst wird eine arithmetische Charakterisierung a11er singularer Moduln fur beliebiges imaginar-quadratisches K gesucht (Kapitel I).

Zweitens so11 eine Zuordnung von gewissen singularen Moduln und Untergruppen von Idealklassengruppen von CM-K6rpern L:::J K mit [L:KJ = 3 gegeben werden (Kapitel III). Die Klassifizierungsaufgabe der singularen Moduln wurde fUr Picardsche Modulgruppen von der analogen aufgabe fUr Hilbertsche Modulgruppen, die in der Klassengleichung auftritt, abgelesen.

Von Interesse sind die automorphen Formen qyf bzw. G i auch dadurch, da(3 sie die Umkehrung von Picards Abbildung ([7J, [8J) vermitteln (siehe [5J):

Fur

und

3

~ = (a l :az :a3)EA sei C~: y3 = X TI (X - a;) i= 1

die zugeh6rige Picard-Kurve (vom Geschlecht 3 und mit K-Multiplikation). Die Periodenmatrix von C bzgl. einer bis auf Monodromie von A fixierten

Basis liefert eine mehrdeutige Abbildung in ,n"3; bzgl. der oben erwahnten automorphen Einlagerung 1'/: B c.,n"3liegt das Bild in B, und die 'Monodromie

Gruppe' is gerade r K(.j=3), d.h. <1>: A ---+ Bjr K(.j=3) ist eindeutig, wobei 1m <1> =

Bjr K(.j=3) \ D· r Kjr K(.j=3) fur D = {(a, 0) II a\ < 1} c B.

KlassenzahlformelJiir singuliire Moduln 89

Die Umkehrung von wird durch die drei Modulformen cp~ vermittelt, und bei geeigneter Normierung der Nullstellen von CIll -l(3) gilt

4 4

C<Il-1(3): y3 = n (X - X;(r)) = X 4 + G2X 2 + G3X + G4 , ;=1

I X;(3) = 0, ;= 1

d.h. die oben genannten G; treten als Koeffizienten auf. 1m zweiten Kapitel wird nun nach Aussagen iiber C(q>~(3):q>~(3):q>~(3)) gefragt,

speziell iiber die Einfachheit der Jacobischen.

o. Definitionen

Sei B = {(Cl' c2)lc;EC, ICl12 + IC212 < 1} C C2 die zwei-dimensionale komplexe Einheitskugel, so wirkt U((2, 1), C) 'gebrochen rational' auf B, d.h. sei C2 in P~ = PC3 eingelagert durch CP(Cl' c2) = (c 1 : C2: 1), so wirkt U((2, 1), C) linear auf P~, und fiir rEB, Y E U((2, 1), C) gilt y' r = cp -1. Y' cp(r). Sei auf C3 bzgl. Einer fixierten Basis die hermitesche Form < , ) durch die Matrix

gegeben, d.h. U((2, 1), C) = U( < , ), C). Sei K imaginar-quadratischer Zahlkorper und K3 in C3 bzgl. dieser Basis eingelagert, so ist o~ ein hermitesches, unimodulares Gitter bzgl. < , ) und U((2, 1), OK) = r K seine Automorphismengruppe (Isometrien), eine arithmetische Untergruppe von U((2, 1), C), die auf B eigentlich diskontinuierlich wirkt.

Fiir z = (Zl' Z2)EB ist z = t(Zl' Z2' 1)EC3 , d.h. cpz = cpz. Sei z = (Zl' Z2)EB algebraisch, d.h. Q(Zl, Z2) algebraischer Zahlkorper und (J

ein Automorphismus eines normalen Zahlkorpers iiber Q(Zl' Z2), so ist zO" = t(z1, z~, 1). Z E B hei~t K-singularer Modul, falls ein Y E U((2, 1), K) = UK existiert, fiir das z isolierter Fixpunkt ist.

I. Arithmetische Charakterisierung singularer Moduln

LEMMA 1. Sei LjK ein CM-Korper, K imaginiir-quadratisch, so existiert ein

vEL mit vii = 1, K(v) = L. Beweis. Sei mEL mit L= K(m), m '# iii (falls m = iii, so gehen wir zu

90 J an Feustel

m' = m + Ie, leEK, Ie i' j(; iiber) und 3rEN:

K (~) = K (;: ~) (da nur endlich viele Zwischenkorper existieren). Wir nehmen an, daj3 aus vEL, vv = 1 folgt K(v) i' L. Sei

m+r Vz =-

iiI+r'

so gilt Vi Vi = 1 (i = 1,2) und K(v 1) = K(vz) = M '* L. Also existieren n1, nz E M mit m = n1 ii1, m + r = nz(iiI + r) und damit m(1 - (ndnz)) = ((n 1/n Z ) - n1)r. Da m If M, ist nz = 1, nz = n1 und damit m = iii, ein Widerspruch. D

THEOREM 1. Sei Z = (Zl' zz)EB, so ist zein K-singuliirer Modul genau dann, wennfiir Lz = K(Zl' Z2) und M z, den Galoisabschluj3 von L iiber K, gilt

(1) [Lz:K] ~ 3 (2) Lz ist CM-Korper (3) Fiir aile ITEGal(Mz/K), ITIL i' Id gilt (z, zo") = O.

Beweis. =>Sei y E UK mit yz = z, z isolierter Fixpunkt von y, so ist yz = 2z, 2 erfUllt das charakteristische Polynom von y und XA: = 1.

Daraus folgt sofort wegen der Einfachheit des Eigenwertes 2 von y (z ist isolierter Fixpunkt von y!) Lz = K(2) und damit [Lz: K] ~ 3.

Sei IT E Gal(M z/ K) wie in 3) gewiihlt, so ist 2" Eigenwert von y mit Eigenvektor z", und damit gilt 2uAu = 1. Die 2u dieser Form erzeugen M z, und es gilt AU = 1/2u = (1/2)" = (A)" Damit kommutiert die komplexe Konjugation mit allen Einlagerungen von L z in C, und es ist L z CM-Korper nach dem Kriterium aus [6] S.6.

Weiterhin ist (z, z") = (yz, yz") = 2AU(z,zU) = 2/2U(z,z"), und da wegen ITIL. i' Id 2 i' 2" ist, gilt (z, z") = O. <= Wir miissen Fallunterscheidungen machen:

(a) Lz = K: Sei y gegeben durch yz = z, ylz 1. = - Id, so ist yK3 = K3 und (be, ye) = (e, e) fUr alle e E K3, also ist y E UK und z isolierter Fixpunkt von y.

(b) [Lz : K] = 2. Damit ist L z normal iiber K und Gal(Lz/K) = (1, IT). Es gilt offensichtlich {z, ZU}.l = z.l n zU.l = Cv, V E K3. Sei y gegeben durch yz = 2z, yz" = 2"zu und yv = v, wobei 2 nach Lemma 1) gewiihlt ist mit 2A = 1, Lz = K(2). Man sieht unmittelbar, daj3 (Lzz E9 LzzU) n K3 = {pz + pUzulp E Lz} und damit y. K3 = K 3, aulJerdem gilt wegen (ye, ye) = (e, e) fUr alle e E C3 y E UK (da Z, ZU und v eine Basis von C3 bilden), und z ist isolierter Fixpunkt von y.

(c) [L : K] = 3. Es ist [M z : K] = 6 oder 3, denn M z ist Zerfiillungskorper eines Polyno~s vom Grade 3 iiber K. Sei ITEGal(MzlK) gegeben mit ordlT = 3 und

KlassenzahlJormel fur singuliire Moduln 91

damit (J\L z oF Id. Sei A E Lz wieder nach Lemma 1) gewahlt, so sei y gegeben durch yz"j = A"j ~ (Jj (j = 0,1,2). Dabei ist K3 = {IJ=op"JZ"J\pELz }, ergo yK3 = K3 und <ye, ye) = <e, c) fur aile e E C3, d.h. y E UK und Z ist isolierter Fixpunkt von y (denn wegen <z"J, z";) = 0 fur i oF j sind die zj eine Basis von C3). 0

PROPOSITION 1: Sei L CM-Korper mit K c L, [L: K] ~ 3, so existiert ein K-singuliirer Modul z mit Lz ~ L.

Beweis. [L: K] = 2: Da L CM-Korper ist, ist L biquadratisch, L = K(fi),

K E Q, K > O. Sei w = ~ + fi + 1 fUr K = Q(~), sei weiterhin A = w/w" sowie cp: L 4 C als Einlagerung so vorgegeben, da~ fi < 0; es ist

ww - w"w" = 4fi oF 0 fUr O"EGal(L/K). Es gilt wegen AW" = A(1 - ~) - Afi = 1 + ~ + fi = w sichtlich A ¢ K, also L = K(A). Man sieht unmittelbar, da~ fur z = (tA, ~A, 1) E C3 gilt <z, z) = AA - 1 < 0, (2, z") = 0, d.h. z ist K-singularer Modul.

[L: K] = 3: Hier wird der Beweis komplizierter, da sich keine explizite Erzeugung von L/K angeben la~t, die 'handhabbar' ist.

Sei {zJ eine K-Basis von L (als Vektorraum), i = 1, 2, 3, so setzen wir H = (z'tl))EM3(L) fUr (JEGal(M/K), ord (J = 3. (M sei wieder der Galoisabschlu~ von L/K).

Sei k = L n R, so setze ich fUr

l-PI 0

pi EkHp ' = 0 pi"

o 0

Es existiert ein p E k mit Tr p = 0, o.B.d.A. sign H p = (2, 1) Wir setzen nun

p* = p( -Nk/Q(p)-l det H detH). Es gilt fur N = tHH;;.lH:

(1) N = (TrL/K(ziZ)P*))EMiK) (2) sign N = sign M p* = (2, 1)

(3) detN = -(Nk/Q(p)/(detHdetH))2 = -1 mod NK/Q(K) (denn

detHdetH = dettHH = det(TrL/Kzizj)EK nR = Q.)

Nun sieht man aber sofort, da~ dieselben Eigenschaften (1)-(3) auch fur N- 1 = H-1Hp.tH- 1 gelten, und nach dem Satz von Landherr (siehe [14], Kap. 10, §1, 1.6: (iv)). gilt nun

f1

N- 1 ~ 1 Gb(K) 0

o ltA, -lJ

92 Jan Feustel

AEGI3(K). Sei nun (Zt, Z2' Z3) A = r(z~, Z2, 1), r, Zit, z2EL, so ist (Z~,Z2)EB K-singuHirer Modul nach Theorem 1). D

II. Singuliire Moduln und Abelsche Mannigfaltigkeiten

Shimura hat in seinen Arbeiten [12J und [13J gezeigt, wie jedem imaginarquadratischen Zahlkorper K und jedem Punkt Z E Beine Abelsche Mannigfaltigkeit Az mit Polarisierung e z und einer mit e z vertraglichen Darstellung ({Jz: K ~ EndoAz = (EndA)®zQ zugeordnet wird. Diese Zuordnung ist eindeutig mod r K .

Dies geschieht auf folgende Weise:

o l mit w.j=l > 0 und (w) = 8K/Q

-1

(Differente von K/Q, wobei diese stets Hauptideal in 3(K) ist) sowie heine komplexe Struktur auf M ®zR (M als Z-Gitter der Dimension 6 aufgefaBt), so daB Tfi hermitesch positiv definit ist. h ist dabei durch seinen (- i)-Eigenraum eindeutig bestimmt, d.h. durch des sen normierte Basis t(Zt' Z2' 1), (Zt' Z2) = zEB (und somit h = hz)' Dann sei (M ®zR)hz der dreidimensionale C-Vektorraum, dessen Struktur von hz herruhrt, und Az = (M ®ZR)hz/M ~ C3/Az • Die Riemannsche Form (und damit die Polarisierung ez) ist durch Trk-/Qexn'(ex, qj E K3 C C3) gegeben, ({Jz durch die gewohnliche skalare Multiplikation in K3.

Man macht sich klar, daB (Az, ez,({Jz) ~ (A~, e~, ({J~) genau dann gilt, wenn ein Y E r K existiert mit yz = Z'.

1m weiteren sei (Az, ez, ({Jz) das zu ZEB gehorige Shimura-Tupel genannt.

THEOREM 2. Sei ZEB K-singuliirer Modul mit [Lz:KJ = 3 und (Az, ez> ((Jz) das zugehOrige Shimura- Tupel.

Dann ist Az einfach, und es gilt Endo Az = Lz. Beweis. Sei y E UK mit yz = Z, yz = AZ, Lz = K(A). Da yz = Z, so kommutiert y

als lineare Abbildung auf C3 mit hz und liefert ergo eine lineare Abbildung auf (M ®zR)hz mit den Eigenwerten A, AU, Au2 fUr O"EGal(Mz/K), ord 0" = 3. Man sieht unmittelbar, daB y E Endo (A.).

Nach Konstruktion hat ({JAex) fur ex E K die Eigenwerte iX, ex, ex auf (M ®z R)hz, wobei die Eigenraume fUr A und iX sowie fUr ex und die direkte Summe derjenigen von AU und Au2 ubereinstimmen, da y e z erhalt.

Also existiert eine Darstellung 'P z: Lz ~ End 0 Az mit 'P z IK = ({Jz und 'P z(A) = Yl A z mit dieser Darstellung 'Pz ist in der Terminologie von Lang [6J also vom CM-Typ {Lz : p, 0", 0"2} (p die komplexe Konjugation in Gal(Mz/Q)). Fur S =

KlassenzahlJormel fur singuliire M oduln 93

{p, (J, (J2} gilt aber {IJ E Gal(M z/Q)ISIJ = S} = Id, und damit (nach Theorem 3.5 aus [6]) ist A z einfach, und es gilt Endo A z = L z •

KOROLLAR 1. Sei L CM-Korper uber dem imaginiir-quadratischen Zahlkorper K mit [L: K] :::; 3, D = {(a, 0) I a < I}, D' = r K· D, so existiert ein z E B\D' mit

Lz ~ L;falls [L: K] = 3, so liegt auf D kein z mit L z ~ L. Beweis. Man macht sich unmittelbar klar, daf3 fiir zED und damit auch fiir

zED' A z in das Produkt zweier Abelscher Mannigfaltigkeiten zerfallt, denn auf Kt(O, 1,0) stimmen fiir zED die skalaren Multiplikationen mit K, die von der natiirlichen Struktur von K3 bzw. von hz herriihren, iiberein, also kann Ct(O, 1,0)/oKt(0, 1,0) von Az abgespalten werden. Nach Theorem 2) kann also fUr einen singularen Modul Z E B mit [L z : K] = 3 Z nicht auf D' liegen.

Falls [L: K] = 2, so betrachten wir Z aus dem Beweis von Proposition 1. Sei Z E 15 = na, 0, b) I a, bE C}, so ist yz" = (yz)" ebenfalls in 15, und yz und yz" spannen 15 auf, d.h. 151- = <yz, yz")1- = Cyt(4, - 3,0). Es ist nun 151- n Ok = ok(O, 1, 0), d.h. 151- n Ok enthalt einen Vektor der Lange 1.

Andererseits ist Cyt(4, -3,0) n Ok = (cr(4, -3,0) n ok) = oKyt(4, -3,0), und dies enthalt keinen Vektor der Lange 1, ein Widerspruch. D

KOROLLAR 2. Sei Z E B, Z singuliirer Modul mit [Lz : K] = 3, so ist Jac(C-l(z») einfach. (Zur Terminologie siehe die Einleitung!).

Beweis. Auf der Abelschen Mannigfaltigkeit Az = Jac(C-l(z») ist eine kanonische Polarisierung @z und eine kanonische Darstellung (Pz von OK in EndAz gegeben (die durch die Wirkung des Automorphismus f der Ordnung 3, f(y) = my, f(x) = x, m primitive 3. Einheitswurzel von C-l(Z) auf die Differentiale erster Art definiert ist).

In seiner Arbeit [la] hat der Autor gezeigt, daf3 (AZ' 0 z, (Pz) und (Az, e z, ({Jz) iibereinstimmen. Die Aussage des Korollars folgt nun unmittelbar aus Theorem 2 D

III. SioguHire Modulo uod Ideal-klasseogruppeo

In diesem Abschnitt solI ein Zusammenhang hergestellt werden zwischen den singularen Moduln Z E B mit L z ~ L fUr ein fixiertes Lund gewisse Idealklassengruppen in L. Der Sinn der folgenden Ergebnisse besteht darin, daf3 sie sozusagen erste Schritte auf dem Weg zur Aufstellung konkreter algebraischer Gleichungen fUr die Werte der speziellen Modulfunktionen <>1' <>2 (siehe Einleitung) darstellen, der sogenannten Klassengleichung.

Bier seien kurz die 'parallelen' Ergebnisse Heckes [3] fUr den Fall Hilbertscher Modulflachen rekapituliert. Sei k ein reell-quadratischer Karper, H x H das Produkt zweier oberer Halbebenen, auf die Gli(oK) =

94 Jan Feustel

{y E GI2(ok) I det y »O} wirkt, s: ((, (') = (- (, - (') Abbildung von H x H auf sich, S = <s, Gli(ok)'

Hecke zeigt in seiner Dissertation, dap ((, (') k-singularer Modul ist, d.h. ein YEMi(Ok) = {YEMi(ok)ldety» O} existiert mit y((, (') = ((, (') genau dann, wenn ein biquadratischer CM-Korper K existiert mit (E K, (' E H Wurzel der Gleichung von iiber k mit in k konjugierten Koeffizienten. Jedem singularen Modul ((, (') sei der ok-Modul M~ = Ok' ( Etl 0k'1 in K zugeordnet, der Ideal bzgl. einer ok-Ordnung r f in Ok ist. Dabei sind M ~ und M ~o gleich genau dann, wenn ein y E Gli(ok) existiert mit y((, (') = ((0, (0'), d.h. y( = (0. Die Ordnung r f mit Fiihrer JE 3(k) wird dabei durch die "Primitivdiskriminante' D~ = POi/k (OK/k die Differente) von ( bestimmt.

Das Polynom

<D(X) = f1 (X - F((, (')) ~modGl;(ok)

~EK,D(~j2a~/k

hat fUr bestimmte Modulfunktionen F bzgl. S (die sich aus speziellen Thetanullwerten bei der automorphen Einlagerung von H x H in die obere Siegelsche Halbebene £2 ergeben) rationale Koeffizienten (Heckes Beweis ist allerdings liickenhaft), F((, (') ist also algebraisch, und der Grad der dadurch erzeugten Korpererweiterung ist ablesbar (da <D(X) als irreduzibel bewiesen wird); fUrJ = 1 ist er z.B. Ni(/l (Clk(l)), da die freien ok-Moduln M c K mit ME 3(K) bis auf die Multiplikation mit einer Konstanten aus K (projektive Aquivalenz) den Idealklassen Ni(/UA) fUr gewisse A E CI(k) entsprechen (A Clk(oi/k) = Clk(l)).

Dies Programm solI sozusagen als Vorbild dienen fUr das Kapitel, das die Grundlagen der oben skizzierten Theorie konstituieren solI, namlich die Zuordnung gewisser singuliirer Moduln z B Idealklassen in CI(L). Der Autor setzt seine Hoffnung darauf, dap diese Zuordnung vielleicht auch zu einem polynomialen Gleichungstyp ahnlich der Klassengleichung fiihren kann, wenn wir als Nullstellen die Werte der Funktionen 6i in diesen gewissen singularen Moduln ansetzen. Von diesem Gleichungstyp miipten wir den Grad kennen und zu beweisen vermogen, dap die Koeffizienten in einem bestimmten algebraischen Zahlkorper liegen.

1m weiteren seien der imaginar-quadratische Zahlkorper K und der CMKorper L mit [L: K] = 3 sowie eine Einlagerung cp: N ~ C fiir den Galoisabschlup N von L iiber K fixiert sowie k = L n R.

DEFINITION 1. Sei M ein orModul in L, so gelte:

Ef = {8EL*18M = M}

E~ = {8Ek* IBM = M}

E~ + = {8 E E~ I 8 » O}

KlassenzahlJormel fur singuliire Moduln 95

DEFINITION 2. Sei GL = {MIM freier dreidimensionaler oK-Modul in L,

versehen mit der hermiteschen Form <m, n) = TrL/Kmii, fUr die ein p E k existiert mit <pep) < 0, <p(p") > 0 fUr alle O"EGal(N/K), p" -.f. p, so da~ p'M = M#}.

THEO REM 3. Es existiert eine surjektive Abbildung 'P: {z E Biz K -singuliirer Modul, Lz = <p(L)}/r K ---+ GLmod L* = PGL mit # 'P- 1(M) = [EIY+: N L/k(Er)].

Beweis. 1. Konstruktion der Abbildung. Sei z = (Z1' Z2)EB K-singuHirer Modul mit Lz = <peL). Nach Theorem 1 gilt

fur H = (z'r'') (Z3 = 1, O"EGal(N/K) wie oben) und

fUr p E k, <p(p) < 0, <p(p"') < 0 fUr alle 0"' E Gal(N / K), p'" -.f. p. Damit gilt <D = H- 1H/fj-1 und wegen <D- 1 = <D auch <D = tfjHp_1H, d.h.

(TrL/K(ziZj/p» EGI3(OK)' Setzen wir M = EBT= 1 0KZi = 'P (z), so gilt also fUr M' = EBt=1 0Kri, ri = zJP «Zi' r)EGI3(OK), d.h. jedes lineare Funktional auf M in OK 1ii~t sich durch ein x E M' mit <p(y) = <y, x) fUr alle y E M darstellen, und damit ist M' = M#, also MEGL.

2. Surjektivitiit

Sei M# = pM fur ein p aus Definition 2), so ist fur M = EBt=1 0KZi und DM = (TrL/KPziz) DM eine hermitesche Matrix aus GI3(OK), sign (DM) = (2, 1)

und nach [0] 2.2. Lemma 4 DM ~ <D und damit auch D;/ ~ <D. . GI3(oK) GI3(oK)

Sei H = (zj(,-l) fur ein O"EGal(N/K) mit ord 0" = 3 und

H, ~ r q>(p) q>(p") 0 , J' l 0 <p(p" )

so ist DM = tfjHpH und ergo H- 1H;ltfj-1 ~ <D, d.h. es _ _ GI 3(oK)

AEGI3(OK) mit HA<DtAtH = H;1 = Hp-l. Nach Theorem 1) HA ( 'a('-l) d M' ffi 3 'I ( '/ ' '/ ') B = Zj un = Wi=1 0KZi a so Z = Z1 Z3' Z2 Z3 E

Modul mit 'P(z) == M mod L*, denn M' = M.

3. 'P- 1(M) = [EIY+ :NL/k(Er)]

existiert .ein

ist nun fUr K -singularer

Man sieht aus dem Beweis von 2. unmittelbar, da~ HA (d.h. auch z) modulo

96 J an Feustel

Rechtsmultiplikation mit <r K, E~· Id) (und damit auch z mod r K) eineindeutig durch M mod L* und p mod N L/k(E~} bestimmt ist. D

DEFINITION 3. Sei M ein endlich dimensionaler (torsionsfreier) Modul uber einem Dedekindschen Ring R, so ist M = ® rximi, mi E M ® R QR, rxi E 3(R}, wobei QR der Quotientenk6rper von R sei. Wir setzen CIR(M} = CI(IIrxi}ECI(R).

LEMMA 2. Seien M 1, M 2 zwei oK-Gitter gleicher Dimension mit M 1 => M 2 (d.h. torsionsfreie orModuln) und M tiM 2 ~ oKI¢ fur ein Ideal cP E 3(K}, so ist Clo/M 2) CI(;~/(M 1} = CI(cP) E CI(K).

Beweis. Reiner [9] Theorem 1.2. (v) S. 162.

DEFINITION 4. Sei 3L = {zEBlz K-singuliirer Modul, Lz = cp(L), EBr=1 0KZi ® 0KE 3(L)}.

KOROLLAR 3. #3JrK = [0:+: NL/k(O!)] # {rx/rxECI(L), NL/k(rx) Clk(oL/Kn k) = Clk(1), 0L/K E 3(L) die Differente, N L/K(rx)cP = CIK(1) fur cP = CloK(oL) und es existiert ein p E k mit cp(p} < 0, cp(pIT) > 0 fur aile 0" E Gal(N / K), pIT #- p und ein rx' E 3(L), CIL(rx') = rx, so daf3 N L/k(rx'}(OL/K n k) = (p)}.

Beweis. Nach Theorem 3) ist nur zu zeigen: Sei rx' E 3(L), so gilt

(1) rx'# = {bELl TrL/KbaEoK\faErx'} = (iX·OL/K}-1. (2) rx' freier oK-Modul genau dann, wenn NL/K(CIL(rx'»cP Haupt ideal ist (3) (OL/K n k) ®O.OL = 0L/K

(1) Ergibt sich sofort aus der Definition der Differente. (2) Folgt sofort aus Lemma 2). (3) Zuerst macht man sich klar, da~ OL/K = 0L/K, denn da L CMK6rper ist, ist mit p E 3(K) auch Jt- in L/K verzweigt. Es genugt also zu zeigen, da~ fur cPE3(L) prim, cP/OL/K, cP = <P, (cP n k) ®OkOL #- cP (d.h. cP total verzweigt in L/Q) der Exponent von cP in OL/K gerade ist. Falls N(cP) #- 3, folgt dies sofort aus dem Exponentensatz ([2] S.431) wegen [L:K] = 3. Falls N(cP) = 3, d.h. 3 total verzweigt in L/Q, so ist nach dem Schachtelungssatz fUr die Differente ([2] S.430) OL/KOK/Q = OL/kOk/Q' und da der Exponent von cP in 0L/k nach dem Exponentensatz 1 ist, in OK/Q 3 und in Ok/q gerade, ist der Exponent von cP in OL/K auch gerade. D

THEOREM 4. Es gilt # 3dr K = (h(L)/h(K)h(k» [ot: N L/k(0!)]/8. Beweis. Nach Korollar 3 ist nur Folgendes zu beweisen: (a) Fur aIle v E CI(k), WE CI(K) existiert ein Z E CI(L) mit N L/K(Z) = w,

NL/k(Z) = v. (b) Fur jedes Tupel (i1, i2, i3) mit ij E {O, 1} existiert ein rx E 3(L) mit

N L/k(rx) = (p), cp(pITU-1»( -1}~ > 0 fur ein fixiertes 0" E Gal(N/K), ord 0" = 3.

Zu (a) ist erst einmal zu beweisen, da~ N L/K und N L/K auf CI(L} surjektive Abbildungen sind. Fur N L/k folgt dies sofort daraus, da~ Lund k (der

KlassenzahlJormel fur singuliire M oduln 97

Hilbertsche Klassenkorper von k) linear disjunkt sind. Fur N L/K ist zu zeigen, daJ3 L nicht im Hilbertschen Klassenkorper K von K liegt.

Sei L c K, so ist L/K abelsch und unverzweigt. Nach dem Schachtelungssatz fUr Diskrimanten gilt N K/Q(E0L/K)E0k/Q = Nk/Q(E0L/k)E0f/Q. Da k/Q abelsch mit [k: Q] = 3, so existiert ein p, das in k/Q verzweigt ist. Nach dem Exponentensatz teilt p2 E0k/Q; falls p =f. 2, teilt aber nach demselben Satz p hOchstens in der Potenz 1 die Diskriminante E0K/Q; also muJ3 p N K/Q(E0L/K) teilen, d.h. L/K ist nicht unverzweigt. Falls p = 2, so teilt p mit dem Exponenten 2 die Diskriminante E0k/Q, aber mit einem Exponenten ~2 die Diskriminante E0K/Q, also teilt 2 auch N k/Q(E0L/k), 2 ist voll verzweigt in L/Q, also auch in L/K. In jedem Fall folgt L ¢ K, also sind Lund K linear disjunkt, und N L/K ist surjektiv.

Sei nun rwENL/k(w), rvENL/Nv), so gilt fUr z = r!ri/(wvNL/K(rJNL/k(rw)2) (wobei die Ideale in k bzw. K durch Tensorieren mit 0L als Ideale in L aufgefaJ3t werden):

NL/K(Z) = w4NL/K(rv)3w-3NL/K(rV)-3 = w,

NL/k(z) = v3NL/k(rw)4v-2NL/k(rW)-4 = v.

Am weitaus kompliziertesten gestaltet sich der Beweis von (b).

PROPOSITION 2. Sei a E Gal(N /K), ord a = 3 gegeben. Falls ein rx E k existiert, so daf3 rx » ° und rx prim zu E0L/K ist und fur Ho = {(rx) I rx E k, rx » 0, rx == 1 mod E0L /d gilt (rx)¢NL/k(3f'fiL/k(L))Ho, so exis~~;,t zuj~dem Thpel {iJ,j = 1,2,3, ijE{O, 1} ein 13 E k mit (13) EN L/k(3(L)) und cp(f3l1 )( -1)lj )0, wobei 3f'fiL/k (L) = {rx E 3(L) I rx prim zu E0 L,d.

Beweis. Nach [2a] S. 22-24 gilt dur 3f'fiL/k(L) und 3f'fiL/k(k) (analog definiert) [3f'fiL/.(k): N L/k(3f'fiL/k(L))Ho] = 2.

Falls ein solches rx wie 0 ben existiert, so ist fUr HI = {( rx) I rx E k, rx prim zu E0 L/k' rx»O} und H2 ={(rx)lrxEk, rx prim zu E0L/k} [NL/k(3f'fiL/.(L))H 1:

N L/k (3f'fiL/. (L))Ho] = 2 und damit [N L/k(3f'fi)L)) H2: N L/k(3!'fl',I.(L))H1 ] = 1. Da H2 sichtlich zujedem Tupel {i j } ein 13' enthalt mit cp(f3'aU- 1»)(-1)1 > 0,

gilt die obige Behauptung. D

LEMMA 3. Seien L, K, k wie oben gegeben, so gilt fur eine Primzahl p: Falls p in L/Q nicht total verzweigt ist, so sind die p-Komponenten von E0L/k und E0K/Q identisch. Falls p in L/Q total verzweigt ist, It Verzweigungsideal von p in 3(k), p teilt E0 K/Q mit dem Exponenten n, so teilt It die Diskriminante E0 L/k mit dem Exponenten 3n - 2.

Beweis. Wir benutzen Schachtelungs- und Exponentensatz fUr Diskriminanten und bemerken nur, daJ3 ein It E 3(k), It Zerlegungsideal von p, It verzweigt in L/k mit demselben Exponenten in E0L /k erscheint wie das Ideal It" 1t,,2. D

98 Jan Feustel

DEFINITION 5. Sei fUr aEZq&KIQ = {bEZlb prim zu ~K/Q}

und

( _1)(a-l)/2

( -1)«(a -1)/2) + «( _1)(a-l)/2a-1)j4

+(~K/Q/8)-1)(a-1)/4)

falls 2 f ~ K/Q

falls ~ K/Q == 4 mod 8

falls ~ K/Q == 0 mod 8

ANMERKUNG. Da!da) auf Z", multiplikativ ist, kann ;;LJKfQ

fortgesetzt werden. Man sieht sofort, da~ !K(a) auf Qq&KIQ mod ~ K/Q definiert ist.

LEMMA 4. Falls vEk, v == 1 mod ~L/b so ist!K(Nk/Q(v)) = 1. Falls wEL, W prim

zu ~L/k' so iSt!K(NL/Q(w)) = 1.

Beweis. Wir benutzen Lemma 3, nach dem gilt ~~/k = ~L/k fUr alle

(JEGal(N/Q) (in 3(N)) und somit Nk/Q(v) == 1 mod~L/k.

Ebenfalls aus Lemma 3 folgt (wegen 3n - 2 > n - 1 fUr n E N) damit

Nk/Q(v) == 1 mod~K/Q und somit nach obiger Anmerkung!K(Nk/Q(v)) = 1.

Sei wEL, w' = NL/K(W), W prim zu ~K/Q und o.B.d.A. WEaL' so ist

N L/Q(W) = N K/Q(W'). Fur aile p 1 ~K/Q' P i= 2 folgt sofort (N K/Q(W')jp) = 1. Falls

~ K/Q == 4 mod 8, so ist N K/Q( w') = a2 - b2(~ K/Q/4), a, bE Z, (a, b) prim zu 2, 21 ab,

-(~K/Q/4) == 1 mod 4 und damit GK(N K/Q(W')) = 1. Falls ~K/Q == 0 mod 8, so ist

wie oben-

, {1, 3} I~K/QI 1 NK/Q(w) mod 8E {1, 7} falls -8- == 3 mod 4.

Wir erhalten wieder GK(N K/Q(W')) = 1. o

PROPOSITION 3. Fiir ein vEk, v prim zu ~L/k gilt: Falls (V) E NL/k(3q&L/,(L))Ho,

so iSt!K(Nk/Q(v)) = 1.

Beweis. Nach Lemma 4) genugt es zu zeigen: Falls (v) E N L/k(3q&Llk(L)), so ist

!K(N k/Q(V)) = 1. Nach dem Dichtigkeitssatz von Tschebotarjow existiert in jeder

Idealklasse aus Cl(L) ein Zerlegungsideal ¢ einer total zerlegten Primzahl. Falls

v E N L/k(3q&Llk(L)), so existiert also ein solches ¢ E 3(L) mit N L/k(¢) = (vN L/k(V')),

v' EL prim zu ~L/k und damit nach Lemma 4

K lassenzahlJormel fur singuliire M oduln 99

Aus der totalen Zeriegtheit von q in LIQ folgt nun

fUr

1 = (0JK /Q ) = (_1)(q-l)/2 TI (E)(q) q pl~~Q q

p#2

=(_1)«q-l)/2)+ L (p-1)(q-1)/4) f1 (1)s(q) pl~KJQ pl~K/Q p p#2 p#2

{(2Ia) = (_1)«(-1)"-1l/2a - 1)/4) falls 810JK/Q

sea) = 1 anderenfalls

Es gilt aber

L (p-1)(q-1) == (I0J K /QI/(8, I0JK /QI)-l)(q - 1) mod 2 4 4 pl~L/K

p#2

und damit

{1 falls 0JL/K == 4 mod 8

(-1) L (p-1)(q-1)/4= (_1)(q-l)/2 ill 2}"-;' pl~L/K a s I :::LJL/K' p#2

Damit ist

D

Nach dem chinesischen Restsatz existiert ein a E Z~L/K' a > Omit fKCa) = fK(a 3 ) = fK(Nk/Q(a)) = -l.

Fur dieses a gilt nun nach Proposition 3

das heiJ3t nach Proposition 2 eXlstIert ein [3 E k mit ([3) E N L/d3(L)) und <p([3(J(j-l))(_l)~ > ° fUr ein beliebiges Tupel {i j }, ijE{O, 1}.

Damit ist (b) bewiesen. Es folgt also unmittelbar Theorem 4). 0

100 Jan Feustel

Literaturverzeichnis

[0] Feustel, J.-M., Holzapfel, R.-P.: Symmetry points and Chern invariants of Picard modular surfaces, Math. Nachr. 111 (1983), S. 7-40.

[1] Feustel, J.-M.: Representation of Picard modular forms by theta constants, Rev. Roumaine Math. Pures Appl. 33 (1988), S.275-281.

[la] Feustel, J.-M.: Arithmetik und Geometrie Picardscher Modulflachen, Dissertation B, Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut fiir Mathematik (1987).

[2] Hasse, H.: Zahlentheorie,Akademie Verlag, Berlin (1963). [2a] Hasse, H.: Bericht fiber neuere Untersuchungen und Probleme aus der Theorie der algebraischen

Zahlkiirper, Teil I, B. G. Teubner, Leipzig/Berlin (1930). [3] Heeke, E.: Zur Theorie der Modulfunktionen von 2 Variablen und ihre Anwendung auf die

Zahlentheorie, Math. Annalen 71, (1912), S. 1-37. [4] Holzapfel, R.-P.: Geometry and arithmetic around Euler partial differential equations, Kluwer

Academic Publishers, Dordrecht, Holland, (1986). [5] Holzapfel, R.-P.: An arithmetic uniformization for arithmetic points of the plane by singular

moduli, J. Ramanujan Math. Soc. 3(1), (1988), S. 35-62. [6] Lang, S.: Complex multiplication, New York, Berlin, Heidelberg, Tokyo, Springer (1983). [7] Picard, E.: Sur des fonctions de deux variables independentes analogues aux fonctions

modulaires, Acta Mathematica 2 (1983), S. 114-135. [8] Picard, E.: Sur les formes quadratiques rerneires indefinies et sur les fonctions hyper

fuchsiennes, Acta Mathematica 5 (1884), S. 121-182. [9] Reiner, I.: A survey of integral representation theory, Bulletin of the American Mathematical

Society, Vol. 76, No.2, (1970), S. 159-227. [10] Shiga, H.: On the representation of Picard modular function by 0 constants I-II, Publ. RIMS,

Kyoto Univ., 24 (1988), S. 311-360. [11] Shiga, H.: On the construction of algebraic numbers as special values of the Picard modular

function, Preprint, Chiba University. [12] Shimura, G.: On analytic families of polarized Abelian varieties and automorphic functions,

Annals of Mathematic, 78 (1963) No.1, S. 149-192. [13] Shimura, G.: Arithmetic of unitary groups, Annals of Mathematic, 79 (1964), S. 369-409. [14] Scharlau, W.: Quadratic and hermitian forms, Springer, Berlin, Heidelberg, New York, Tokyo,

(1985).


Chow categories

J. FRANKE Universitiit lena, DDR-6900 lena, Universitiitshochhaus 17. OG, and Karl- Weierstrap-Institut fur Mathematik, DDR-I086 Berlin, Mohrenstrape 39, Germany

Received 3 December 1988; accepted in revised form 15 March 1990

Introduction

This paper arose from an attempt to solve some questions which were posed at the seminar of A. N. Parchin when Deligne's program ([D]) was reviewed. These problems are related to hypothetical functorial and metrical versions of the Riemann-Roch-Hirzebruch theorem. One of the problems posed by Deligne is, for instance, the following construction:

Let a proper morphism of schemes X --+ S of relative dimension n and a polynomial P(ci(E) of absolute degree n + 1 (where deg(ci) = i) in the Chern classes of vector bundles E1, ••• , Ek be given. Construct a functor which to the vector bundles E j on X associates a line bundle on S

(1)

which is an 'incarnation' of Ixls P(ci(Ej » E CH1(S). The functor (1) should be equipped with some natural transformations which correspond to well-known equalities between Chern classes (cf. [D, 2.1]). Further steps in Deligne's program. are to equip the line bundles (1) with metrics, to prove a functorial version of the Riemann-Roch-Hirzebruch formula which provides an isomorphism between the determinant det(Rp*(F» of the cohomology of a vector bundle F and a certain line bundle of type (1); and (finally) to compare the metric on the right side of the Riemann-Roch isomorphism and the Quillen metric on the determinant of the cohomology.

In [D], Deligne dealt with the case n = 1. He considered (1) as a closed expression. It is our strategy to give 'live' to each ingredient of (1). If one tries to do so, the ith Chern functor ci(E) should take values in the ith Chow category CHi(X). It is the aim of these notes to explain what we believe to be the best definition of the Chow category, and to define some of the basic functors between Chow categories.

Our proposal for (1) is the following expression:

(2)

102 J. Franke

where p: X --+ S is the morphism we have in consideration and p* is a pushforward functor which will be introduced in §3. The most complicated ingredient of (2) is the Chern functor ci ( .). Its construction has been outlined in [Frl], and details are contained in the notes [Fr2] which I distributed in June 1988. We shall publish our results on Chern functors together with more considerations about the Riemann-Roch problem in a continuation of this paper.

One of the advantages of the approach to Deligne's program via Chow categories is that it allows us to state the functorial Riemann-Roch theorem in Grothendieck's form. Hence it should be possible to copy the standard prooffor Riemann-Roch theorems. Our proposal for the Riemann-Roch-Grothendieck isomorphism is a canonical isomorphism

(3)

for any local complete intersection p with relative cotangential complex n~/s. The isomorphism (3) should be characterized by certain axiomatic properties.

To explain the ingredients of (3) further I mention that the Chern functor will not be a mere object of the Chow category but an intersection product

Therefore no regularity assumptions for X are necessary to define both sides of (3) as a functor with values in the quotient category CU'(S) ® Q. The remaining ingredient of (3) is the Gysin functor pl. This is our first example of a non-trivial functor between Chow categories, and the most considerations of this paper are directly or indirectly devoted to its construction.

After recalling some basic properties of Quillen's spectral sequence in §1, we define the Chow categories and some of the basic functors in §2 and §3. §4 contains the construction of the Gysin functor. In §5 we use this Gysin functor to outline the construction of a functorial intersection product. As an example which lies outside Deligne's program, we apply the intersection product functor to construct a biextension between certain groups of algebraic cycles. This biextension generalizes the well-known auto duality of the Jacobian, and should be equivalent to a construction of Bloch.

I started my research on Chow categories while I was a postgraduate student in Moscow under the guidance of I. M. Gel'fand. I am much obliged to A. A. Beilinson, Ju.1. Manin, A. N. Parchin, V. V. Schechtman, and the participants of Parchin's seminar for many helpful discussions. In particular, Beilinson and Manin pointed out that the Chow category should provide an alternative construction of Bloch's biextension. Their proposal is carried out, at least partially, in §5.5.

Chow categories 103

Notations

Throughout this paper, schemes are assumed to be Noetherian, separated over Spec(d') and universally catenary. Our notations of K-theory are as usual Ki(X) = Ki(P(X)) and K!i(X) = Ki(M(X)), where P(X) and M(X) are the exact categories of vector bundles and of coherent @x-modules on X.

Products in K-theory are defined by Waldhausen's pairing BQA /\ BQB ...... BQQC (cf. [W], [Gr]). The relation between the product and the boundary of the localization sequence is given by formula [Gr, Corollary (2.6)]. In particular, the boundary of the K-theoretic product of two invertible functions differs by a sign from the tame symbol.

1. The sheaves Gk

For a scheme X, denote by X k the set of points of codimension k (i.e., of points x with dim (@x.x) = k) and by X(k) the set of points of X, equipped with the following topology. U is open in X(k) iff it is Zariski-open and for every x E Xl with 1 < k, we have either x E U or x 1= a. In particular, X(l) = X Zar' For a point x E X, k(x) is the residue field of x.

1.1. Definition of Gk

The descending filtration of M(X) by M p(X) = {coherent sheaves on X with support in codimension ~ p} defines a spectral sequence (cf. [Q,(S.S)] or [G, p. 269]) with initial term

Ef·q(X) = K_p_q(Mp/Mp+l) = U K_p_q(k(x)) (1) XEXp

converging to K'_p_q(X). In particular, E~·q(X) is the homology in the middle term of

U K1-P_q(k(x)) ...... U K_p_q(k(x)) ...... U K_P_q_1(k(x)). (2) XEXp _ 1 XEXp XEXp + 1

We are particularly interested in the groups Zk(X)=E~·-k(X), CHk(X)= E~·-k(X), and Gk(X) = E~-l.-k(X). By (2), they can be defined elementarily, using only Ko, K 1, and K z of fields.

The Ef·q are presheaves on XZar ' Furthermore, one checks easily that the

104 J. Franke

restriction of Gk to X(k) is a sheaf. By (2), there is an exact sequence for U open in

X(k)

where Gk(X - U)O = image(Gk(X) --+ Gk(X - U)) and Zk(X, X - U) =

{XEZk(X)supp(x) c X - U}. The arrow Gk(X) --+ ker(o) has a natural splitting.

1.2. M ayer- Vietoris and localization sequences

If X is the union of its open subsets U and V, we have

and hence

--+E~-l,q(U II V) --+ E~,q(X) --+ E~,q(U) EB E~,q(V)

--+E~,q(U II V) --+ E~+l,q(X)--+. (4)

Let Z c X be closed. We call Z of pure codimension d if X k II Z = Zk-d for k E 7L. Then the exact sequence

gives rise to

--+E~-l,q(X - Z) --+ E~-d,q+d(Z) --+ E~,q(X) --+ E~,q(X - Z)

--+E~+l-d,q+d(Z)--+ ... (5)

1.3. Flat pull-back

If f: Y --+ X is a flat morphism, it defines an exact functor f*: M(X) --+ M(¥) which maps Mk(X) into Mk(Y)' Consequently, we have a homomorphism f*: Ek·q(X) --+ Ek,q(¥) which commutes with the differentials dk, and hence preserves (3), (4) and (5).

1.4. Proper pushlorward

Let f: X --+ Y be a morphism of finite type. We call f of constant relative

dimension dE 7L if for every x E Xl such that dim(f(x)) = dim(x) we have f(x) E YI-d' The proof of the following lemma is straightforward:

Chow categories 105

LEMMA. Let

X' gx) X

Y' ---=g,---+) Y

be a Cartesian diagram in which f is of constant relative dimension d. In each of the following cases, f' is also of constant relative dimension d:

. (i) If g is flat. (ii) If g and gx are l.c.i. (local complete intersections) and for every x E X',

digx) = d!'(x)(g), where dAg) is the relative dimension of the lei-morphism gat x (cf. [FL, p. 89] or [SGA6, VIII.1.9.]).

Proof Since the question is local, we may assume in (i) that X is a closed subscheme of AY. Then f is of relative dimension d if and only if X is of codimension n - din Ay, and this condition remains valid after flat base change. By (i), (ii) is reduced to the case of a regular closed immersion f in which it is trivial. 0

Now we assume that f: X -+ Y is a proper morphism of constant relative dimension d. Then we have exact functors

(6)

defining

(7)

The following theorem is similar to results of Gillet and Schechtman:

THEOREM. (i) The homomorphism (7) commutes with the differential d l of the Quillen spectral sequence. Hence it dejinesf*: E~·q(X) -+ E~-d.q+d(-y)

(ii) The homomorphism f* on the E2-terms is compatible with the localization sequence (3), i.e., if U is open in X(k) and V = Y - f(X - U) then we have a commutative diagram

106 J. Franke

Proof of (i). It is possible to copy the proof in [G, 7.22]. It should also be possible to apply the results of [GN].

Proof of (ii). This follows from (1) and the definition of (3).

1.5. Specialization

This is a modification of [F, Remark 2.3.], cf. also [G, 8.6.]. Let Dc X be a regular embedding of codimension 1, and assume that f is a section of @x in some Zariski-neighbourhood U of D generating the sheaf of ideals defining DcU.

The existence off is a serious restriction to the embedding D c X, for instance it implies the triviality of the conormal bundle of the immersion, which means that we are in the situation described in [F, Remark 2.3.].

We define homomorphisms

as follows. The tensor product P(U - D) x M p(U - D) -+ M p(U - D) defines

where [f] is the class off· in K 1(U - D). Let

M~ = {coherent @u-modules F with codu(supp(F)) ~ p

and codD(D n supp(F)) ~ p}.

(9)

(10)

Then Mp(U - D)/Mp+k(U - D) = (M~/M~+k)/(Mp(D)jMp+k(D)), consequently we have

(11)

If k = 2, a is the boundary homomorphism in (5). We define SPg by the composition of

(12)

On the line p + q = 0, sPf is independent off, and we obtain the homomorphism i* described in [F, Remark 2.3.]. The composition

Chow categories 107

(13)

is also independent of f

1.6. Compatibilities

Let

X' _-,9,-' ---*) X

k f j y' ) Y

9

be a Cartesian square with 9 flat and! proper of constant relative dimension d. Then we have the base change identity

g*!* = !~g'* in Hom(E!:,q(X), Er-d,q+d(y')), k E {I, 2}. (14)

For the diagram of functors

commutes up to a natural transformation. Consider a fibre square

r~· c:

r: D c: X

in which D c: X and D' c: X' are regular embeddings of codimension 1. Let! be the same as in 1.5. If p is proper of constant relative dimension d, the lemma in 1.4 implies that Pv is of the same relative dimension d. We have

kE{I,2}. (15)

108 J. Franke

If in the same fibre square p is flat, we have

(16)

(15) is a consequence of the commutative diagram

The commmutativity of (A) follows from the fact that the diagram of bilinear functors

PxM p*(P)®M

P(X -D) x (Mp/Mp+1)(X' - D') (Mp/Mp+1)(X' - D')

.. ]Id.'. ,"M~P"M j '. P(X - D) x (Mp-d/Mp+l-d)(X - D) ) (Mp-d/Mp+l-d)(X - D)

commutes up to a natural transformation, and (B) commutes because p* maps sheaves on X' whose support is of codimension p and meets D' in codimension p to sheaves on X with the similar property, and hence defines a morphism between the quasi-fibrations used to define (11).

The proof of (16) is similar. If we have a commutative diagram

D )X

\} z

in which g and h are flat and D, X, and f satisfy the assumptions of 1.5, then

(17)

This can easily be reduced to the following general situation:

LEMMA. Let the following objects be given: (i) A sequence d ~ fJI ~ qj of exact functors between exact categories such that

ab ~ 0 and BQd -+ BQfJI -+ BQqj is a fibration up to homotopy.

Chow categories 109

(ii) An exact functor P --+ pi between exact categories, an object Y of P and an

endomorphism f: Y --+ Y in P which becomes an isomorphism in P'. (iii) An exact category!!fl, biexactfunctors <8>: P x!!fl--+ fA and pi x!!fl--+ CC such

that

commutes up to a natural transformation, and an exact functor G:!!fl --+ d such that there is a functorial exact sequence in fA:

0--+ Y<8> A J® IdA) Y<8> A __ 11----+) G(A)--+ 0 (A E Ob(!!fl».

Let [f] E K 1 (P') be the class off viewed as an automorphism in P'. Then

(18)

where a: K;+l(CC) --+ K;(d) is the boundary defined by the fibration (i), u: K 1(P I

) x K;(!!fl) --+ K;+ l(CC) is the pairing defined by <8>, and

G*: K;(!!fl) --+ K;(d) is defined by G. To derive (17)from (18), we put d = M p (D)/Mp +1(D), fA = M~/M~+l (cf. 1.5.)

CC = M p (X)/Mp +1(X),!!fl = M p (Z)/Mp +1(Z),P = P(X),P' = P(X - D), Y= (r)x, andf=multiplication by f Furthermore we put G = g* and define <8>: P x!!fl--+ fA by (M, E) --+ h*(M) <8> E.

Proof of Lemma. The class [fJ is given by the homotopy class of the map S2 --+ IBPQ'I defined by the diagram

o y~

Y J) Y

~/.; . (19)

o

Here we use the usual notations for morphisms in QPI , and Oy = Y --+ 0, oy = 0 --+ Y. To get S2 from the diagram (19), identify its left and right boundary.

Consequently, the homotopy class ~2IBQ!!flI-+ IBQQCCI obtained by applying Waldhausen's pairing to [fJ can be defined by the geometric realization of the

110 J. Franke

map which associates to A E!!) the following diagram of vertical morphisms in

QQ~

° VO~~ ""vO~®A Y® A V(f®IdA),>y® A

v ~ ~®A Or'®A~ /vo7 . 0

(20)

(the left superscript v denotes vertical morphisms in QQ) and to a morphism in Q!!) the similar diagram of bimorphisms in QQ~. The diagram (20) has an obvious lifting to QQB6':

(21)

(Tim(f)] = vertical morphism from ° to Y® A defined by the subobject im(f® IdA) C Y® A). The diagram ofbimorphisms corresponding to (20) has a lifting to QQrJI which is similar to (21). Our task is now to compute the difference between the two homotopy classes ~DBQ!!)I-+ IBQQrJll defined by the arrows on the left and the right boundary of (21). Because V[im(f)] is equal to the composition

the map ~ IBQ!!)I -+ IBQQrJll defined by the vertical morphisms on the right boundary of (21) and the related bimorphisms is homotopic to the map defined by the diagram

~G(A)

~r~ 1"' (22) Y®A

and the similar diagram of bimorphisms. By the commutative diagram

G(A)

r 1 ~VOh(A) VIt" ~

_______ 0,

yiC\ A~o-!! 101 Y® A

(22) is homotopic to

VoP(~G(A)~ohlAl / ~ 0A 0. v Y@A v !

o! Y®A~

Chow categories 111

The bottom half of this diagram coincides with the left boundary of (21). By the

very definition of the boundary operator a, we conclude that o([f]u .) is the

homotopy class of the map L IBQ~I - IBQQdl given by the diagram

and the similar diagram of bimorphisms. This is, however, the composition of

G*: IBQ~I -IBQdl with the map L IBQdl -IBQQdl defined in [W, p. 197].

The proof of the lemma and of (17) is complete.

1.7. A relation between two specializations

Let Di C X be regular immersions of codimension one such that the sheaf of

ideals defining Di is in some neighbourhood of Di trivialized by J;, iE {1; 2}. We

suppose also that Dl (l D2 C X is a regular immersion of codimension two, i.e.,

thatJ;IDj is not a zero-divisor if i "# j. Then we have the identity

(23)

This follows from the commutative diagram

112 J. Franke

1.8. A relation between specialization and restriction to closed subschemes

Let X be a regular scheme satisfying Gersten's conjecture. Then we have an

isomorphism

(24)

defined by the well-known acyclic resolvent

(25)

of the sheaf :K q associated to U --+ Kq(U). In (25), tf~·q is the Zariski-sheaf

U --+ E~·q(U).

If i: Z c X is a closed regular subscheme of X, the composition

(26)

defines a homomorphism

(27)

PROPOSITION. Let Z and D be closed regular subschemes of a regular scheme

X satisfying Gersten's conjecture. We assume that Z n D is regular and of

codimension one in Z and that (X, D,J) satisfies the assumptions of 1.5. Then we

have

(28)

where i and iD are the inclusions Z c X and Z n D cD.

Proof. This follows from the commutative diagram

E~.q(X - D) [fJu·) E~.q-l(X _ D) a ) E~·q(D)

1 B (A) 1 B (B) 1 B

Chow categories 113

In this diagram, the arrow (a) and its symmetric counterpart (a') are defined by the purity isomorphism

p _{Oif P >1 .tt D(X, :f{' q,X) -.u' 'f - 1

Jt q -l,olP- , (29)

where .ttl) is the derived functor of the sheaf of sections with support in D. The only non-zero isomorphism in (29) is normalized by the commutativity of

The commutativity of the squares (B) and (B') is therefore obvious. For the commutativity of (C), we denote by P the category of sheaves F on U with the property

TorfX({Dz, F) = 0 if i > O.

There is an obvious diagram

BQP(U n D) BQP BQP(U - D)

111 (31)

BQM(D n Z n U)-----+) BQM(Z n U)--) BQM«Z - D) n U)

in which the rows are fibrations up to homotopy. Since the boundary homomorphism of the top row coincides with the left vertical arrow in (30), (31) implies the commutativity of

Kq(U - D) ---+) Kq(Z n U - D)

l' l' K q- 1(U n D)---+) K q_1(Z nUn D)

for every Zariski-open U in X. By (30), this proves the commutativity of (C). The

114 J. Franke

commutativity of (A) and its counterpart (A') follows from the diagram of resolvents

:f{' q.X ~ B~·-q ) 81·- q --~~ ...

1 [nu 1 [flu 1 [nu

:f{' q+ l.X ) 8~·-q-l ) 81·- q- l

---~ ...

in which all the squares except the first one are anti-commutative. The commutativity of the other squares in the diagram is obvious. The proof of (28) is complete.

1.9. Homotopy invariance

Let p: E --+ X be the projection of a vector bundle to its base. Then p*: E~·q(X) --+ E~·q(E) is an isomorphism.

Proof By the localization sequence and the five lemma, we may reduce the assertion to the case that all connected components of X are irreducible and hence eguidimensional. In this case the assertion follows from [G, Theorem 8.3].

2. Definition of the Chow category by means of cycles

On a normal locally factorial scheme X, every Weil divisor BEE}·-l(X) defines a line bundle O(D), and isomorphisms between O(D) and O(D') correspond to rational functions f with div(f) = D' - D. We try to generalize this to higher codimension.

Let CH~(X) be the following category. Objects of CH~ are cycles z E E1 -i(X). Homomorphisms between z and z' are elements of the factor set

The composition Hom(z, z') x Hom(z', z") --+ Hom(z, z") sends the equivalence classes off and f' in (1) to the class off + f'. It is easy to see that CH~ is a Picard category in the sense of [D, §4.1] if the sum is given by

zEElz'=z+z' (2)

class off EEl class off' = class off + 1'.

The commutativity and associativity law are simply identities between functors.

Chow categories 115

To admit also non-invertible arrows, we mention that E\:-i (being the free group generated by Xi) carries a natural ordering ~, and define the extended Chow category CH~(X)e which has the same objects as CH~(X) and

as morphisms between z and z'. The composition of arrows is defined by adding f and f'. (2) defines a sum in CH~(X)e'

If X is normal and locally factorial, then CH;(X) is (via D --+ O(D)) equivalent to the category of line bundles and isomorphisms, while CH;(X)e is equivalent to the category of line bundles and inclusions of line bundles on X. The sum EB corresponds to the tensor product of line bundles.

By the results of §1, there are flat pull-back, proper push-forward, and specialization functors between the categories CH~. If, for instance, p: Y --+ X is flat, the functor p* sends the object z to p*(z) and the class off in (1) to the class of p*(f). Using these functors, we could try to establish a functorial analogue of the usual intersection theory. We shall, however, prefer another definition of the Chow category which defines CHi(X) as the category of principal homogeneous sheaves for Gi on X(i)' We shall see in §3 that this definition is essentially equivalent to our previous definition. The advantages of the definition in §3 are that it is similar to the equivalence between line bundles and (!)~-principal homogeneous sheaves, that it is sometimes convenient to prove the commutativity of diagrams by computing images of so called 'rational sections', and that (in the case of manifolds over q it provides an easy definition of what a metric on an object of the Chow category should be.

3. The categories CHk(X) and CHk(X)

3.1. Definition

Let k ~ 1. Recall that X(k) is a topology on X consisting of sufficiently large Zariski open subsets. Let X(k) be the pretopology (cf. for instance [M]) on the category of open subsets in X(k) in which the Vi form a covering of V if and only if V - Viis of co dimension ~ k + 1 in V. Then Gk = E~ - 1, - k is a sheaf on both X(k) and X(k)' We recall from [M] that if G is a sheaf of groups over any site, then a G-principal homogeneous sheaf is a sheaf X of sets over this site which is equipped with a G-action such that the homomorphism

GxX --+XxX

(g, x) --+ (x, gx)

116 J. Franke

is an isomorphism in category of sheaves of sets. A morphism in the category of G-principal homogeneous sheaves is a morphism in the category of sets which is compatible with the G-actions, such a morphism is automatically an isomorphism.

Let CHk(X) (resp. CHk(X» be the category of Gk-principal homogeneous sheaves on X(k) (resp. X(k»' If A is an object of one of these categories and if U is open in X(k)' then the set of sections of A on U is denoted by A(U). CHk(X) is a full subcategory of CHk(X), and an object of CHk(x) belongs to CHk(X) if and only if X(k) has a covering Ui such that A(Ui) is not empty.

It is clear that the operation

Gk AEBB=AxB (1)

defines the structure of a Picard category (in the sense of [D, §4J) on CHk and CHk. The commutativity law A EB B~ B EB A sends a EB b to b EB a, and the associativity law (A EB B) EB C ~ A EB (B EB C) sends (a EB b) EB c to a EB (bEB c) if a, b, and c are sections of A, B, and C on U. The zero object is Gk , and the isomorphism Gk EB A ~ A sends g EB a to ga, where ga is the action of g e Gk(U) on aeA(U).

To admit also non-invertible arrows we define the following extended Chow category. Let G:(U) be the semi-group of self-homomorphisms of the zero object of CH~(U)e' If A e Ob(CHk(X», put

(2)

Homomorphisms from A to B in CHk(X)e (resp. rnk(X)e) are sheaf morphisms between Ae and Be respecting the G: -action.

Now we discuss the fundamental properties of these Chow categories.

3.2. Relation to Line Bundles

Since X(l) = XZar> the natural homomorphism (9I --+ Gl defines a functor Cl : Oine bundles on X and isomorphisms) --+ CH1(X) and Cl : (line bundles and (9xlinear maps which are isomorphisms at the maximal points Xo) --+ CH1(X)e' This functor maps ® between line bundles to EB in CH1. It is faithful if X is reduced and an equivalence if X is normal.

3.3. Rational sections and their cycles

Let A be an object of CHk(X). We define its sheaf of rational sections by

Ar(U) = uA(V), (3)

Chow categories 117

where the union is over all V which are open in U(k) and meet every irreducible component of U. It is easy to check that Ar(X) is not empty.

Every a E Ar(U) defines a cycle c(a) as follows. Choose a representative a' EA(V) for a. There exist a covering Uj of U(k) and sections bjEA(Uj ). Then a'lvr'lu} - bjlvr'luj = CjEGk(Vn Uj)' Let Zj = O(Cj) EZk(Uj ), cf. 1(3). Then z;!u,rlUj = zjlu,rluj' consequently (since Zk is a sheaf on X(k» there exists Z E Zk(U) with zluj = Zj' We put c(a) = z.

We have

A(U) = {aEAr(U)Ic(a) = O} Ae(U) = {aEAr(U)lc(a) ~ O}

If X is irreducible, Ar is a constant sheaf.

(4)

We will often use (4) to construct objects of the Chow category by first constructing their sets of rational sections, then specifying the cycle map C on the set of rational cycles, and then defining the object itself by the first equation in (4).

An example is the group of rational sections of Gk. For an open Gk(U) = ker(E~-I.-k(U) -+ E~·-k(U»/im(E~-2.-k(U) -+ E~-I.-k(U». Because points of X

of codimension larger than k are elements of U, replacing X by U does not change E~-I.-k(U) or E~-2.-k(U). However, every element of E~·-k(X) vanishes on some open and dense subset U of X(k)' Consequently, (Gk).(X) = E~-I.-k(X)/E~-2.-k(X). It is easy to see that on this set C is given by the E I-differential.

3.4. Relations between the several definitions of CHk

Let k > O. Then there is an equivalence of categories

O( .): CH~(X) -+ clik(X}

O(z)(U) = HomcH~(U)(O, z). (5)

Gk(U), being the automorphism group of any object of CH~(U), acts on the right side of (5). It is clear that a homomorphism from Z to z' in CH~(X) defines a homomorphism from O(z) to O(z') in rnk(X), that O( . ) is compatible with $, and that 0(·) defines an equivalence of CH~(X)e and rnk(X)e.

An inverse to 0(·) may be constructed as follows: For every object A of CHk(X), fix a rational section a A of A. The inverse functor associates the cycle c(aA)EE~·-k(X) to A and the element aA , -q>(aA)E(Gk)r(X)=E~-I.-k(X)/ E~-2.-k(X) to a morphism q>: A -+ A'.

118 J. Franke

To investigate the relation between CH and CH, consider the following assumption:

It is clear that (LF k) is true if the local rings of X satisfy Gersten's conjecture. If X is regular, it satisfies (LF k) up to torsion by the result of [S], and (LF 1) by the Auslander-Buxbaum theorem. Let X satisfy (LFk)' We want to prove CHk(X) = CHk(X). For every object A ofCHk(X), we have to find a covering U i

of X such that A has a section on Ui' By the above remark, it suffices to do this if A = O(z) for a codimension k cycle z on X. By (LF k), for every x E X there exists gxEE~-l,-k(Spec(9x,X> such that (gx) = ZISpeclVx.x ' It is clearly possible to extend gx to g~ E E~ - 1, - k(X). Let U x = X - supp(z - a(g~». Then g~ defines a section of O(Z) on U x' By our choice of gx and g~, X E U x' Consequently, the U x form a covering of X(k) on which O(z) has sections.

3.5. Convention

For k :::;; 0, we put CHk(X) = CHk(X) = CH~(X). If A is an object in CHk(X),

k :::;; 0 and U open in X(k)' Ar(U) consists of a single element denoted by p. We put c(P) = A E E~'O(X) = E?'O(X) = ZO(X) if k = 0 and c(P) = 0 if k < O. A(U) and Ae(U) are defined by (4).

Note that CHk(X) = CHk(X) consists of only one zero object if k < O.

3.6. Definition of a fibred Picard Category

Recall from [D, §4] that a commutative Picard category is a groupoid P together with a functor E9: P x P -+ P, an associativity law aA,B,C. (A E9 B) E9 C~ A E9 (B E9 C) and a commutativity law CA,B: A E9 B~ B E9 A satisfying the compatibilities [DM, (1.0.1) and (1.0.2)], such that the translation functor X E9 . is an equivalence of categories for every X E Ob(P). It follows that P has a zero object, which we assume to be fixed.

An additive functor is a functor F: P -+ P' between commutative Picard categories together with a functor-isomorphism F(A E9 B)~ F(A) E9 F(B) satisfying the additive analogues of [DM, Definition 1.8]. An additive functormorphism is a natural transformation F -+ G satisfying the additive analogue of [DM, (1.12.1) and (1.12.2)].

Let K be a category. A fibred Picard category over K consists of:

(i) For every object X E K, a commutative Picard category P x· (ii) For every homomorphismf: X -+ Yin K, an additive functor f*: Py -+ Px ·

(iii) For every pair of compos able arrows!, 9 in K, an additive functormorphism

Chow categories 119

K f.g: (fg)* ~ g*f* such that for every X t y /!- z 1- U in K, the diagram

(fgh)*(A) Kj,gh ) (gh)*(f*(A))

<". j h*(gf)*(A)

j <" h*(Kj,g) ) h*(g*(f*(A)))

(6)

commutes.

Let P be a second fibred Picard category over K with pull-back functors / and natural transformations K f.g for composable arrows in K. An admissible functor of fibred Picard categories is a pair (F, cp), consisting of:

(i) For every X E Ob(K), an additive functor F x: P x ~ P x (ii) For an arrow f: X ~ Y in K, an additive

cp f: F x 0 f* ~ f' ° F y such that the diagram

F xo(gf)*

F~,."l j 'Pgi ) (gf)' ° F z

j '., F x 0 f* 0 g* / ° g' ° F z

.\ I(~·I f' ° Fy ° g*

commutes for every sequence X L y ~ Z in K.

functor-morphism

(7)

If P and pi are fibred Picard categories over K and F, F' are admissible functors from P to pi, an admissible functor-isomorphism from F to F' is a family (t/J x: F x ~ F:rhEOb(K) of additive functor-isomorphisms such that for every arrow f: X ~ Y in K the diagram

Fxf* 'Pj ) f'Fy

•. j 'Pj

k("" F:rf* ) f'F~

commutes. A cofibred Picard category over K is a fibred Picard category over lOP. The

120 J. Franke

definition of an admissible functor between cofibred Picard categories is similar to the fibred case.

3.7. Flat pull-back

For a continuous mapping f: X -+ Y we denote the pull-back functor from sheaves on Y to sheaves on X by f+. Iff: X -+ Y is flat, the map f: X(k) -+ Y(k) is continuous. In §1 we defined a flat pull-back morphism f+: f*Gk y-+ Gk X.

If A EOb(CHk(y)) (resp. CHk(Y)),f+(A) is af+(Gk,Y}-torser on X(k) (re~p. on .i(k»)' We define f*(A)EOb(CHk(X)) (resp. CHk(X)) to be the image of f+(A) under f*:f+(Gk y) -+ Gk X. This defines a functor from CH\y) to CHk(X) and from CHk(Y)e t~ CHk(Xi, and similar for CH. Every a E A(U) (resp. AAU), resp. Ar(U)) defines f*(a) E (f* A)(f -l(U)) (resp. f*(a) E (f* A)e(f -l(U)), resp. f*(a) E (f* A)r(f-l(U)) with c(f*(a) = f*(c(a))).

If k ~ 0, f* is defined to be the functor introduced in §2, and we putf*(P) = f3 (cf. Convention 3.5).

Let X L Y ~ Z be flat morphisms. There are a natural isomorphism

f*(A EB B)~ f*(A) EB f*(B) sending f*(a EB b) to f*(a) EBf*(b), and a natural isomorphism f*(g*(A)) ~ (gf)*(A) sendingf*(g*(a)) to (gf)*(a). These data define on CHk and rnk the structure of a fibred Picard category over (schemes, flat morphisms), and the functors

can be extended to admissible functors of fibred Picard categories. If f: X -+ Y is flat and A E Ob(CHk(y)), we put A(X) = (f* A)(X), Ae(X) =

(f* A)e(X), and Ar(X) = (f* A)r(X), By the previous remarks, these are presheaves on YJpqc '

3.8. Ifj: U -+ X is an open immersion, we often write Iu instead ofj*. With this notation, we have

PROPOSITION. Let k > ° and A E Ob(rnk(X)). If X Zar has a covering by open subsets Ui such that Alui E Ob(CHk(U;)), then A E Ob(CHk(X)).

Proof Let r E Ar(X) be a rational section. By replacing U i by a covering of (U;)(k) , we may assume A(Ui) -# 0, with section aiEA(UJ Let ai = bi + rlui with biE(Gk)r(Uj ), Then bi can be represented as the image of Ci = (Ci.X)XE(U;)k-1 EE1- 1,-k(U;) in (Gk)r(UJ

We define C;= (C;,X)XEXk-l EE1- 1,-k(X) by

c~ = {o I,X Ci,x

if x¢ U i

ifxEUi'

Chow categories 121

Let b; be the image of c; in (Gk),(X). Then c(b;)lu, = c(b;), hence c(b; + r)lu, = O. If V; = X - supp(c(b; + r)), then V; is open in X(k) and contains U;, consequently the V; form a covering of X(k) on which A is trivial.

3.9. PROPOSITION. Let p: E -+ X be the projection of a vector bundle E to its base X. Then p*CDk(X) -+ CDk(E) and p*: CHk(X) -+ CHk(E) are equivalences of categories.

Proof In §1, we proved that p* is an isomorphism between the E2-terms of the Quillen spectral sequences for X and E. Since the group of isomorphism classes of objects of CHk is E~' -t, and since the group of automorphisms of any object of CDk is E~-l,-t, the result follows for CDk.

To prove the proposition for CHt, it suffices to prove that A E Ob(CHk(X)) and p* A E Ob(CHk(E)) implies A E Ob(CHk(X)). By Proposition 3.8. We may assume E is trivial, i.e., E = A~. By induction on d, we may also assume d = 1. Every A E Ob(CHk(X)) is isomorphic to O(z) for some Z E Zk(X). Let x EX. If p*(O(z)) E Ob(CHk(E)), there exists a cycle z' E Zk(E) with (x, 0) If. supp(z') and [p*z] = [z'] in CHk(E). If t is the coordinate on Ai and SPt the homomorphism defined in §1, this implies [sPt(z')] = [sPt(P*(z))] = [z] in CHk(X). But (x, 0) If. supp(z1, hence x If. supp(sPt(z')), consequently O(z) is locally trivial.

3.10. Proper Push-Forward

Let f: X -+ Y be proper of constant relative dimension dE 7L.. We define pushforward functors f*: CHk(X) -+ CHk-d(y) in the following manner. If k - d < 0 and A E Ob(CHk(X)), f*(A) is the only object of CHk-d(y), every arrow in CHk(X) is mapped to the identity arrow, and f*(a) = P for every a E Ar(X),

Let k = d. Then every object in CHk(X) defines its class [A]EE~,-k(X). We define f*(A) to be the object of CHO(y) defined by f*([A])EE~'O(Y) = EY'O(y). Every arrow in CHk(X) is mapped to the identity arrow, and f*(a) = P for aEAr(X),

Let k < 0 and k - d > O. For aEOb(CHk(X)) (there is only one object, and only its identical arrow), we put f*(A) = Gk,y and f*(P) = O. Finally we consider the case k ~ 0 and k - d > O. For 9 E Gk,x(U), we have f*(g) E Gk-d,Y(Y - f(X - U)), hence f*: (Gk,X),(X) -+ (Gk-d,Y )r(Y)' Also, we have f*: Zk(X) -+ Zk-d(y), and f* is compatible with 1(3), hence with c. Let A E Ob(CHk(X)). We define (f*A),(Y) to be the set of equivalence classes of pairs (g, a) with gE(Gk-d,Y).(Y) and aEAr(Y)' Two pairs are equivalent if they are of the form (g + f*(h), a) and (g, a + h) for some hE(Gk,x).(X), The group (Gk-d,Y).(Y) acts on (f*A)r(Y) by the rule h: (g, a) -+ (g + h, a). We define the cycle of an element of (f*A)r(Y) by c«g, a)) = c(g) + f*c(a). Now the sheaves (f*A)e and f*A can be defined by (4). For a E Ar(X), the class of (0, a) in (f*A)r(Y) is denoted by fia). If aEA(U), thenf*(a)E(f*A)(Y - f(X - U)). If cp: A -+ B is a morphism in CHk(X), then f*(cp) sends f*(a) to f*(cp(a)).

122 J. Franke

In the remaining part of this paper we use the abbreviations c.r.d. for the condition 'constant relative dimension' and CHO(X), ClnX) for the direct sums of categories

00

CHO(X) = EB CHk(X), (8) k= - 00 k= - 00

It is easy to see that the functors f* constitute the structure of a cofibred Picard category over (schemes, proper morphisms of c.r.d.) on cll". The transformation f*(A) EEl f*(B) --+ f*(A EB B) maps f*(a) EElf*(b) to f*(a EEl b), and the isomorphism g*(f*(A)) --+ (gf)*(A) maps g*(f*(a)) to (gf)*(a). It is easy to verify that the axioms of a cofibred Picard category are satisfied.

In the following cases, f*(A) belongs to CHk-d(y):

(i) If Y satisfies (LFh-d' (ii) If k > 0, f is a closed immersion and A belongs to CHk(X).

3.11. Definition of a bifibred Picard category

For a bicategory K, we denote by Hor(K) (resp. Ver(K)) the category of objects of K with horizontal (resp. vertical) morphisms of K as morphisms. A bifibred Picard category over K consists of the following data:

(i) For every object X E Ob(K), a commutative Picard category Px ·

(ii) The structure of a cofibred Picard category over Hor(K) for the Px ' We denote the push-forward functor of a horizontal morphism g by g*.

(iii) The structure of a fibred Picard category, with pull-back functorsf*, over Ver(K) for the Px '

(iv) For every bimorphism A in K with boundary

X' ---.:g,-' ~) Y'

r] (A) ]~ X --'g'----~) Y

an additive functor-isomorphism CPA: f*g* --+ g~f'*. The following compatibility assumption must be satisfied: If A, B, Care bimorphisms fitting into a diagram

X Y

,] 9

]. (B)

Z' h'

X' Y'

1] f'] g'

V (C) (A)

Z X Y h 9

Chow categories 123

then the following diagrams commute:

can. f* h ) g* * 'P" ) g' f'*h

*j * 'Pc (9) can.

g~h~J*

and a similar diagram for the vertical composition Av 0 B. In (9), can. denotes isomorphisms canonically defined by the datum (ii). If we are given a second bifibred Ricard category P over K with pull-back and push-forward functors!, and g! and base-change isomorphisms (P A, an admissible functor of bifibred Picard categories consists of:

- For every object X, an additive functor F x: Px --+ Px.

- For every horizontal morphism g: X --+ Y, an additive functor-isomorphism !x.g : F y 9 * --+ g!F x such that F becomes an admissible functor between cofibred Picard categories over Hor(K).

-For a vertical morphismf: X --+ Y, the isomorphism {3I: Fxf* --+ !,Fy making F an admissible functor between fibred Picard categories over Ver(K).

The following compatibility between !X. and {3 must be satisfied: For every bimorphism

X' _--,9,--' _) Y'

rj A jf, X ) Y

9

the following diagram commutes:

If G: P --+ P is another admissible functor between bifibred Picard categories, then the collection t/I of additive functor-isomorphisms t/lx: F x --+ Gx is called a

124 J. Franke

biadmissible functor-isomorphism if it is admissible for fibred Picard categories over Hor(K) and cofibred Picard categories over Ver(K).

Let F: P --+ P be an admissible functor between bifibred Picard categories over K. If for every object X the functor F x is an equivalence of categories, it has an inverse Fx 1 together with a natural transformation F xFx 1 --+ Id, and this transformation determines Fx 1 up to unique functor-isomorphism. There is a unique way of giving F- 1 the structure of an admissible functor between bifibred Picard categories such that the transformation F F - 1 --+ Id is biadmissible. In a similar fashion, admissible functors between fibred or cofibred Picard categories which are equivalences may be inverted.

In this paper we restrict our attention to bicategories K of the following type. Let C be a category and S*, S* be two distinguished families ofmorphisms in C which contain the identify morphisms and are closed under composition. We also assume that for 9 E S * and f E S* the fibre product

y' g' )y

f'1 if f' ES*, g' ES*

x' g )X

exists. We denote by (C, S*, S*) the bicategory which has the same objects as C, elements of S* (resp. S*) as horizontal (resp. vertical) morphisms and precisely one bimorphism for every fibre square as above (with boundary given by g,

g' E S* andf,f' E S*). The composition of morphisms and bimorphisms is defined in the obvious manner.

3.12. Base change

Consider a cartesian diagram

X' f' ) y'

1-y

1, f ) X

with f flat and 9 proper of constant relative dimension. For every A E Ob(CHk(y')), there is a base-change isomorphism f*g*A ~ g~f'* A sending f*(g*(a)) to g~(f'*(a)) for a E Ar(X), That this definition is correct follows from the base change identity we proved for E~·q (and hence for Gk) in §l.

Chow categories 125

It is easy to check that these base-change isomorphisms constitute the structure of a bifibred Picard category (in the sense of 3.11.) on clI· over (schemes, proper morphisms of c.r.d., flat morphisms).

3.13. Specialization

Let Dc X be a closed regular immersion of codimension 1, and that the ideal defining D is generated by f in some neighbourhood of D. We define a functor sPf: CHk(X - D) --+ CHk(D) in the following manner. Let k ~ O. Then on the set of objects sPf is defined using Convention 3.5 and the homomorphisms sPf

defined in §1. On the set ofmorphisms sPf is trivial since there are only identity morphisms. Furthermore, put sPf(P) = p.

Let k > 0 and A EOb(CHk(X - D)). We define (sPf(A))r(D) as the set of equivalence classes of pairs (g,a) with gE(Gk,D).{D) and aEAr(X - D). Two pairs are equivalent iff they are of the form (g + sPf(h), a) and (g, h + a)

with gE(Gk,D).{D), hE(Gk,x).{X - D), aEAr(X - D). We put c«g, a)) =

c(g) + sPf(c(a)). The sheaves (sPfA)e and spfA are defined by (4). For aEAr(X - D), we denote the equivalence class of (0, a) by spAa)E

(SPf (A))r (D). We have c(spAa)) = sPf(c(a)). If aEA(U), then sPf(a)E

(spf(A))(D - (D n (X - U)).

Let Ksp be the following category. Objects of Ksp are triples (D, X, f) satisfying the assumptions of 1.5. A morphism between (D, X,f) and (D', X',!,) is a Cartesian diagram

D' X'

1,· 1p (10)

D X

such that!, = f o p = p*(f). A morphism is called flat (proper) if p is flat (proper). If in the latter case p is of constant relative dimension, the lemma in 1.4. implies that PD is of the same relative dimension.

As usual, we define an isomorphism sPf(A Ei3 B) --+ sPf(A) Ei3 sPf(B) which sends sPf(a Ei3 b) to sPf(a) Ei3 sPf(b). If in (10) p is proper and of constant relative dimension, we define an isomorphism sPf(p*(A)) --+ PD*(sPr(A)) which sends sPf(p*(a)) to PD*(SPr(a)). If p is flat, the isomorphism sPr(p*(A)) --+ p1)(spf(A)) sends sPr(p*(a)) to P1)(sPf(a)). By (1.15) and (1.16), these definitions are correct.

By the results of 3.7,3.10 and 3.12, the categories CH·(X - D) and ClI·(D) are bifibred Picard categories over (Ksp , proper morphisms of c.r.d., flat morphisms). It is easy to check that sPf is an admissible functor (in the sense of 3.11) between bifibred Picard categories over Ksp-

126 J. Franke

3.14. LEMMA. Let f: X -+ Y be flat. For a Zariski-open U c Y, let fu:f -l(U) -+ U be the restriction of f If for every Zariski-open subset U c Y we are given an additive functor-automorphism <(Ju of £t: CHk(U) -+ CHk(f-l(U)) such that the restriction of <(Ju to f -1(V) is <{Jv for every V c U, then <{Ju = Id for every u.

Proof It suffices to prove <(Jy = Id. Let A EOb(CHk(Y)), then there is a covering U j of ~k) such that the restriction of A to U j is trivial. Since <{J is additive, <{Juj,AluJ = Id. By our assumption, this implies that t~e action of <(JY,A on f*(A)lr'(U,) is trivial. Since thef-1(Uj) form a covering of X(k)' this proves the result.

3.15. Some natural transformations involving sp

If (D, X,J) E Ob(Ksp) and if in the commutative diagram

D ) X

\/ (11)

Z

g and h are flat, then there is an additive functor-isomorphism

(12)

sending sp f(h*(a)) to g*(a). By (1.17), this definition is correct. It is easy to check that r satisfies the following compatibility with flat and proper base-changes Z' -+ Z: It is clear that (since g is flat) the assumptions to (11) remain true after arbitrary base change Z' -+ Z. In particular, c'lf(Z'), CH"(X xzZ') and CU"(D X z Z') are bifibred Picard categories over (Z-schemes, proper morphisms of c.r.d., flat morphisms).

Both sides of (12) are admissible functors between bifibred Picard categories. The compatibility with base-changes Z' -+ Z is the fact that r is a biadmissible transformation. By Lemma 3.14, r is characterized uniquely by this compatibility. This implies a compatibility of r with flat maps Z -+ Z' and flat basechanges X' -+ X.

If in the diagram

Chow categories 127

the triples (Dl' X, fl), (Dl n D2, D2, flID2)' (D2' X, f2), (Dl n D2, Dlo f21DJ belong to Ob(Ksp), we have a natural transformation

(15)

sending sPIt(spJ,(a)) to sPJ,(sPIt(a)). By (1.23), this is a correct definition. The following compatibilities are satisfied:

(A) Compatibility with fiat and proper base-changes. Let K~p be the category whose objects are 5-tuples (Dl' Db X,fl,f2) as in (14), and morphisms from (D~, D~, X',f'I, f~) to (Dl' D2, X,fl,f2) are morphisms X' ..!!. X such thatf; = /; 0 hand D; = D; xxX'. In K~p the fibre product of two morphisms exists if one of them is flat, hence the bicategory (K~p, proper morphisms of c.r.d., flat morphisms) is well-defined. The Picard categories rnO(X), clr(D;), and Clr(D I n D2) are bifibred over (K~p, proper morphisms of c.r.d., flat morphisms), and both sides of (15) are biadmissible functors. Then the transformation (15) is biadmissible.

(B). wft,J,wh.It = Id.

(C) The Coxeter equality (W12W23)3 = Id. If (DI' X,/;)EOb(Ksp) for iE{I;2;3} and if for i ¥= j the embeddings Di n Dr-+ Dj and Dl n D2 n D3 -+ Di n Dj are regular of codimension one, then the following diagram, in which each arrow is in an .obvious manner constructed from the transformations (15), commutes:

SPJ ID D sPJ ID sPJ --~) sPhID,nD2sPItID2sPJ, "c 'j " . j

'PhID.CDThID. 'Ph SPI "D'CDT "ID,'Ph (16)

SPf2ID,nD3SPItID3SPh ) SPf.lD2 nD3 SPf2 1D3 SPh

(D) Compatibility between wand 't. Let (Y, D,f) and (X, D2,f2) be objects of Ksp , and let p: X -+ Y be a flat projection whose restriction P2 to D2 remains flat. Then (Dl' D2, X,fl,f2) belongs to K~p, where Dl = p-l(D) and fl = fo p. The following diagram commutes:

~. I * ) sPit D2P2

~* /P12SPJ'

/~., sPJ ID sPJ p*--~) sPJ2ID 1 P*ISPJ ' 2 1 1

(17)

128 J. Franke

where P1 and P12 are the restrictions of p to D1 and D1 n D2. In the special case (Y, D) = (D2' D1 n D2), (such that D2 --+ X is a section of p), P2 is the identity, and (17) simplifies to

SP[,iD2SPh P* SP[,iD2

j j (18)

sPhiD!SP['P* ) sPh iD!pisp['iD2

4. Functoriality with respect to local complete intersections

For a better understanding of what follows, we suggest the reader to read the formulation of theorem 4.7. first.

4.1. LEMMA. Let

DnZ c Z

n n (1)

D c X

be a Cartesian diagram in which the horizontal inclusions are regular embeddings of codimension one and the vertical inclusions are regular of codimension d. Let X;nD be the blow-up of X along Z n D. For a regular embedding A ~ B, let N~ be the normal bundle of j. For a vector bundle E, let [P>(E) be its projective fibration.

(i) We have (X;nD) Xx D = D;nD UP>(N¥~D) [P>(N~nD)' the components of the union are glued along the hyperplane [P>(N~nD) of [P>(N~nD)' Let (X;nDt = (X;nD) - D;nD' Then the preimage of D in (X;nDt can be identified with the vector bundle N~nD ® (N~nD)-l = E.

(ii) There exists a unique lifting j: Z --+ X;nD of the embedding Z --+ X. The immersion j is regular of codimension d, and its restriction to Z n D is (in the notations of (i)) the zero-section of the bundle E.

(iii) Let Z' c X be a regular inclusion of co dimension d - e containing Z, and suppose that Z c Z' is regular of codimension e. If Z' is the closure of Z' n (X - D) in (X;nDt, then Z' = (Z' n (X - D)) u N~~n/} ® (N~nD) -1, and the inclusion Z --+ Z' is regular of co dimension e.

(iv) If X --+ X is a morphism of schemes, we denote the base-change of objects on X to X by~. If the assumptions to the diagram (1) remain satisfied, with the same

d, after base-change to X, then (XinDt = (XZnDt xxX( = ((XZnDt)). If furthermore the assumption of (iii) remains satisfied, with the same e, after base-change to X, then Z' = (z'f.

Chow categories 129

(v) Ifz n D and X are flat over a common base scheme S, then X;nD isflat over S. If furthermore in (iii) Z' is flat over S, then so is Z'.

Proof. The first part of(ii) follows from the universal property of the blow-up ([H, II.7.14]), since Z n Dc Z is (by assumption) regular of codimension one. The proofs of (i), (iii), (iv), and of the second part of (ii) are straightforward computations involving Micali's theorem (cf. [FL, §IV.2]).

To prove the first part of (v), it suffices to prove that Jk is flat over S, where J is the sheaf of ideals defining Z n D on X. Since a locally free (DznD-module is flat over S, this follows by induction on k from the sequence 0 ~ Jk+ 1 ~ Jk ~ Sk(N:nD) ~ O. The proof of the second part of (v) is similar.

4.2. Deformation to the normal bundle

d (cf. [F, §5.1]). Throughout 4.2-4.4, the symbol Y c X denotes a regular embedding of codimension d.

LEMMA. Let n ~ O. To each sequence X 0 ~ X 1 ~ ... ~ X n of regular immersions, one can construct a commutative diagram

(2)

with the following properties:

(i) On M i , there is an action of the affine group Affl (X n) = {g E r(X n' &'(§ 2 2) I g( (0) = oo} which is compatible with the action of Affl(Xn) on Pt·

(ii) The restriction of'ltj to 'ltj-l(Ai) is an isomorphism 'ltj-l(Ai)-~ Aii , and 'lto is an isomorphism everywhere: 'lto: Mo~ Pio'

(iii) For a section t E P1(Xn), put M!t) = 'lti-1(t). Then M!OO) ~ M i , and M(oo) is the

sequence

X d, NX' d2 NX2 dn NXn o c Xo C Xo C ••• c Xo'

(iv) Ifp: Xn ~ Z is aflat morphism such that the restriction ofp to Xi isflatfor every i, then Mi is flat over Z for every i.

(v) For a morphism Xn.!4 X n, we use - to denote base change to X n. If we have - d, - d-Xo C Xl'" c! Xn (with the same di as in (2)!), then we can define an

isomorphism M<p: M j XXnXn~ MiX.) (the last symbol means the construction of M i , applied to the sequence X.) On M)oo) xxnXn, M<p is an

130 J. Franke

isomorphism of vector bundles. If we apply a second base change

Xn.!4 Xn ~ X n, then the diagram

M j XXnXn--~) (M j XXnX) xx"Xn

M~ j j M."j,

MAX.) (M", MAX.) xx"Xn

commutes. Proof Let M n be the blow-up of IPL along CXl(X 0), where CXl is the infinity

section of IPL' Mo is the image of the immersionj constructed in 4.1(ii), and M j

for 0 <j < n is constructed by applying 4.1(iii) to Z' = IP.L (of course, with D = CXl and Z = IP}J The Lemma follows from the results of 4.1.

4.3. An awkward proposition

Let Kd" ... ,dn be the following category. Its objects are sequences

A homomorphism from X, to y, is a morphism from Xn to y" such that Xi = 1'; xYn X n' it is called flat (proper) if X n -+ y" is flat (proper). Since in X, the fibre product of a flat and an arbitrary morphism exists, the bicategory (Kd" ... ,dn, proper morphisms of c.r.d., flat morphisms) is well-defined.

Let Kd" ... ,dn,sp be the following category. Objects are triples (D', X',f) where D' and X'. form a Cartesian diagram

Do d,

Dl d2

D2 dn

Dn c c c··· c

n l n l n l n l

Xo d, Xl

d2 X 2

dn

Xn C C c··· c

andf is a section of (!JXn in some neighbourhood of Dn such that (Di' Xi' fix,) belongs to Ob(Ksp) for every i. A morphism from (D!, X!,!,) to (D', X',f) is a morphism X~ to Xn such that X; = Xi XXnX~, D; = Di XXnX~, and!, = (pullback of f to X~), it is flat (proper) if X~ -+ X n is flat (proper). The bicategory (Kd1, ... ,dn,sp, proper morphisms of constant relative dimension, flat morphisms) is well-defined.

By the results of §3, clr(x;) (cf. (3.13)) are bifibred Picard categories over (Kd" ... ,dn, proper mor. of c.r.d., flat mor.). Over (Kd" ... ,dn,sp, proper mor;..,of c.r.d., flat mor.), Cff(Xi) and rn'(D;) are bifibred, and sPflx,: clr(x i ) -+ CH'(D i) is biadmissible.

We are now ready to formulate our awkward proposition. To stimulate the

Chow categories 131

reader's patience, we mention beforehand that the functor F x. on K d , will play the role of the Gysin functor j! for regular immersions of codimension d 1, that its uniqueness up to an additive functor-isomorphism on K d ,.d2 will be used to construct the isomorphism (jj)! -+ j'j!, and that its uniqueness up to unique natural transformation on K d,.dz.d3 will be used to verify (3.6).

PROPOSITION. On Kd, ..... dn' there exists an admissible functor between bifibred Picard categories (cf 3.11.) F X.: clr(x n) -+ Cff(X 0), which preserves the graduation, together with the following data:

(i) If in the commutative diagram

X dnx dnx OC lC'''C n

~/ (3)

Z the projections ri are fiat, then there is an additive functor-isomorphism

Cir : F x. or: -+ r6 (4)

which satisfies the following compatibility with base-changes Z' -+ Z: If X; = Xi X z Z' and if r;: X; -+ Z' is the natural projection, then (since the ri

are fiat) X! is in Kd, ..... dn' cl.qx i) and clr(z') are bifibred over (Z-schemes, proper mor. of const. reI. dim., flat morphisms), and the functors r;* and F x'. are biadmissible. The condition is that Cir ' is a biadmissible transformation.

(ii) On Kd, ..... dn.sp' there is a biadmissible transformation

(denoted by f3 f.X .• D.) (5)

with the following properties:

(ii.l) Let (DP), X.,jl) and (DF), X.,jz) be objects of Kd, ..... dn.sp such that (D!lZ), D!l), fzID~I)) and (DPZ), DF), fdD~2») are both objects of Kd, ..... dn' where D~lZ) = D~l) n D~Z). Then the diagram

SPr, ID~"(f3 J ,.X .. D'/' sPf,IDi,2)SPf2IxoF x. )

j w",".""

SPf2iDi,')SPf,ixoF x.

jsPf;IDh"lfij"X"D" ,)

F D!12)SPf,ID:I)SPf2

j F ,. ,(wI",1

132 J. Franke

commutes. The transformations OJ have been defined in 3.15. (ii.2) If (D, Y, F)EOb(Ksp), X.EOb(Kd, .... ,dJ' and rn: Xn --+ Y is a flat

projection whose restriction ri to Xi remains flat, we put Di = Xi xyD and denote the projection of Di to D by rD,i' Then the following diagram commutes:

(7)

UNIQUENESS. The data (i) and (ii) determine F x up to a unique biadmissible transformation between biadmissible functors over Kd1, ... ,dn '

REMARK. In the special case Y = X 0, (7) simplifies to

(8)

Proof of Proposition

4.4. Construction of Fx., For Xo e ... eXn, let Mo e ... e Mn be the result of construction 4.2. The projections nn have been defined in (2). We put

qn = composition of Mn no) IJJ>L --+ Xn

qn,a = restriction of qn to n; l(AlJ (9)

Furthermore, let

(10)

be the bundle projection (cf. 4.2 (iii)). By 4.2 (v), the categories CH'(Ml OO ») are bifibred over Kd" ... ,dn , and p! is a biadmissible functor. By 3.9, we know that it is also an equivalence of categories. Consequently it has a biadmissible inverse (p!)-l (cf. 3.11). Finally, letfo be a rational function on lJJ>§pec(Z) whose only zero is a simple zero along 00 and whose pole does not intersect 00. We denote the pull-back of fo to IJJ>L by f and put

(11)

Chow categories 133

By 4.2(v) and our previous remarks, this is a biadmissible (over Kdl ..... dJ functor. Let un: Xn -+ Z be a flat morphism whose restriction Ui to Xi remains flat.

4.2(iv) proves that Mi -+ Z is flat, hence we can define lXu by

MM(<<» F u* = (P*) - lSp q* u* tunq ••• , n' • l (p*) - l(U P )*

X n 00 ,,:(1) n,a n 00 p 00

(12)

It is biadmissible over (Z-schemes) since (12) contains only biadmissible transformations.

Let (D., X., g) E Ob(Kd, ..... dn,sp). We denote by M. and M~D) the construction 4.2, applied to X. and D. respectively. The analogues for D. of the morphisms (9) and (10) are denoted by a superscript (D). By 4.2(v), we have M!D) ~ Mi XXnDn c Mi' There is a unique biadmissible (over Kd" ... ,dn,sp) transformation <p: spg(p!) - 1 -+ (p~)*) - lSpp~(g) such that the diagram

commutes. We also mention that p*(g) = q:(g)IM~«» and that the sheaf of ideals defining Dn c Mn is in some neighbourhood of Dn trivialized by q:(g). We define the datum (ii) by

(13)

Since (13) contains only biadmissible transformations, it defines a biadmissible (over Kdl, .... dn,sp) functor-isomorphism. It is straightforward to check that (6) follows from (3.16) (applied to the three functions n:(f), q:(g(1» and q:(g(2» on M n) and (7) follows from (3.17) (applied to a fibre diagram of the form

134 J. Franke

4.5. Uniqueness of F

Let G be a functor with the properties (i) and (ii), and let F be the functor constructed in 4.4. We use the notations of 4.4 and put MIa) = n;-1(A1J Since qo

is flat and MbOO ) = Xo, (3.12) provides us with a biadmissible (over K d" ... ,dJ isomorphism

(14)

We define <Dx.: Gx . ~ F x. by

( * )-1 * - F ~ Poo sp,,~(J)qn,a - x.' (15)

In (15), (a) is base-change with respect to the flat morphism qn,a, and (b) is canonical. Since (15) contains only biadmissible transformations, <Dx. is biadmissible (over Kd" ... ,dJ We have to check that it is compatible with the data (i) and (ii).

If we are given a commutative diagram (3), we define an additive functorisomorphism Xz: rt ~ rt by

r* o (l,)-1 ) G r*

x. n x. ) F r* __ (l,,-,~) ro*. x. n

Since this definition contains only biadmissible (over Z-schemes transformations, we have Xu = Xzlro'(U) for every Zariski-open U c Z. By 3.14, Xz is the identity, and <D is compatible with the datum (i).

Diagram (16) implies that <D is compatible with the datum (ii).

spgGx. SPg(t/t\ sp sp q* G ) sPgsPfGM'O)q:,a 13 ) spgGM!OO)sPfq:'a 9 f O,a x.

\ ro,") NT 1 Wg,f 113

SPfSpg~t,a Gx. ) sPfSPgGm'O)q:,a GM!D.OO)SP9sPfq:,a p 113 (C) g,f l Wg,f

sPfq~!*sPgGx. (B) sp f G M!D.O)Spgq:.a 13 ) G M(D·oo)SP fSPgq:,a

131 1 NT 1 GD,sPg

t/t ) sPfq~!*GD.SP9 ) sp f G M(D,olq'D)*Sp ) GM'D'OO)SPfSP9q~~J* • n,a 9

Chow categories 135

SpPM~oo)SP fq:.a spgG Moo) P!,(P!,) - lSp fq:.a ) Spg(p*) -lSp fq:.a

Ii! NT !Ii NT !q> GM~D,oo)SPgSp fq:.a ) GM~D,oo)SPgP!'(P!') - lSp fq:.a rx (p~)*) - lSpgSp fq:'a

W g ,} ! (D) ! NT ! W g.!

! \ !q> NT !

\ jm" /1 N\ G WO'.'P!::)·(P!::;') - 'sp jsp,q:"j

GM~D,oo)P~)*(p~)*)-lSPfq~~J*SPg (16)

Glue the right boundary of the upper diagram with the left boundary of the lower diagram.

The notations in (16) are the same as in (13). To save space, the indices at the transformations IX and f3 have been omitted, and the various pull-backs ofJand g have been denoted by the same letter J or g. The vertical arrow at the left boundary of (16) is the isomorphism f3 for G., and the outer right column of (16) is the sequence (13). The top and the bottom row of (16) are (15) for X. and D .. The squares marked by 'NT' are commutative because they are of the form

F(A) ~A) G(A)

F(,) j j G(,)

F(B)-~) G(B) ~.

where ~: F -+ G is a natural transformation between functors and 17= A -+ B is a morphism.

The commutativity of (A) is (3.18). (B) commutes because f3 is admissible with respect to flat pull-backs in Kd1 ... , .dn.sp (cf. the diagram at the end of 3.6). (C) is of type (6) and (E) of type (8). The commutativity of (D) is easily derived from the fact that sp is an admissible (with respect to flat pull-backs) functor.

4.6. Uniqueness up to unique Junctor-isomorphism

Suppose we are given a biadmissible automorphism <1> of F which is (in an obvious sense) compatible with the data (i) and (ii). From the compatibility with

136 J. Franke

(i) it follows tht <l>x. = Id if X. is a sequence of vector bundles with base Xo. In particular, <l>M(OO) = Id, where M(oo) is the same as above. Since

Fx. ) sPfq~.aF x. ) spfF M(a)q:.a ) F M(OO)SPfq:.a

j ., NT j .. "".,.1 (A) j .. ".M"') (B) j .M','~ld Fx. ) sPfq~.aF x. ) spfF M(a)q:.a ) F MlOO)SPfq:'a

commutes, we have <l>x. = Id, and the proof of 4.3 is complete. NT has the same meaning as in (16), (A) commutes since is biadmissible, and (B) commutes since is compatible with the datum (ii).

4.7. The Gysin~functor

We are now ready to prove the main theorem of §4. Throughout this paper, 'lei' will be an abbreviation for local complete intersection. A morphism X ~ Y is called a smoothable lei-morphism (abbreviated: slei-morphism) if it has a factorization X ~ S ~ Y where S ~ Y is smooth. Then it follows ([SGA 6, Exp. VIII] or [FL, IY.3.l0.]) that X ~ S is a regular immersion. The relative dimension of a lei-morphism at XEX has been defined in [SGA 6] and [FL].

Let K lci be the following category: Objects are triples (f, X, Y) withf: X ~ Ya slei-morphism. A morphism from (f', X', Y') to (f, X, Y) is a Cartesian diagram (which we denote by (Px, py))

X' _"---1'-----+) Y'

Px j j py

X _-,J,---~) Y

such that for every x E X' dAf') = dpx(x)(f), where dAf) is the relative dimension of the lei-morphism f at the point x. A morphism in Klci is called flat (resp. proper of c.r.d.) if so is py (and hence Px, cf. the lemma in 1.4). The bicategory (Klci , proper morphisms of c.r.d., flat morphisms) is well-defined and will be denoted by IK lci '

Let K lci •sp be the following category. Objects are 5-tuples (f, X, Y, D, A) such thatf: X ~ Y is a slei-morphism, DeY is a regular immersion of codimension one, A is a section of Cl!y in some neighbourhood of D which generates the sheaf of ideals defining D, and Dx ~ X is a regular immersion of codimension one, where Dx = f -l(D). A morphism in K lci •sp from (I', X', Y', D', A') to (f, X, Y, D, A) is a morphism (Px, py): (f', X', Y') ~ (f, X, Y) in K lci such that D' = pyl(D) and

Chow categories 137

A' = pt(A). A morphism is said to be flat (proper), if so is py. The bicategory (Klci.sp , proper morphisms of c.r.d., flat morphisms) is denoted by !Klci,sP'

Let Klci.com be the category whose objects 5-tuples (f, g, X, Y, Z) where f: X -4 Y and g: Y -4 Z are lci-morphisms such that g and gf (and hencef too) are slci. A morphism from (f', g', X', Y', Z') to (f, g, X, Y, Z) is a triple (Px, py, Pz), Px: X' -4 X, py: Y' -4 Y, pz: Z' -4 Z such that (py, pz): (g', Y', Z') -4 (g, Y, Z) and (Px, py): (f', X', Y') -4 (f, X, Y) are morphisms in Klci . It is flat (proper) if so is pz (and hence Px and py too). As usual, !Klci,com refers to the bicategory (Klci,com, proper morphisms of c.r.d., flat morphisms).

The main result of §4 is the construction of an inverse image functor!, for local complete intersections f Unlike the functors constructed in §3, it is no longer possible to define this functor directly. Instead, we describe it as a certain biadmissible functor, equipped with certain natural isomorphisms described in 4.7.1-4.7.3 which have to satisfy certain conditions explained in these paragraphs. The system of functors!, (for slci-morphisms f) is unique in the sense that, given another system of functors f? together with similar natural transformations satisfying the same conditions there exists a unique functorisomorphism!, -4 f? respecting the natural transformations 4.7.1-4.7.3.

It should also be mentioned that the notions of a natural transformation and of a natural isomorphism are equivalent if applied to functors between groupoids, in particular to functors between Picard categories. Consequently, these two notions are used synonymously in the following text, and natural transformation is often abbreviated to transformations because confusions are impossible.

The main result of §4 is

THEOREM. Let us denote objects of !Klci by (f, X, Y), such that Clr(X) and clI"(Y) are bifibred Picard categories over Klci . Then there is a biadmissible , "'. "" . functor f: CH'(Y) -4 CH'(X) between the bifibred Picard categories over

!KlciCH"(Y) and CH"(X), together with the following data:

4.7.1. For each flat morphism h: Y -4 Z such that hf is flat, we are given an isomorphism Yj,h: fh* -4 (hf)* satisfying the following compatibility with flat and proper base changes Z' -4 Z. For every Z' -+ Z, 1': X' =

Z' X z X -+ Z' X z Y = Y' is slci (this is so because hf is flat), clI"(X') clI"(y'), clI"(Z') are bifibred over (Z-schemes Z', proper morphisms of c.r.d., flat morphisms), and f', h'* and (h'f')* are biadmissible functors between these categories. The condition is that Y r,h': f"h'* -+ (h'f)* is a biadmissible functorisomorphism.

4.7.2. If we denote objects of !Klci,sp by (f, X, Y, D, A) and put Dx = f- 1(D), then clI"(X), clJ"(Y), ClI"(D), and CU"(Dx) are bifibred Picard categories over !Klci ,sp'

.......... . ........... ........... "" . , The functors SP.l.: CH (y) -+ CH (D), sPf*(.l.): CH (X) -+ CH (Dx ), f:

138 J. Franke

rn·(y) -+ CH·(X), and f~: clf(D) -+ cl-f(Dx) are biadmissible (fD is the restriction of f to f -I(D». The datum we require is a biadmissible functorisomorphism

which satisfies the following properties:

4.7.2.1. If Cf, X, Y, Di, Ai) (i E {1, 2}) are objects of KICi.Sp such that the immersions D12 = DIll D2 -+ Di and DX,12 = DX,11l DX,2 -+ DX,i (iE {1, 2}) are regular of codimension one, then the diagram similar to (6) commutes:

sP"r' f' ~ sp"f'sp" ~ r'r'sP"

SPA,SPA2t -+ SPA, tSPA2 -+ tSPA,SPA2

(17)

(For the sake of simplicity, the various pull-backs of Ai and restrictions off have been denoted by the same letters.)

4.7.2.2. Let (D, Z, A) E Ob(Ksp), P: X -+ Ya ski-morphism and q: Y -+ Z be a flat morphism such that qp is flat. We denote by Dx and Dy the pre-images of D in X and Y, by PD the restriction of P to Dx , and by qD the restriction of q to Dy. The condition is that the following diagram (which is similar to (7» commutes:

, * ' * ' * p;,qrp, ~ P'DSP~",q ~ sP'''''r·q (18)

(qDPD)*SPA ) sp(qP)*(A)(qp)*

4.7.3. Let us denote objects of !Klci com by (J, g, X, Y, Z), such that cl-f(X), clf(Y), Clf(Z) are bifibred Picard ~ategories over !Klci,com, and f i, g!, and (gf)' are biadmissible functors between them. The datum we need is a biadmissible functor-isomorphism

subject to the following conditions:

4.7.3.1. If (J, g, X, Y, Z) is an object of !Klci,com and (D, Z, A) E Ob(Ksp) such that (g-I(D), Y, g*(A» and «gf) -I(D), X, (gf)*(A» are objects of Ksp. Then the

Chow categories 139

following diagram commutes:

f ' I f' I ... I gr' ~ ·sp~("g;, ~ 'Pl'fT'DgD (19)

(gf)!SPA ) SP(9f)*(A)(gDfDY

The restrictions off and g to (gf) -1(D) and g - 1(D) have been denoted by fD and

gD'

4.7.3.2. The analogue of 3.(6), applied to lci-morphismsf, g, h such that h, hg, and hgf exist and are slci, commutes (of course, * is replaced by!).

4.7.3.3. If we have a Cartesian square

X' g'

I] X 9

with f smooth and g slci, then the following diagram is commutative:

'*f' base change I g . --------'=----~) f'g*

] f,." -, ] r'(f •. ,,) - •. (20)

gdf' ~ (fg')! = (gf')! ~ fd g!

By 'base change' we mean the base change isomorphism defined by the coadmissible structure of f!over Ver(lKlcJ

4.7.4. UNIQUENESS. The data 4.7.1-3, determine a biadmissible functor f' over IKlci up to unique biadmissible functor-isomorphism.

Proof The proof will be carried out in steps 4.8-4.17.

4.8. i! for regular closed immersions. Let i: X 0 --+ X 1 be a regular closed immersion. Since the co dimension of i is locally constant, it suffices to construct i! if i is of constant codimension d. Then X. = (X 0 c X 1) is an object of K d , and we put i! = Fx. (cf. 4.3). The data 4.7.1 and 4.7.2 are given by the isomorphisms (J. and {3 in Proposition 4.3.

To construct Gi,j: i!i' --+ (ji)!, we may assume that i and j are of constant codimensions d 1 and d2 , Then X. = (X 0 ~ X 1 ~ X 2) E Ob(Kd" d2)' and both i!i'

140 J. Franke

and (ji)! are candidates for F x.' Hence, Proposition 4.3 implies that there is a unique biadmissible (over K d1 ,d2) isomorphism ei'/ iii' --+ (ji)! which is compatible with the datum 4.7.2, i.e., which satisfies 4.7.3.1 (that such an isomorphism is compatible with the datum 4.7.1 follows from 3.14).

If we denote objects of K d1 ,d2,d3 by X. = (X 0 ~ Xl ..4 X 2 !<.. X 3), then (kji)! and i!j'k! satisfy the conditions for F x., hence a 'good' isomorphism between them is unique. This is 4.7.3.2. in the case of regular closed immersions.

4.9. The isomorphisms ({)j,p

Let

s

/j, X ) Y

be a commutative diagram with regular closed immersions i andj and a smooth morphism p. We construct an isomorphism

(21)

as follows: Let S = X Xy S, p: S --+ X be the natural projection, and i: S --+ S be the base-change of i. The inclusion j determines a section J of p:

Jet/I, x )Y

(22)

i

Then

j'p*--~) J'I'p* --~) J'p*i! (23)

determines ({)j,p'

We need the following compatibilities:

(i) If k: Z --+ K is a regular immersion, we have

(24)

Chow categories 141

(ii) If we have a commutative diagram

with regular imbeddings i, j, k and smooth morphisms p, q, then

(iii) Compatibility with flat and proper base changes Y' --+ Y. (iv) If we have a commutative diagram

S __ i_l S

/1· 1, X lY--_lZ

j

(25)

(26)

(27)

in which the square is Cartesian, i andj are regular immersions, and p is smooth, then the following diagram commutes:

(fk)!p* ~ k'T'p* ~ k!p*i!

···1 1··· (28)

(ij)! j'i!

Since (23) contains only biadmissible transformations, (iii) is clear. Proof of (i). We need the following

SUBLEMMA 1. Let k: Z --+ X be a regular closed immersion, S -4 X a smooth morphism, S L Z the restriction of S to z, J X --+ S a section of p, Tits restriction to Z:

(29)

142 J. Franke

Then

1 .rr (30)

kTp* --~) j'l{!p*

commutes. Proof of Sub lemma 1. For A E Ob(CHi(X)), let bX •A be the unique automorph

ism of k!A making (30) commutative. We have to show that bX,A = O. First we assume k: Z ---+ X is the zero-section of a vector bundle. Then k! is an

equivalence of categories, hence bX,A = k!(b~,A) for a unique automorphism b~,A of A. Since (30) contains only biadmissible transformations, (b~,A)lu = b~,Alu for every open U eX. By 3.14 (applied to f = Id), this implies bX,A = O.

Now we return to the general situation. Let Mo ---+ M 1 be the deformation of Z ---+ X to the normal bundle. We use the notations qn and Poo as in (9) and (10), and also the isomorphism (14). By (14), we have bX,A = sPJq~(bX,A) as elements of Gi(Z). Since the transformations in (30) are compatible with flat base change, sPJq~(bX,A) = SPJ(bM~'l,qt,A)' Of course, bM('l,B and bM(oo),B are defined by means of the pull-backs of S to Mla) and MlOO ). Since the transformations in (30) are compatible with specialization, we get

by our previous considerations, and the proof of the sublemma is complete. We return to the proof of (i). The composite diagram of (22) and (29) is

7 CI' :JY.l k

We have a diagram

, , (a) -, -,-, * (b)

kjp'~r :~ k'j'I'p* --'--'~) k'j'p*i!

j'i{!p*i!

1 (c) ) j'p*k!i!

1 (~ (c') ) k!i!

Chow categories 143

The commutativity ofthe triangle on the left is 4.7.3.2 for closed immersions, and the square on the right side is (30). It is clear that the composition of (a'), (b'), and (c') is k!(<pj.p). Since S = Z Xy S, the composition (d)(c)(b)(a) is <Pjk.p' The proof of (i) is complete.

Proof of (ii). We consider the diagram

in which the squares are Cartesian. Our first aim is to show that the diagram

[(!(pq)*

j (31)

k!q*p*

commutes. Since (31) contains only isomorphisms which are compatible with restriction to open subsets of X, this follows from 3.14.

Now we consider the diagram

~!(Pq)*i! (b») [(!q*I'p* (c) ) [{Tq*p*

\., ;'~) j.fA NT j ... ~ J'p*i! (b') J'T'p* --~

(c')

(B) kq*p*

,/ j"p*

It is easy to check that (a')(b')-l(c') is <Pj,p and that (a)(b)-l(c)-l(d)-l is <Pk,pq,

such that (26) follows. It remains to prove that (32) commutes. For (A), this is an application of (31). For (B), form the fibre square

144 J. Franke

where T= X Xs T= X xsT (with X --+ S and X --+ S given by j and ]), and consider the isomorphisms

(b) ) l{'q*j'r (e) "1'1"1'1 1 )JT--~)l

Their composition is o/k.q' and the composition (c)(b)(a) is o/k,q' which proves the commutativity of (B).

Proof of (iv). With the same notations as in (27), put S' = S X z X ~ X and consider the diagram

f([/[ __ ' J j

We have a commutative diagram

(ik)'p* ) k'i'p* ) k'p*i'

j (A) j NT j k '(ij)'p* ) k 'j'i'p* ) k'j'p*i'

j (B) j k 'p'*(ij)' ) k 'p'*j'i'

j NT j (ij) , _1.1

JT

(A) is 4.7.3.2., applied to the special case of regular immersions in which it has already been proved. (B) is the fact that ei.i (cf. 4.7.3) is biadmissible at least in the special case of closed immersions. If we take the outer contour of this diagram and delete the vertices of the two middle rows, we get (28).

4.10. Now we are ready to construct t. We choose a factorization

(j: X--~) S ----'p'---~) Y

Chow categories 145

off, where p is smooth and i is a closed (and hence regular) immersion. We put

f~ = i!p*. (33)

Our task is to define the 'change offactorization'-isomorphism. First we assume we are given a new factorization b' and a smooth morphism r: S' -+ S making

S'

~lK X~/,Y S

commutative. We defineCPd'.d.r by

i"p'* ) i"r*p* <P,',,) i !p*.

The following facts are consequences of (i)-(iv) in 4.9:

(i) If k: Z -+ X is a regular immersion and if O'k is the factorization of fk by zE:. S.4 Y, then

(34)

(ii) If u is a factorization off sitting in the top row of the diagram

S

then we have

(35)

(iii) Compatibility with flat and proper base change Y -+ Y.

If 0'1 and 0'2 are two factorizations off, we denote by O't X 0'2 the factorization

146 J. Franke

by r 1,2: Sl Xy S2 --+ S1,2 the natural projections, and define CP"!"12: f~! --+ f~2 by the composition of

(36)

The identity CP"!,"2CP"2,"3 = CP"!,"3 follows from the diagram

III which each arrow is in an obvious manner of the form cP"',,,,,. The commutativity of the small triangles follows from (35).

The isomorphisms (36) enable us to identify the functors f~ with a unique functor [' By fact (iii) above, this functor is biadmissible over 1K1ci '

The construction of the data 4.7.2. and 4.7.1 is easy. If q: Y --+ Z is flat and if qf is also flat, then for every factorization (J as in (33) we have an isomorphism f~q* --+ i'(qp)* --+ (qf)*.1t is easy to see that the isomorphism f'q* --+ (qf)* defined this way is biadmissible in the sense explained in 4.7.1. By 3.14, such an isomorphism is unique, hence it is independent of the choice of the factorization.

Let (f, X, Y, D, A)EOb(K1ci,sp)' We choose a factorization X ~ S -A Y off, put Dx = f -l(D), Ds = P -l(D), and denote by iD and PD the restrictions of i to Dx and P to Ds. By our assumptions to objects of K1ci,sP' the restrictionfD off to Dx is lei, andfD = PDiD is a factorization offD into a smooth morphism and a regular immersion. We define

Since our definition of the transformations CPj,n cp",,,',r" and CP"!,"2 contains only isomorphisms compatible with specialization, b).,j is independent of the factorization. The verification of 4.7.2.1 and 4.7.2.2 is easy.

4.11. Let (f, g, X, Y, Z)EOb(K1ci,com)' For two Z-schemes A, B we denote A x z B --+ B by AB --+ B. By our assumption, there exists a closed immersion ko

of X into a smooth Z-scheme A. We have an induced immersion k: X --+ Ay and a factorization X ~ Ay ~ Y of f, which we denote by Ct. We choose a

Chow categories 147

factorization 0": Y ~ S -4 Z of g and consider the diagram

As

y~

A)\, X >Y >Z

J g (37)

Let IX * 0" be the factorization of gf over As. We define

(38)

Let io be another closed immersion of X into a smooth Z-scheme B, with induced factorization {3: X ~ By -4 Y of f, and let r: Y -4 T ~ Z be another factorization of g. Consider the morphisms

m' A y >AsxT

~) As

IT ~ 1[' n"

S

Y/·\SXT

ASXT

k /j. X -------+> Ay > As

(39)

We get a commutative diagram

r::~: -(-A-) ~. k 'm' r"'··-(-B-) ~. (m'kJr"" (40)

k 11[*1 1 k 11' 11['* (I'k) 11['*

(A) commutes by 4.9(iii) since in the left diagram (39) the upper triangle is the base change of the lower triangle by the flat map As x T ~ S X Y, and (B) follows from 4.9(i), applied to the right side of (39).

If we apply (40) to p*A for A EOb(CH"(Z)), we get a diagram whose outer contour is

f:': (--~. (fgJl "I (41)

f~g~A > (fg)~*(JA

148 J. Franke

where (fg)!*(u x <) --+ (fg)!*u is defined by the commutative diagram of factorizations

Let us consider the morphisms (A x B means A X z B)

m"

· 1-/

(AXB)Y

X ) Ay ASXT --~) s x T--~) S----i> Z . k m' 1!" q p

By 4.9(iv), the diagram

(mr--~) n!m"!p"*--~' ·"pr" (m'k)! k!m!

commutes. Applying this to objects of the form 1t"*q*p* A, A E Ob(CH"(Z)), we get

(42)

Gluing (41) and (42) and using 4.10(ii), we get

(43)

Chow categories 149

Applying (43) another time, with the roles of 0( and (J and of f3 and T

interchanged, and using the definition (36), we arrive at the commutativity of

f~ g~ --~) (gf)~. (J

j ,.,I" .. J j (44)

fftg~ ) (gf)ftor

This proves that the transformations (38) fit together and define "'f,g: fig! -4 (gf)!.

4.12. We omit the proof of 4.7.3.1 since it is straightforward. To prove 4.7.3.2, we consider lci-morphisms U ~ X !!., Y ~ Z such that U, X, and Y admit closed immersions into smooth Z-schemes. We want to prove

f'r-~'f'r' (45)

(gf) !h ! ) (hgf)!

We choose closed Z-immersions of U, X, and Y into smooth Z-schemes A, B, and S. Then we have the following factorizations off, g, and h:

0(: U ) Ax p )X

f3: X ) By q )Y

Y k

) S ) Z. (J:

It suffices to prove

j j (46)

(gf)~. f3 h~ --~) (hgf)~. f3. (J

We consider the morphisms (A x B means A xzB):

(A x B)s !9' "'('

(A x B)y Bs

I/ P'~Y ~ Ax By S

Y "{. /j?' ~ Y ~ U )X ) Y )Z.

f g h

150 J. Franke

We have a commutative diagram

P'i'jk'C NT' P'i'kC"C -(-A-) ~) p*(k'j)'q'*C

j"p'*q*k'C--~. i"P'·lq··c

(B) r'k"''j'q .. c NT' (k"i'l'rq··c

i"(qp')*k'C--~) jt!k"'(q'p")*C--~) (k"j')'(q'p")*C

(47)

The commutativity of (A) belongs to the conditions which were used to characterize the isomorphism j 'k" --+ (k'j)' defined in 4.8, and (B) is of type 3(7) (applied to the biadmissible functor F = k I). If we insert C = r*(.) in (46) and apply ii, we get a diagram whose outer contour can be identified with (46).

Now we prove 4.7.3.3. Since (20) is clear for a smooth morphism g, it suffices to consider the case of a regular closed immersion g. In this case, the proof consists of two parts:

SUBLEMMA 1. Ifin 0': X ~ Y -4 Zi is a regular immersion and p is smooth, then

i 'p* ( j'(YP.ld) ., , I"P'

1 !"

(pl)~ ) (pi)'

commutes.

SUBLEMMA 2. We suppose that in a Cartesian diagram

i' X'--~) y'

1 1, X )y

(48)

Chow categories 151

P is smooth and i is a regular immersion. Then IT: X' ..£. y' .4 Y is an admissible factorization of the lei-morphism ip'. With these notations, the diagram

, base change , F'" ----"'---~) i' 'p*

p' 'i' --~) (ip')' +- (ip')~

(49)

commutes. It is clear that (20) for a regular immersion g follows from (48) and (49).

Proof of Sub lemma 1. In 4.11, we choose for ko the immersion of X into the smooth Z-scheme Y, and put S = Y. Then (37) becomes

YxY / '\op,

YxY Y

V ~,/~ X ) Y ) Z

P

(Y x Y = YXz Y, and PI = projection to the first factor). Hence ei,p is

i'p* -+ i'ptp* -+ i'(PPI)* = (pi)~x<T -+ (pi)'

II (pi)~

(50)

By the definition made at the beginning of 4.10, the isomorphism (pi)~ -+ (pi)~ x <T in (50) is Cf>;;;'<T,<T,p,' So it remains to prove Cf><TX<T,<T,Pl = Cf><TX<T,<T' By (4.10(ii), the following diagram commutes:

(pl")lT! lPaX(1XI'1,O"l,Pl (')' ( pi <T x II X <T

1·······, 1 , ...... ~;:,;:. (pi)~ x <T '-----( ")' ~"'"'P' .,... pi <T x <T X<T

CPa x (1 x (110' X 0'1P23

where P23: Y x Y x Y -+ Y x Y is projection to the last two factors and S12:

Y x Y x Y -+ Y x Y x Y interchanges the first two factors. Since the triangle on the right side commutes, the right vertical arrow is the identity. Using this and

152 J. Franke

definition (36), we see that the commutativity of the square implies the desired

equality ({J(JX(J,(J,Pl = ({J(JX(J,(J'

Proof of Sub lemma 2. By a special choice of factorizations in the application of 4.11 to the composition X'.4 X ..!.. Y (namely, A = Y' and S = Y), (37) becomes

Y' = Y' Xy Y

/~ X Xy Y' = X' Y

/PX/i~ X' ) X ) Y,

and (49) follows from definition (38).

4.13. Our previous considerations in 4.8-4.12 prove the existence of functors fl with the properties required in 4.7.1-4.7.3. It remains to prove the uniqueness assertion 4.7.4. Let e be another collection of functors satisfying the same conditions. We proceed in several steps:

4.13.1. Let i be a regular closed immersion of codimension of codimension d, defining an object X 0 ..!.. Xl of K d ; we denote this object by X .. Then both i 1 and f satisfy the conditions for F x. in 4.3. By the uniqueness result in Proposition 4.3,

there is a unique biadmissible functor-isomorphism i 1 --+ f which respects the datum 4.7.2. By 3.14, this isomorphism automatically respects 4.7.1. Since it was mentioned in 4.8 that the composition law i1f --+ (ij)1 is the unique one which is biadmissible and respects the datum 4.7.2, it follows that our isomorphism i 1 --+ f respects the composition law 4.7.3. Since the co dimension of a regular immersion is locally constant, we get the isomorphism i 1 --+ f for any regular closed immersion i. If in the next considerations an isomorphism i 1 --+ f is used without any comment, it is supposed to be the isomorphism constructed here.

4.13.2. Let

be a commutative diagram in which p is smooth and i is a regular closed immersion. We want to prove that the transformation

(51)

Chow categories 153

must be (21). Since confusions are impossible, the transformations 4.7.1-3 for f! and ( are denoted by the same letters y, 0, 6. In the following diagram

:~l: rr'(:1 fp? ~( --- J?J?p? ~T ,'IP!I\

~\ .'1 r

All notations are the same as in (22)! The commutativity of the pentagon (A) is (20), commutativity of (B) follows

from 3.14, and the commutativity of (C) follows from 4.7.3.2. If we identify k? with k! by 4.13.1, 6j,pYj~I~ becomes (51) and YJ,p(b)6J,i is (21), which proves our claim.

4.13.3. Let/: X --+ Y be a slci-morphism with a factorization 0": X ~ Z -4 Y into a smooth projection and a regular immersion. We define ra: t --+ r' by

(52)

By the result of 4.13.2· and the d~finition of CfJura,r in 4.10, if~: xLi 1. Y is another factorization of/and r: Z --+ Z is smooth such that ri = i and pr = p, then

commutes. By definition (36), this implies CfJa"a2r a, = r a2 for any two factorizations 0"1 and 0"2 off Consequently, (52) defines an unique isomorphism (--+r'. This isomorphism is biadmissible over 1K1ci since its definition contains only biadmissible transformations. In the case of a closed immersion we obtain the isomorphism constructed in 4.13.1. In paragraph 4.13.4, any isomorphism ( --+ f! will be the isomorphism constructed here.

4.13.4. It remains to prove that the isomorphism ( --+ f! is compatible with the isomorphisms 4.7.1-4.7.3. For 4.7.1, this follows from 3.14. To prove com-

154 J. Franke

patibility with the isomorphisms cp.q' i.e., with 4.7.3, it suffices by 4.7.3.2 to consider the following four cases:

(a) p and q are smooth. Then the assertion follows from 3.14.

(f3) p and q are regular immersions. Then the compatibility follows from the considerations in 4.13.1.

(y) p = i is a regular closed immersion, and q is smooth. Then the assertion is a trivial consequence of definition (52).

(0") q = i: Y ----> Z is a regular closed immersion, A.4 Z is smooth, X = Ay = A X z Y, and p: X ----> Y is the natural projection:

A i'. j,

z

Case (l5) is a consequence of the following diagram:

P*i~P"'d ~,.,

pT

NT ! ,.?

pT /

~~';'

flat base change

(A)

ep,i ) (ip)! = (p'i')! ( ei"p'

(?) j (C)

ep,i ) (ipf = (p'i'f ( ei',p'

(D)

(E) Yp,ld

p*i?------------------------------» i'?p'* flat base change

The outer contour commutes by the axiomatic description of i ~ f in 4.13.1 (biadmissibility). (A) and (D) are (20), (B) and (E) commute by 3.14, applied to p and p'. By case (y), (C) commutes. It follows that (?) commutes, which is the desired result. The proof of case (<5) is complete.

It remains to prove the compatibility off! ----> f? with the datum 4.7.2, i.e., with the isomorphisms <5;.,J. Iff is smooth, this follows from 4.7.2.2. Iff is a regular

Chow categories 155

immersion, this is among the conditions characterizing the isomorphism defined in 4.13.1. By 4.7.3.1, the general case follows from these two cases since it has already been proved that our isomorphism commutes with Sp,q'

4.14. An alternative description ofi!A

Let Z c X be a closed regular submanifold of a regular quasi-projective manifold X over an infinite field F. Since in this case Gersten's conjecture is known to be true, we have a homomorphism i*: E~,q(X) ~ E~,q(Z) (cf. 1.8.). For A E Ob(CHk(X)) = Ob(CHk(X)) and U c Z(k) open we put

(53)

where:

- The triple (g, V, a) consists of 9 E Gk(U), an open subset V c X Zar such that Vn Z = U, and aEA(V).

- Two triples (g, V, a) and (g', V', a') are equivalent if and only if i*(a' - a) = 9 - g', where a - a' is defined as an element of Gk(V n V').

It is clear that (53) defines a separated presheaf on Z(k)' By the moving lemma, every point Z E Z(k) has a neighbourhood U in Z(k) such that (i !,O A)(U) is not empty. The group Gk(U) acts on (i !,O A)(U) by y: (g, V, a) ~ (y + g, V, a), and it acts simply transitive if (i!,OA)(U) is not empty. For aEA(V), we denote the class of (0, V, a) in (53) by i!(a).

Let i!A be the sheaf associated to the presheaf i!'O A on Z(k)' From the previous remarks it is clear that i!A is an object ofCHk(Z). We want to prove that this new functor i! is canonically isomorphic to the old one constructed in Theorem 4.7. A careful examination of the proof of 4.3 shows that in 4.5 and 4.6 we do not leave the category of regular manifolds over a field if the closed immersions we consider belong to this category. Therefore it suffices to equip the functor i!

defined by (53) with the data 4.7.1-4.7.3:

4.14.1. Let

be a commutative diagram of regular manifolds over F, with a closed immersion i and flat morphisms p and q. If W is open in l(k)' then U = p -1(W) and V = q-l(W) are open in Z(k) and X(k) and satisfy the assumptions of (53). There is

156 J. Franke

a unique isomorphism y: P* A --+ i !q* A with the property that for every a e A(W) we have y(p*(a)) = i!(q*(a)).

4.14.2. Let i: Z --+ X be a closed immersion of codimension d between regular manifolds, D c X a regular submanifold of codimension one such that Z 11 D is also regular and of codimension one in Z, A. a section of (f)x in some neighbourhood of D which trivializes the sheaf of ideals defining D.

Let xeD and A e Ob(CHk(X - D)). Since the restriction CHk(X)--+ CHk(X - D) is surjective, it is possible to extend A to an object of CHk(X). Therefore the moving lemma implies the existence of an open V c X - D such that A(V) is not empty and K = (closure of X - D - V in X) meets Z 11 D in codimension ~k and x¢K. Let U=Z-D-(KI1Z), W=D-(KI1D), Y= Z 11 D - (K 11 Z 11 D). If aeA(V), then

spia) e (sp;.A)(W) ib(spia)) e (ibsp;.A)(Y) i!(a)e(i!'O A)(U) c (i!A)(U) sp;.lz(i!(a))e(sp;.lzi!A)(y),

where in is the inclusion Z 11 D --+ D. There exists a unique isomorphism 15: ibsp;.A --+ sp;.lzi!A with the property

(54)

Indeed, it follows from Proposition 1.8 that the Gk(Y)-equivariant isomorphism t5:(ibsp;.A)(Y)--+(sp;.lzi!A)(Y) characterized by (54) is independent of aeA(V). Consequently, definition (54) is correct.

4.14.3. If j: Y --+ Z is another embedding satisfying our assumptions, then Bi,i: j !i ! A --+ (ij)! A is characterized by j !i!( a) --+ (ij)!( a).

It is easy to see that the isomorphisms 4.14.1-3 satisfy all assumptions of Theorem 4.7. If we replace the word 'slci-morphism' by 'locally closed embedding of regular manifolds over a field'. Consequently, (53) is an alternative description of the functors i! defined in 4.7. We shall use this description in a forthcoming paper when metrics on objects of CHk are investigated.

5. Intersection products. Bloch's biextension

Here we restrict our considerations to the case of regular manifolds over a field F. For the sake of simplicity we shall also assume that these manifolds are connected. By working with methods similar to [F, §20.2], we could also deal with the case of smooth schemes over a Dedekind domain. The smoothness assumption is absolutely essential only for 5.3.

Chow categories 157

5.1. The cross product

Let X and Y be manifolds and P1,2 the projections of X x Y to X and Y. The biexact functor

(1) (A; B) --~) p!A®p!B

defines a product

(2)

which satisfies

(3)

Because of Ef+ 1, - P = 0, (3) implies that the following product is well-defined:

x: (Ef- 1, - P(X)/d1(Ef- 2, - P(X)) x E1' -q(y)

--+Ef+q- 1,-p-q(X x Y)/im(d 1).

In our terminology, this is a pairing

(3) implies:

c(g x z) = c(g) x z.

In a similar manner, we get

with the property

c(z x g) = z x c(g).

(4)

(5)

(6)

(7)

For A EOb(CHk(X)), BEOb(CH1(Y)), we define A ~ BE Ob(CHk+ l(X X Y)) as follows:

(A ~ B)r(X x Y) = {(g, a, b)}/~, (8)

158 J. Franke

where:

- The entries of the triple are 9 E (Gk + 1).(X x Y), a E Ar(X), bE Br( Y). - (g, a, b) ~ (g', a', b') if and only if

g' - 9 = c(a) x (b - b') + (a - a') x c(b')

= c(a') x (b - b') + (a - a') x c(b). (9)

That the two expressions on the right hand side of (9) agree is a consequence of the identity

c(g) x h = 9 x c(h) in (Gk+ 1)r(X X y) (10)

if gE(Gk)r(X), hE(Gdr(Y)' To prove (to), we choose representatives ')IEE~-1.-k(X), I1EEi-1,-I(y) for 9 and h. Then a representative for c(g) x h - 9 xc(h) is

by (3). This proves (10) and completes the explanation of (8). We define c: (A ~ B)r(X x Y) -+ Zk+1(X X Y) by

c«g, a, b)mod~) = c(g) + c(a) x c(b). (11)

By (9), (5), and (7), this is independent of the choice of the representative. Finally we define (A ~ B)(U), U c (X X Y)(k+ 1) open, by formula 3(4):

(A ~ B)(U) = {aE(A [g] B)r(U)Ic(a)lu = O}. (12)

For a E Ar(X), bE Br(Y) we denote the equivalence class of (0, a, b) in (8) by a x b.

Then c(a x b) = c(a) x c(b). If U c X(k)' V c Y(1) are open and a E A(U), bE B(V), then ax bE(A ~ B)(X x Y - (X - U) x(Y - V)).

5.2. Natural transformations involving ~

We give a mere list of them without going into detail.

5.2.1. 'Biadditivity. For A, BE Ob(CHk(X)), C, DE Ob(CH1(y)), we have isomorphisms

A ~ (C EEl D) -+ A I2SI C EEl A I2SI D

(A EEl B) ~ C -+ A [g] C EEl B ~ C

(13)

(14)

Chow categories 159

which satisfy the additivity condition [DM, 1.8] in each of the two variables and make the diagram

(A E8 B) ~ (C E8 D) A ~ (C E8 D) E8 B ~ (C EB D)

I I (\5)

(A E8 B) ~ C EB (A E8 B) ~ D -----+) A ~ C E8 A ~ D E8 B ~ C E8 B ~ D

commutative. If a, b, c, d are rational sections of A, B, C, D, then the isomorphisms (13) and (14) send a x (c E8 d) to a x c E8 a x d and (a E8 b) x c to a x c EB b x c.

5.2.2. Symmetry. Let s: X x Y ---+ Yx X be the permutation of the factors. Since s interchanges the two expressions on the right hand side of (9), there is a canonical isomorphism s*(A ~ B) ---+ B ~ A sending s*(a x b) to b x a.

5.2.3. Compatibility with pull-back. If f: Z -+ X is flat, A E Ob(CHk(X)), BEOb(CH1(y)), then there is an isomorphism (f*A)~B-+(fxIdy)*(A~B) which maps f*(a) x b to (f x Idy)*(a x b).

Similarly, if i: Z -+ X is a closed immersion and if X and Yare quasiprojective, there is an isomorphism (i'A) ~ B -+ (i x Idy)'(A ~ B) which maps ilea) x b to (i x Idy)'(a x b) if a E Ar(X) is a rational section whose cycle meets Z properly (cf. 4.14).

Finally, if X, Y, and Z are quasi-projective regular manifolds, we may factor g: Z -+ X into Z ~ S ..!. Y, with a closed immersion i and f smooth and quasiprojective. We get an isomorphism

(g'A) ~ B -+ (g x Idy) '(A ~ B)

by composing

g'A ~ B -+i'f* A ~ B -+ (i x Idy)'(f* A ~ B)

-+(i x Idy)'(f x Idy)*(A ~ B) -+ (g x Idy) '(a ~ B).

It is easy to see that this isomorphism does not depend on the factorization of g.

5.2.4. Compatibility with push-forward. Iff: Z -+ X is proper, A E Ob(CHk(Z)), BE Ob(CH1(y), there is an isomorphism

which maps f*(a) x b to (f x Idy)*(a x b). There are some obvious compatibilities between the isomorphisms 5.2.1-4.

We do not list them explicitly because they are not essential for our approach to Deligne's program.

160 J. Franke

5.3. The intersection product. Let X be a regular manifold, A E Ob(CHk(X», BE Ob(CH1(X». By ~: X -+ X x X we denote the diagonal. This is a regular closed immersion. We put

Au B = A!(A 129 B)EOb(CHk+l(X». (16)

If a and b are rational sections of A and B, we put au b = ~!(a x b). This is a rational section of A u B if the supports of c(a) and c(b) intersect properly. If a E A(U), bE B(V), then au bE (A u B)(U u V).

The following isomorphisms are derived from 5.2:

5.3.1. Biadditivity. Isomorphisms Au (C EB D) -+ A u C EB A u D and (A EB B) u C -+ A u C EB B u C. They are additive in each variable and satisfy the analogue of (15).

5.3.2. Symmetry. An isomorphism s: A u B EB B u A defined by composing

Au B = A !(A 129 B) = (sA)!(A ~ B) -+ A !s*(A 129 B) -+ A !(B 129 A) = B u A.

The symmetry s: A u A -+ A u A may be different from the identity!

5.3.3. An associativity law and compatibilities with pull-back and pushforward. We do not define them explicitly because we shall not need them later.

5.4. Example

Let p: X -+ S be a proper morphism of relative dimension one between regular manifolds over a field. To line bundles L, M on X, Deligne associates the line bundle <L, M) on S which is Zariski-locally on S generated by sections <1, m), where 1 and m are rational sections of Land M whose divisors do not intersect. The relations

<gl, m) = g(div(m»)(l, m); <1, gm) = g(div(l)(I, m)

are satisfied. Cf. [D, 6.1].

Since CH1(X) is the category of line bundles on X, we can consider the intersection product L u ME Ob(CH2(X». There is a canonical isomorphism

<L, M) -+ p*(L u M)

<1, m) -+ p*(1 u m).

Chow categories 161

5.5. Bloch's biextension

By the identification CHP(X) = HP(X, % p) and Cq(X) = Hq-l(X, % q), there is a product CHP(X) x Gp(X) --+ Gp+q(X). Following the methods of [Gr], it is easy to show that this product coincides up to sign with the product defined by the biadditive functor u. More precisely, if g E Gq(X) is viewed as an automorphism of BE Ob(CHq(X)), then for any A E Ob(CHP(x)) the automorphism IdA u g of Au B is given by [A]' g E Gp+iX), where [A] E CHP(X) is the class of A.

If f: X --+ S is a proper morphism of relative dimension n between regular manifolds over a field, we put

CHP(X/S)V = {aECHP(X)1 For every Zariski-open U c S and every gEGn+1_p(f-1(U)), f*(a'g) = 0 in G1(U) (i.e., = 1 in (D( U)*)}. (17)

Let p + q = n + 1, a E CHP(X/S)", bE CHq(X/S)v. We choose representatives A EOb(CHP(X)), BEOb(CHq(X)) for a and b. We have a line bundle

(18)

on S. If g E G p(X) is any automorphism of A, then f*(g u IdB) = f*([B] . g) = 1. Consequently, if A' is another representative for A, then all isomorphisms A --+ A' define the same isomorphism L A •B --+ LA'.B' Since the same is true for representatives of b, the LA,B can be identified with one line bundle La,b'

By the biadditivity of our intersection product, we get isomorphisms

(19)

satisfying the axioms of [SGA 7.1, Exp. VII, 2.1]. For instance, it follows from the analogue of (15) for the functor u that the isomorphisms (19) satisfy [SGA 7.1, Exp. VII, (2.1.1)].

If C(jYfP(X/S)V is the sheaf on SZar associated to the presheaf U--+ CHP(f-1(U)/U)V, we get a biextension of C(jYfP(X/S)V X C(jYfn+ 1-P(X/S)" by (Dt.

This biextension is unique up to unique isomorphism of biextensions. Using the crucial lemma 1 in Bloch's paper [B], it is easy to see that

CHP(X? S)V is contained in the group of cycles which are homologically equivalent to zero on each geometric fibre (i.e. of cycles z on X such that for every geometric point s of S and every prime 1 prime to the characteristic of k(s), the fundamental class of z in H 2P(Xs ' d'1(P)) vanishes). If Bloch's biextension is

162 J. Franke

also defined (i.e., if S and X are smooth and quasi-projective over a field), then there exists a canonical isomorphism between Bloch's biextension and the biextension constructed here.

References

[B] [D]

[DM]

[F] [FL] [Frl]

[Fr2] [G]

[GN]

[Gr]

[H] [M] [Q] [S] [SGA6]

[SGA 71]

[W]

S. Bloch, Cyles and Biextensions, Contemporary Mathematics vol. 83, 1989, p. 19-30. P. Deligne, Le determinant de la cohomology, Contemporary Mathematics 67 (1987), p. 93-177. P. Deligne and J. Milne, Tannakian Categories, Lecture Notes in Mathematics 900, p. 101-228. W. Fulton, Intersection Theory, Springer-Verlag, 1984. W. Fulton and S. Lang, Riemann-Roch Algebra, Springer-Verlag, 1985. J. Franke, Chow categories, Proceedings of the 28. Mathematische Arbeitstagung, Bonn (MPI) 1988. J. Franke, Chow categories, Manuscript, June 1988. H. Gillet, Riemann-Roch theorems for higher algebraic K-theory, Advances in Math. 40 (1981),203-289. C. Giffen and A. Neeman, K-theory of triangulated categories, Preprint, University of Virginia. D. Grayson, Products in K-theory and intersection algebraic cycles, Inv. Math. 17 (1978) 71-83. R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977. J. Milne, Etale cohomology, Princeton University Press, 1980. D. Quillen, Higher algebraic K-theory I, Lecture Notes in Math. No. 341, p. 85-147. C. Soule, Operations en K-theorie algebrique, Can. J. Math., 37: 3 (1985),488-550. P. Berthelot; A. Grothendieck and L. Illusie, Theorie des Intersections et Theon!me de Riemann-Roch, Lecture Notes in Mathematics 225. A. Grothendieck, Groupes de Monodromie en Geometrie Algebrique, Lecture Notes in Mathematics 288. F. Waldhausen, Algebraic K-theory of generalized free products, Part 1, Annals of Mathematics.

Compositio M athematica 76: 163 -196, 1990. © 1990 Kluwer Academic Publishers.

Complex analytic compactifications of C3

MIKIO FURUSHIMA Kumamoto National College of Technology, Nishi-goshi-machi, Kiguchi-gun, Kumamoto, 861-11, Japan; Current address: Department of Mathematics, Ryukyu University, Nishihara-cho, Okinawa, 903-11, Japan)

Received 5 September 1988; accepted in revised form 28 August 1989

Introduction

The purpose of this paper is to give a proof of the announcement [7]. Let X be an n-dimensional compact connected complex manifold and Y an

analytic subset of X. We call the pair (X, Y) a complex analytic compactification of en if X - Y is biholomorphic to en. By a theorem of Hartogs, Y is a divisor onX.

In this paper, we will consider only the case of n = 3. Let (X, Y) be a complex analytic compactification of e3 . Assume that Y has at most isolated singularities. Then Y is normal. Thus, by Peternell-Schneider [18] (cf. Brenton [2]), X is projective. In particular, X is a Fano 3-fold of index r(l ~ r ~ 4) with b2 (X) = 1. In the case of r ~ 2, such a (X, Y) is completely determined (cf. [3], [5], [6], [18]). In the case of r = 1, by a detailed analysis of the singularities of the boundary Y, we can prove that such a compactification (X, Y) does not exist. Thus, we have:

THEOREM. Let (X, Y) be a complex analytic compactijication of e3 . Assume that Y has at most isolated singularities. Then X is a Fano 3-fold of index r(2 ~ r ~ 4) with b2 (X) = 1, and

(1) r = 4 => (X, Y) ~ (!P 3 , !P2 ),

(2) r = 3 => (X, Y) ~ ((iJ3, (iJ6), where (iJ3 is a smooth quadric hypersurface in

!p4 and (iJ6 is a quadric cone in !p3, (3) r = 2 => (X, Y) ~ (VS, H s ), where Vs is a complete intersection of three

hyperplanes in the Grassmannian G(2, 5) ~ !p9 , and H 5 is a normal

hyperplane section of Vs with exactly one rational double point of A4-type.

This paper consists of five sections. In Section 1, we will prove that (X, Y) ~ (V22, H 22 ) if such a (X, Y) exists in the case of r = 1, where V22 ~ !PH°(V22 , (1)( -KV22 )) ~ !p13 is a Fano 3-fold of degree 22 in !p13 (index 1, genus 12) and H 22 is a normal hyperplane section which is rational (Proposition 1.13). In Section 2, we will determine the singularities of Y = H22 (Proposition 2.5). In Sections 3 and 4, we will prove that such a (X, Y) = (V22, H 22 ) does

164 Mikio Furushima

not exist. In Section 5, we will refer to a recent work of Peternell-Schneider [18] (c.f. Peternell [19]) on a projective compactification (X, Y) of C3 with b2 (X) = 1 (especially, the case where the boundary Y is non-normal), and prove that there is a compactification (X, Y) of C 3 with a non-normal boundary Y in the case of the index r = 1.

Notations

a canonical divisor on a projective manifold M. the ith Betti number of M.

• N qM : the normal bundle of C in M. • Cd~M): the first Chern class of a locally free sheaf ~M on M. • m((Dy,J: the multiplicity of the local ring (Dy,x at x.

1. The structure in the case of r = 1

1. Let (X, Y) be an analytic compactification of C 3 such that Y has at most isolated singularities. Assume that the index r = 1. Then X is a Fano 3-fold of index 1 with Pic X ~ /fc i (Dx(Y) ([3], [9], [18]). Then, by Proposition 1, Proposition 2 and Proposition 3 in [3], we have:

LEMMA 1.1. (1) Ky = 0, (2) HI (Y, (Dy) = 0, H2(y, (Dy) ~ C, (3) HI(Y;/f) = 0,H2(Y;/f) ~ /fcdNYlx),

Let Sing Y be the singular locus of Y and put S:= { Y E Sing Y; y is not a rational singularity}. Let n: Y -+ Y be the minimal resolution of singularities of Y and Z be the fundamental cycle of S associated with the resolution (Y, n).

We put E:= n-I(Sing Y), C:= n-I(S) = Ur= 1 C i (C;'s are irreducible).

LEMMA 1.2. S #- <p Proof Let us consider the following exact sequence (see [2]):

-+ HI(y; IR) -+ HI(y; IR) -+ HI(E; IR) -+ H2(y; IR)-+

-+ H2(y; IR) -+ H2(E; IR) -+ 0. (1.1)

Assume that S = <p. Then we have Kr = (Dr and HI(E; IR) = 0. By Lemma 1.1(3), HI(y; IR) = 0. Thus Y is a K-3 surface. Since b2 (Y) = 1 and Y is projective, we have b + (Y) = 1. On the other hand, by Brenton [2], b + (Y) = b + (Y). Thus we have b + (Y) = 1. This is a contradiction. Therefore S #- <p. Q.E.D.

Complex analytic compactijications of C 3 165

Thus, by Umezu [22], we have:

LEMMA U. (1) K y = -~~=l niCi(ni>O, niEZ), and thus iT is a ruled surface over a non

singular compact algebraic curve R of genus q = dim HI(y; (!Jr)(namely, Y is birationally equivalent to a !pI-bundle over R),

(2) if q =1= 1, then S consists of one point with pg:= dim(R1n* (!Jr). = q + 1, (3) if q = 1, then S consists of either one point with pg = 2 or two points with

Pg = 1. Moreover, in the second case of (3), both of the two points are simply elliptic.

LEMMA 1.3. S consists of one point with Pg = q + 1 and b2 (y) = b2 (E) + 1. Proof. Assume that S consists of two points. By Lemma U(3), these two points

are simply elliptic, and C = n-I(S) = C 1 U C2 , where C 1, C2 are distinct

sections of Y. Since b2 (y) = 1, by (1.1), we have bl(Y) = bl(E). Since

we have a contradiction. Therefore S consists of one point with Pg = q + 1. Since

bl(Y) = bdE) and b2 (y) = 1, we have b2 (Y) = b2 (Y) + b2 (E) = 1 + b2 (E). Q.E.D.

2. Let U be a strongly pseudoconvex neighborhood of C in Y. A cycle D on U is an integral combination of the C;, D = ~di C i (1 ~ i ~ s), with di E Z. We denote the support of D by I D 1= U C;, di =1= O. We put {!JD = (!Ju/{!Ju( -D). Let Ku be a canonical divisor on U. We put X(D):= dim HO(U, (!JD) - dim HI (U, (!JD)' Then, by the Riemann-Roch theorem [21],

X(D) = - H(DoD) + (DoKu)}. (1.2)

For two cycles A, B, we have, by (1.2),

X(A + B) = X(A) + X(B) - (A 0 B). (1.3)

LEMMA 1.4. (1) q = 0 => Y is a rational surface, and Kr = Ku = - z. (2) q =1= 0 => there is an irreducible component Cit of C such that Cit is a section of

Yand the rest C - Cit = Ui .. it Ci( =1= cjJ) is contained in the singular fibers ofY, and -Kr = Z + Cit.

Proof. (1) Since q = 0, we have pg = 1. Thus S consists of a minimally elliptic singularity. By Laufer [11], we have Kr = Ku = -z.

166 Mikio Furushima

(2) Since K y = - -r.t = 1 nj Cj (nj > 0, nj E Z), for a general fiber f of Y, we have

s 2 = (-Kyof) = L nj(Cjof).

j= 1

Thus we have the following: (i) there is an irreducible component Ch of C such that nh = 2, (Ch of) = 2

and (Cjof) = 0 (i =F id, (ii) there are two irreducible components C b C2 of C such that nl = n2 = 1,

(Cjof) = 1 (i = 1,2), (Cjof) = 0 (i ~ 3), and (iii) there is an irreducible component Cl of C such that nl = 1, (Cl of) = 2,

(Cjof) = 0 (i =F 1).

Claim 1. The case (ii) can not occur. Indeed, by the adjunction formula, the curve Cj (i = 1,2) is a non-singular

elliptic curve with (Clo C 2 ) = 0 and there is no other irreducible component of C which intersects Cj (i = 1,2). Thus C = Cl U C2 (Cl (") C2 = cf», namely, S consists of two points. This contradicts Corollary 1.4.

Claim 2. The case (iii) can not occur. Indeed, by the adjunction formula, Cl is a non-singular elliptic curve and there

is no other irreducible component of C which intersects Cl. By Corollary 1.4, we have C = Cb hence, Ky = - Cl. This contradicts Lemma 1 and Lemma 2 in Umezu [22].

Thus we have the case (i). In particular, Ch is a section of the ruled surface Yand C/s (i =F i l ) are all contained singular fibers of Y. We also have C - Ch =F cf>

by the same reason as above. Since nh = 2, we have -Ky = 2Ch + -r. j" jl njCj

(nj > 0). We remark that the genus of Ch is equal to q = hl«(9y) ~ 1.

Claim 3. -Ky = Ch + z. Indeed, since (- K y - Cjl ) 0 Cj ~ 0 (1 ~ i ~ s), by definition of the funda

mental cycle, -Ky - Cjl ~ Z. Now, assume that -Ky = Ch + Z + D, where D > O. For a general fiber f of Y, we have 2 = (-Kyof) = (Ch of) + (Z of) + (D of). Since Cjl C 1 Z I, we have (Ch of) = (Z of) = 1, and (D of) = O. This means that the support 1 D 1 is contained in the singular fibers of Y. Since

by the Riemann-Roch theorem, we have

o ~ - dimHl(Y,(9y(-Z)) = t(ZoZ + ZoKy) + 1- q.

Since KM=Ku, by (1.2), we have X(Z)~l-q. Since HO(U; (9z)~1C (cf. [11,

Complex analytic compactijications ojC3 167

p. 1260J), X(Z) = 1 - dim Hl(U, (9z) ::s;; 1. Since S does not consists of a rational singularity, X(Z) "* 1 by Artin [1J. Thus we have

(1.4)

Since 1 - q = X(C;.) = X(-Ku - C;.) = X(Z + D) = X(Z) + X(D) - (DoZ), we have

x(Z) = -X(D) + 1 - q + (DoZ). (1.5)

Since (D 0 Z) ::s;; 0, by (1.4), (1.5), we have X(D) ::s;; O. On the other hand, the support I D I is contained in the singular fibers of Y. Thus, the contraction of I D I in Y yields rational singularities. Hence X(D) ~ 1. This is a contradiction. Therefore D = 0, namely, -Ky = Z + C i ,. Q.E.D.

COROLLARY 1.5. Assume that q "* O. Then (1) (C;. oZ) = 2 - 2q. (2) (Z 0 Z) ::s;; (C;. 0 C;.).

Proof Since Ky = - Z - Cip by the adjunction formula, we have (1) and (2).

LEMMA 1.6.

(1) q"* 0=>b2 (Y)::S;; 9 - 4q +)9 + Sq. (2) q = O=> 11 ::s;; b2 (Y)::S;; 13.

Proof (1) By the Noether formula, we have

Since Ky = - Z - Cip we have

By (1.6), (1.7) and Corollary 1.5, we have

b2 (Y) = 6 - 4q - (ZoZ) - (C;. oC;.).

::s;; 6 - 4q - 2(ZoZ).

Since S = {one point} is a hypersurface singularity, we have

-(ZoZ)::S;; n:=m({9y,s) (Wagreich [23J).

Q.E.D.

(1.6)

(1.7)

(1.S)

(1.9)

(1.10)

168 Mikio Furushima

Pg ~ t{n - l)(n - 2) (Yau [24]). (1.11)

Since Pg = q + 1, by (1.10), (1.11), we have

(1.12)

By (1.9), (1.12), we have the claim. (2) By the Noether formula, we have

(1.13)

Since Pg = 1 and S is a hypersurface singularity, by Laufer [11], we have - 3 ~ (Z· Z) ~ - 1. By Lemma 1.4(1) and (1.13), we have the claim. Q.E.D.

COROLLARY 1.7. 0 ~ q ~ 3. Proof Assume that q #- O. By Lemma 1.6(1), we have

2 ~ b2 (Y) ~ 9 - 4q + )9 + 8q. This implies q ~ 3. Q.E.D.

3. By the classification of Fano 3-Folds with the second Betti numbers one due to Iskovskih [9] (see also Mukai [15], [16]), we have:

(Table 1)

g 2 3 4 5 6 7 8 9 10 12

!b3(X) 52 30 20 14 10 7 5 3 2 0

where g:= t(K~) + 1 = t(y3) + 1. Since 2q = b1 (Y) = b3 (Y) = b3 (Y) = b3 (X) (cf. [2]), by Corollary 1.7, 0 ~

tb 3 (X) ~ 3. Thus, by the Table 1 above, we have (g, q) = (9,3), (10,2) or (12,0).

LEMMA 1.8. q #- 3. Proof Assume that q = 3. By Lemma 1.6(1), we have 2 ~ b2 (Y) ~ - 3 +

J33 < 3, namely, b2 (Y) = 2. Hence, Y is a OJ>l-bundle over a smooth compact algebraic curve R of genus 3. Therefore, Y is a cone over R. This is a contradiction, by Table (I) in [3]. Q.E.D.

LEMMA 1.9. Assume that q #- O. Then there is exactly one exceptional curve of the first kind in every singular fiber of the ruled surface Y, and the other irreducible components of the singular fiber are all contained in E := 7t - 1 (Sing Y).

Proof Since q #- 0, the rest E - C il must be contained in the singular fibers

Complex analytic compactijications of C 3 169

of f. Let F 1, ... , F, be the singular fibers of f, 1 + (Xi ((Xi> 0) the "number" of the irreducible components of Fi and bi the "number" of the irreducible components of Fi which are not contained in E. Then we have

{I,+ t ~ b,(l') ~ 2 + ,t, .,. L (1 + (Xi - bi) + 1 = t. i= 1

Thus we have ~~= 1 (1 - bi) = O. Since each singular fiber Fi contains at least an exceptional curve of the first kind, we have bi ;?; 1 for 1 ~ i ~ r, hence, bi = 1 for 1 ~ i ~ r. Q.E.D.

LEMMA 1.10. Assume that q = 2. Then the dual graphs of all the exceptional curves in f look like the Figure 1 below.

Proof By Lemma 1.6(1), we have 2 ~ b2(f) ~ 6. Since Y is not a cone (see Table 1 in [3], we have b2 (f) :1= 2. If b2 (y) = 3, then f contains two exceptional curves ofthe first kind in a singular fiber. This contradicts Lemma 1.9. Hence we have 4 ~ b2(f) ~ 6. Thus, by (1.9), (1.12), we have -4 ~ (Z·Z) ~ -3. We put n := (C;. • CiJ < O. Then, by Lemma 1.3 and (1.8), we have

(i) b2(Y)=6~(n,t)=(-4,5) and (Z·Z) = -4. (ii) b2(f)=5~(n,t)=(-3,4) and (Z·Z) = -4.

(iii) b2(f)=4~(n,t)=(-3,3) and (Z·Z) = -3, = ( - 2, 3) and (Z· Z) = - 4.

Thus, we have (a) (Z· Z) = -4 ~ (n, t) = (-2,3), (-3,4), (-4,5). (b) (Z·Z) = -3 ~(n, t) = (-3,3).

Since Sing Y - S consists of rational double points, by Lemma 1.9 and (a), (b), the configuration of the exceptional curves of f can be easily described. Thus we have the lemma. Q.E.D.

Notation. The vertex !II represents a non-singular compact algebraic curve of genus 2 with the self-intersection number - k, (which is corresponding to the section C;. of f), ® a non-singular rational curve with the self-intersection number -k. We denote 0 simply by O. Adjacent to the graph, we write a basis {ei} (0 ~ i ~ t) of H2(f:Z), where t = dimH2(E; ~).

LEMMA 1.11. Assume that q = 2. Then there is a canonical curve D of genus 10 and deg D = 18 such that

(i) Sing Y (1 D = </>, (ii) (1Jy(Y) = (1Jy(D).

170 Mikio Furushima

Proof Since q = 2, by Table 1, X is a Fano 3-fold of degree 18 in [p>11 and Y is a hyperplane section of X (see [9]). For a sufficiently general hyperplane section H, we put D = HoY, which is desired. Q.E.D.

4. We put 15 := n -1 (D) ~ Y. Since D n Sing Y = ¢ by Lemma 1.11, we have 15 ~ D (isomorphism), (15 0 15) = 18 and (15 oEj) = 0 for each irreducible component Ej of E = n- 1 (Sing Y).

Let {ei}(O ~ i ~ t) be a basis of H2(y; Z) ~ zt+ 1 (see Fig. 1). Then, we have

t

C1((()(15)) = L lXiei (lXiE Z), i~O

where C1 ((()(15)) E H2(y:Z) is the first Chern class of (()(15), and (i) the intersection number eiOej is determined by the graph in Fig. 1,

(ii) C1((()(15))oei = 0 (0 ~ i ~ t), (iii) cd(()(15))oC1((()(15)) = 18, (iv) dio := cd (()(15)) 0 eio #- 0, where eio is a class corresponding to the exceptional

curve of the first kind.

(I) -0 (k~2,3)

(2)

(3)

(4)

Complex analytic compactifications of 1[3 171

(5)

(6)

(7)

(8)

e,

(Fig. 1).

By (*) and (i)--(iv) above, for each graph in the Fig. 1, we have the equations of !Xi (0 ::;; i ::;; t) and dio over 7L below:

Case (1) !Xl - k!Xo =0

!X3 - 2!Xl + !XO =0

!X3 - 2!X2 =0 (C-l)

!X2 - !X3 + !Xl = d3

!X3 • d3 = 18

.'. !X6 = 36/2k - 1 (k = 2,3). Hence !xo ¢ 7L.

172 Mikio Furushima

Case (2).

OC1 - 3oco =0

OC4 - 3OC1 + OCo =0

OC3 - 2OC2 =0

OC4 - 2OC3 + OC2 =0 (C-2)

OC1 - OC4 + OC3 = d4

oc4·dS = 18

••• oc5 = 247• Hence OCo ¢ Z.

Case (3).

OC1 - 3oco =0

OC3 - 2OC1 + OCo =0

OC3 - 2oco =0

OC4 + OC1 - 2OC3 + OC2 =0 (C-3)

OC3 - OC4 = d4

ocs·ds = 18

••• oc5 = 8. Hence OCo ¢ Z.

Case (4).

OC1 - 4oco =0

OCs - 4OC1 + OCo =0

OC3 - 2OC2 =0

OC4 - 2OC3 + OC2 =0 (C-4)

OCs - 2OC4 + OC3 =0

OC1 - OCs + OC4 = ds

OCs • ds = 18

••• oc5 = 2S4. Hence OCo ¢ Z.

Case (5).

OC1 - 4oco =0

OC4 - 3OC1 + OCo =0

OC3 - 2OC2 =0

OC4 - 2OC3 + OC2 =0 (C-5)

OCs + OC1 - 2OC4 + OC3 =0

OC4 - OCs = ds

ocs· ds = 18

:. oc5 = %. Hence oco¢ Z.

Complex analytic compactijications of ([;3 173

Case (6).

=0

OC3 - 2OC1 + OCo = 0

OC3 - 2OC2 = 0

OC4 + OC1 - 2OC3 + OC2 = 0 (C-6)

OCs - 2OC4 + OC3 = 0

OC4 - OCs = ds

as'ds = 18

... oc6 = 6. Hence OCo ¢ 7L.

Case (7).

OC1 - 4oco = 0

OCo - 2OC1 + OC3 = 0

OC3 - 2OC2 = 0

OCs + OC1 - 3OC3 + OC2 = 0 (C-7)

OCs - 2OC4 = 0

OC4 - OCs + OC3 = ds

ocs'ds =18

... OC6 = 136 . Hence OCo ¢ 7L.

Case (8).

OC1 - 4oco =0

OC3 - 2OC1 + OCo =0

OC3 - 2OC2 =0

OC1 - OC3 + OC2 = d3

OCs - 2OC4 =0 (C-8)

OC4 - OCs = ds

OCo + OCs =0

ocsds + oc3d3 = 18

... OC6 = 6. Hence OCo ¢ 7L.

By the computations (C-IHC-8), we find that these equations have no integral solutions. Thus, we have:

LEMMA 1.12. q =F 2.

By Lemma 1.8, Lemma 1.12, and Table 1, we have the following

PROPOSITION 1.13. (cf. [18], [19]). Assume that the index r = 1. Then,

174 Mikio Furushima

(X, Y) ~ (V22' H 22 ), where V22 is a Fano 3-fold of degree 22 in \p13(index 1, genus 12) and H 22 is a hyperplane section of V 22 which is rational.

REMARK 1.14. Among Fano 3-folds of degree 22 in \p13(index 1, genus 12), there is a special one, V22 4 \p 12 , which has been overlooked by Iskovskih [8] (see Mukai-Umemura [14]).

Recently, Mukai has succeeded in classifying Fano 3-folds of index 1 with b2 (X) = 1, applying the theory of vector bundles on K-3 surfaces (see [15], [16]).

2. Determination of the boundary

1. Let (X, Y) = (V22, H 22 ) be as in Proposition 1.13. Since q = 0, by Lemma 1.3, S consists of one point x with pg = 1, namely, x is a minimally elliptic singularity. We put Sing Y - {x} =:{Yl> ... ,Yk} (k ~ 0), and B:=1t- 1 ({Yl> ... ,Yk}). Then y/s are all rational double points.

By Lemma 1.6(2), we have:

(2.1)

(2.2)

(2.3)

2. Let To (resp. T;) be a contractible neighborhood of x (resp. Yi) in Y. We may assume that To, Ti (1 ~ i ~ k) are disjoint. We put T:= Uf=o Ti and oT:= Uf=o oT;, where OTi is the boundary of Ti. We put T*:= T - Sing Yand y* := Y - Sing Y. Since T* ~ oT (deformation retract), by the Mayer-Vietoris exact sequence, we have

-+ Hi(oT; Z) -+ Hi(Y*; Z) EB Hi(T; Z)-+

-+ Hi(Y; Z) -+ Hi- doT; Z) -+. (2.4)

Since Sing Yis isolated in Y, we have H2(Y*; Z) ~ H2(y, E;Z) ~ H2(y, SingY; Z) ~ H2(y; Z) ~ Z. On the other hand, since X = V22 is a Fano 3-fold of index 1 and the genus g = 12, we have H 3 (X; Z) = 0 (cf. [8], [15], [16]). Thus we have H3(Y; Z) ~ H3(X;Z)~ H3(X; Z) ~ 0, and H 1(Y*; Z) ~ H3(y; Z) ~ H 3(X; Z) ~O (cf. [2], [3]). Therefore we have finally the Poincare's exact sequence:

(2.5)

By Lemma 2.5 in [18], we have

Complex analytic compactijicatioils of C3 175

LEMMA 2.1 (cf. Peternell-Schneider [18]). H 1 (aT; Z) = H 1 (aTo; Z) E9 Yt' ~

Z22, where Yt' = E9~=lHl(aTi;Z), namely, we have:

(Table 2)

H1(iJTO;Z) Z22 Zll Z2 0

jf 0 Z2 Zll Z22

LEMMA 2.2. For the rational double point y j E Sing Y - {x} (1 ~ j ~ k), we have:

(Table 3)

Type of Yj An D2n(resp. D2n + 1) E6 E7 E8

H 1(iJTj ;Z) Zn+1 Z2 EB Z2(resp. Z4) Z3 Z2 0

Proof. Apply Lemma M below.

LEMMA M (Mumford [17]).

Let S be a smooth complex surface and consider a divisor C = Uf= 1 Ci (Ci : a smooth rational curve) with normal crossings. Let aT be the boundary of a tubular

neighborhood T of C in S. Then, H 1 (aT; Z) is generated by Yl,' .. , Yn with the fundamental relations:

n

L (Ci'Cj)'Yj (j= 1,2, ... ,n), (#) j= 1

where Yj is a loop in aT which goes around Cj with positive orientation.

REMARK. By Lemma M, one can easily compute the homology group H 1(aTo: Z) for each exceptional divisor C in Table L-I-Table L-9 below.

By Lemma 2.1, Lemma 2.2, and (2.1), (2.2), we have easily the following

LEMMA 2.3. (1) (Z'Z) = -1 ~Hl(aTo: Z) ~ 0, Z2 (2) (Z'Z) = -2~Hl(aTo;Z) ~ O.

LEMMA 2.4. (1) The case of (Z' Z) = -1. We have:

(i) H 1 (aTo: Z) ~ Zl1 ~ b2(C) = 1,3,9 (Table L-l) (ii) H 1 (aTo: Z) ~ Z22 ~ b2(C) = 2,10 (Table L-2)

(2) The case of (Z' Z) = - 2. We have:

(i) Hl(aTO:Z) ~ Z2~b2(C) = 1 (Table L-3) (ii) Hl(aTO;Z) ~ Zl1 ~b2(C) = 2,4, 10 (Table L-4)

(iii) H 1 (aTo; Z) ~ Z22 ~ b2 (C) = 3,11 (Table L-5)

176 Mikio Furushima

(3) The case of (Z' Z) = - 3. We have: (i) H 1 (8To: E) ~ 0 => b2 (C) = 1 (Table L-6)

(ii) H d8To; E) ~ E2 => b2 (C) = 2 (Table L-7) (iii) H1(8To;E) ~ Ell =>b2 (C) = 3,5, 11 (Table L-8) (iv) H d8To; E) ~ E22 => b2 (C) = 4, 12 (Table L-9)

Proof We will prove for the case (3)(iii). The proof for the other cases are similar. Since (Z'Z) = -3 and H1(8To;E) ~ Ell, we have b2 (B) + b2 (C) = 12 by (2.3), and :Yt':= El3i=l H1 (8Ti ; E) ~ E2 • By the Table 3 Sing Y - {x} = {A 1 -type}, {ETtype} or {A 1 -type + Es-type}, hence, b2 (B) = 1,7,9, respectively. By (2.3), we have b2 (C) = 3,5,11. Pick out the possible types of the dual graphs with b2 (C) = 3,5,11, from the Table 3 in Laufer [11], we have finally the Table L-8. We remark that there is no dual graph with b2 (C) = 4.

Q.E.D.

From Lemma 2.4 and the Table L-I-Table L-9, we have directly the following

(Table L-1)

~ Dual graph A*'A* Hl(ilTo:Z)

1 Cu -1 0

2 Tr -2, -2,-3 Z3

3 As.**** -2, -2, -2,-3 Z2 ffi Z2

4 £8,* -3 0

(Table L-2)

~ Type of x A*'A* HdilTo; Z)

5 Ta -2, -3 Z2

6 A 6 ,**** -2, -2, -2,-3 Z4

(Table L-3)

~ Type of x A*'A* Hl(ilTo; Z)

7 Cu -2 "7L 2"

(Table L-4)

.~ Dual graph A*'A* H 1(ilTo; Z)

8 Ta -2, -4 Z~2

9 Ta -3, -3 Zs

Complex analytic compactijications of C3 177

10 A6,**** -2, -2, -2,-4 lL2 E9 lL4

11 A6,**** -2, -2, -3,-3 lL2 E9 lL6

12 A6,**** -2, -3, -3,-2 ZI7

13 A*,o + A*,o + A*,o -2,-2,-2 Z~4

+ A4,**o -2,-2

14 A*,o + A*,o + E7 ,o -2,-2 Z~2

A*,o + An,**o -2, -2, -2, lLT2 if (m, n) = (1,3) 15 + Am,**o -2, -2 Zf4 if (m, n) = (2,2)

(m + n = 4)

16 A*,o + D7 ,*,o -2,-2 Z~4

17 A' 7,**.0 -2, -2 Zs

18 A2,**,o + A3,**o -2, -2, -2, -2 Z2 E9 Z6

19 AI,**,o + Ds,*,o -2, -2,-2 Z2 E9 Z4

20 Ao,**o + E7 ,o -2, -2 Z2 E9 Z2

(Table L-5)

~ Dual graph A.·A* H l(oTo; lL)

21 Tr -2, -2,-4 Z6

22 Tr -2, -3,-3 Zs

23 A7 ,**** -2, -2, -2, -4 lL~3

24 A 7 ,**** -2, -2, -3, -3 Z4 E9 Z6

25 A 7,.*** -2, -3, -3, -2 Z6 E9 lL9

26 A*,o + A*,o + -2, -2 Z~2 E9 Z4

+ A*,o + As,*.,o -2, -2,-2

A.,o + An,**,o + -2, -2,-2 27 + Am,**,o -2, -2 lL~2 E9 Z4

(m + n = 5)

28 A3,.*,o + A3,**,o -2, -2, -2,-2 Z4 E9 Z6

29 A2,*.,o + Ds,.,o -2, -2,-2 Z~3

30 AI, •• ,o + E7 ,o -2, -2 Z4

31 D9,.,o -2 Z2

178 Mikio Furushima

(Table £-6)

~ Dual graph A.'A. H 1(ilTo;Z)

32 Cu -3 Z3

(Table £-7)

~ Dual graph A.'A. H 1(ilTo;Z)

33 Ta -2, -5 Z6

34 Ta -3, -4 Z8

(Table £-8)

~ Dual graph A.'A* Hl(ilTo;Z)

35 Tr -2, -2,-5 Zf2

36 Tr -2, -3,-4 Z13

37 Tr -3, -3,-3 Zt2

38 A 1 ••• ** -2, -2, -2, -5 Z:2

39 A 1.* •• * -2, -2, -3, -4 ZfIJ2 10

40 A 1 •• *** -2, -3, -3, -3 Z:2

41 A 7.***. -2, -2, -2,-5 Z:2

42 A7 .**** -2, -2, -3, -4 ZfIJ2 10

43 A7 .**** -2, -3, -4, -2 ZfIJ2 16

44 A7 .**** -2, -3, -3, -3 ZfIJ2 15

45 A*.o + A •. o + A •. o + -3, -2, -2, Z:2 + As .• *.o -2, -2

46 A •. o + A •. o + A*.o + -2, -2,-2 ZfIJ2 + As .••. o -2, -3 26

47 A*.o + A •. **o + -3, -2,-2 Z:2 + Am.* •. o (m + n = 5) -2, -2,

ZfIJ2 18 if (m, n) = (1,4)

48 A •. o + A •. * •. o + -2, -2,-2 Z10 ffi Z10 if (m, n) = (2,3) + Am .••. o (m + n = 5) -2, -3 ZfIJ2 if (m, n) = (3,2) 22

Z4 ffi Z6 ffi Z 12 if (m, n) = (4, 1)

49 D9 ••• o -3 Zs

50 A3 .**.o + A3 .• *.o -3, -2, -2, -2 Z33

Complex analytic compactijications of 1[3 179

51 A3 ,**,o + A3,**,o -2, -2, -2, -3 l"E!lz 14

52 Az,**,o + Ds,*,o -3, -2,-2 l"E!lZ 10

53 Az,**,o + Ds,*,o -2, -2,-3 l"E!lZ 30

54 A1,**,o + E7 ,o -3, -2 1"9

55 A1,*,o + A1,*,o + -2, -2 l"~z + E6 ,o

56 A1,*,o + A 7,*,o -2, -2 1"3 Efl1"9

57 A4 ,*,o + A4 ,*,o -2, -2 l"~z

(Table L-9)

~ Dual graph A*'A* H l(oTo; 1")

58 A 8,**** -2, -2, -2,-5 1"IZ

59 A8,**** -2, -2, -3,-4 1"20

60 A 8,**** -2, -3, -4,-2 1"34

61 A8,**** -2, -3, -3, -3 1"48

62 A*,o + A*,o + A*,o + -3, -2,-2 1"E!l3

+ A6 ,**,o -2, -2 8

63 A*,o + A*,o + A*,o + -2, -2,-2 1"E!l3

+A6 ,**,o -2, -3 14

64 A*,o + A.,**o + -3, -2,-2 1"E!l3

+ Am,**,o (m + n = 6) -2, -2 8

1"36 if (m, n) = (1, 5) 1"E!l3 if (m, n) = (2,4)

A*,o + A.,**,o + -2, -2,-2 10 65 1"44 if (m, n) = (3, 3)

+ Am,**,o (m + n = 6) -2, -3 1"4 Efll"E!l2 if (m, n) = (4,2) 1"13 Efll"S2 if (m, n) = (5, 1)

66 A +A' 4,**,0 3,**.0

-3, -2, -2,-2 1"6 Efll"12

6: A +A' 4.**,0 3.**,0 -2, -2, -2,-3 l"Z8

68 A 3,**,o + Ds,*,o -3, -2,-2 "£:22 "

69 A 3,**,o + D s,*,o -2, -2,-3 1"20

70 A 2 ,**,o + E7 ,o -3, -2 1"10

71 A1o,**,o -2 1"4

72 A4 ,*,o + E6 ,o -2 1"6

180 Mikio Furushima

PROPOSITION 2.5. Let (X, Y) = (V22,H22 ) be a compactijication of c3 as in • Proposition 1.13. Then, (a) Sing Y = {x}, where x is a minimally elliptic singularity of A3.**0 + D 5.*.0 - Type

(Table L-9, (68)), or (b) Sing Y = {x, y}, where x is a minimally elliptic singularity of eu-type (Table

L-3, (7)) and y is a rational double point of AlO-type.

In the Table L-l-Table L-9, we use the same terminology as that of the Table I-Table 3 in Laufer [11, p. 1287-1294].

3. Non-existence of the case (a)

Assume that there is a compactification (X, Y) = (V22,H22 ) of the case (a) in Proposition 2.5. Let n: Y -+ Y be the minimal resolution of the singularity x := Sing Y and Z the fundamental cycle of x associated with the resolution (Y, n). By assumption, we have Ky = -Z and (Z'Z) = -3. The dual graph ofn- 1(x) looks like the Fig. 2, where we denote by 0 (resp. 8)) a smooth rational curve with the self-intersection number -2 (resp. -3). We can represent Yas a ruled surface v: Y -+ plover pl (see Fig. 3), where

12 is a section (3.2)

{(D' e)y = 2, (D' B)y = 3, where D = n(D) is a canonical hyperplane section such that Pic Y ~ 7L. (9y(D), in particular,

\ deg D = (D' D)y = 22. (3.4)

10

n- 1(x) = U fi U 11 U 12, (3.5) i= 1

Z = f4 + 2f3 + 2f2 + 2fl + 11 + 212 + + 3f5 + 4f6 + 2f7 + 3f8 + 2f9 + flO' (3.6)

Complex analytic compactijications ofC3 181

2 2 4 2

Fig. 2.

LEMMA 3.1. (1) there is no line in X through the point x = Sing Y EX. (2) Co := n(C) eYe X is a unique conic on X through the point x.

Proof. Since the multiplicity m(lPy,x) is equal to 3 by Laufer [11] and Pic X ~ 7L ·lPx(Y), any line or any conic through the point x must be contained in Y. Now, since (C ·[h, = (Co· D)y = 2 and D is a hyperplane section, Co is a conic on X. Let F be a line or a conic on X through the point x. Then, we have FeY.

(-1)(;

12 (-3)

1 --------~--------------------~~-IPI

Fig. 3.

182 Mikio Furushima

Let F be the proper transform of F in Y. Since D can be written as follows:

D = 2C + 414 + 613 + 211 + 612 + 611 + + 612 + 1215 + 1816 + 917 + 151s + 1219 + 9110 + 6B, (3.7)

we have (D· F) =f. 1, and also have (D· F) = 2 if and only if F = C. This proves (1) and (2). Q.E.D.

2. Let us consider the triple projection of X = V22 from the singularity x = Sing Y EX. For this purpose, we will consider the linear system IH - 3xl :=

l(Ox(H) EB m~.xl, where H is a hyperplane section of X and mx.X is the maximal ideal of the local ring (Ox,x' Since the multiplicity m(Oy,x) is equal to 3, we have Y E IH - 3xl (c.f. [16a]).

Let 0' 1: Xl -+ X be the blowing up of X at the point x, and putE1:= 0' - l(X) ~ [p2. Let Y1 be the proper transform of Y in X l' Since - Kx = Hand Y E IH - 3xl, we have

(3.8)

(3.9)

By the adjunction formula, we have

(3.10)

LEMMA 3.2. HO(X 1 ,(Ox.(O'iH - 3Ed) ~ 1[4, and Hi(X 1 ,(Ox,(O'iH - 3Ed) = 0

for i > O. Proof. Let us consider the exact sequence

(3.11)

Since Y1 = O'iH - 3E 1 and Hi(Xt.(Ox,) = 0 for i > 0, we have only to prove Hi(Y1 , (Oy,(Yd) = 0 for i > 0 and HO(y1 , (Oy,(Yd) ~ 1[3.

By (3.10), we have

(Oy,(Yd = (Oy,(O'i H - 3E1 )

= (Oy,(D 1 + 3Ky,), (3.12)

where D1 := O'i Riy, is linearly equivalent to the proper transform of D in Y1.

Complex analytic compactijications of (:3 183

Claim. (9y,(D + 2Ky,) is nef and big on Y1. Indeed, there exists a birational . morphism Ill: Y ~ Y1 such that 1t = (ally,)o Ill. Then, we have lli(D 1 + 2Ky ,) = D - 2Z. It is easy to see that D - 2Z is nef and big on Y (see (3.6), (3.7)). Thus (9r.(Dl + 2Ky ,) is nef and big on Y1 •

By the Kawamata-Vieweg vanishing theorem, we have H i(y1 , {9y,(D 1 + 3Ky,)) = 0 for i> 0, namely, H i (Y1 , (9y.) = 0 for i> O. On the other hand, since Hi(y, (9r(D - 3Z)) = 0 for i > 0, by the Riemann-Roch theorem, we have HO(y, (9r(D - 3Z)) ~ (:3. Q.E.D.

By Lemma 3.2, the linear system IH - 3xl defines a rational map cD:= cDIH-3xl: X ---~ p3, called a triple projection.

Let {gl> g2, g3} be a basis of HO(y, (9r(D - 3Z)) such that

(gl) = 11C + 10f4 + 9f3 + 211 + 6f2 + 3fl

(g2) = 5C + 4f4 + 3f3 + 2f2 + fl + 2fs + + 4f6 + 2h + 4fs + 4f9 + 4fl0 + 4,8

(g3) = 8C + 7f4 + 6f3 + 11 + 4fz + 2fl + + fs + 2f6 + f7 + 2fs + 2f9 + 2flO + 2,8

(3.13)

Since 2(g3) = (gl) + (g2), g:= (gl: g2: g3) defines a rational map g: Y---~ Q of Yonto a conic Q:= {w~ = wowd 4P2(WO: Wl:W2). This implies that cD(Y) = Q ~ pl and W = cD(X) is a quadratic hypersurface in p3. Thus we have the following

LEMMA 3.3. Let cD:= cD1H _ 3xI : X - - - ~ p3 be the triple projection from the point x. Then the image tv:= cD(X) is an irreducible quadric hypersurface in p3, and Q = cD(Y) is a smooth hyperplane section, namely, a conic in p2.

3. Next, we will study the resolution ofthe indeterminancy of the rational map cD: X ___ ~ p3.

Let cDl!~H-3Ed: Xl---~ p3 be a rational map defined by the linear system laiH - 3Ell. Then we have the diagram:

Let ~ c X be a small neighborhood of x in X with a coordinate system

184 Mikio Furushima

(Zl,Z2,Z3). By Laufer [11], we may assume that

{L\n Y= {Z2°Z~ = Z~oZ2 + ZlZ~ + ZlZt}, x = (0, 0, 0) E L\.

(3.14)

By an easy computation, we find that Y1 has two rational double points q1 of A4-type and qo of D6-type (cf. [11, Theorem 3.15]). Let J1.1: Y -+ Y1 be the birationa1 map as above. Then we have J1.11(qd = f1 U f2 U f3 U f4, J1.11(qo) =

UJ2sfj. We put W):= J1.1(li), and C1:= J1.1(C) ~ Y1 eX 1. Then C1 is the proper

transform of Co in X 1> in particular, C1 is a smooth rational curve in Y1 eX 1 . h A- h 1(1) 21(1) h 1(1) 1(1) WIt q1EC1,qO'FC1·Moreover,we aveY1oE1 = 1 + 2,were 1'2 are

d·· 1· . E 1T1>2 d 1(1) 1(1) X two Istmct mes m 1 ~ u- , an 1 n 2 = q1 E 1. By (3.13), the base locus Bsl(DdYdl = C13 q1. Since H1(X 1> (Dx,) = 0 by

(3.12), we have the base locus BslO"i H - 3El13 C1 3 q1. Since Pic X ~ Z(Dx(Y), the linear system IH - 3xl has no fixed component.

Thus, we have the following

LEMMA 3.4. The linear system 100i H - 3E lion X 1 has no fixed component, but has the base locus BslO"iH - 3E11 = C13 q1.

We need the following

LEMMA Mo (Morrison [13]). Let S be a surface with only one singularity x of An-type in a smooth projective 3101d X. Let E eSc X be a smooth rational curve in X. Let J1.: S -+ S be the minimal resolution of the singularity of S and put

n

J1.-1(X) = U Cj, j= 1

where C/s (1 ~j ~ n) are smooth rational curves with

(CjOCj)S = -2

(Cjo Cj + ds = 1

(CioCj)s = 0

(1 ~j ~ n),

(l~j~n-l),

if Ii - jl ~ 2.

Let E be the proper transform of E in S. Assume that (i) NElS ~ (DE( -1),

(ii) deg N EIX = - 2,

Complex analytic compactifications of(;3 185

where N £IS (resp. N Elx) is the normal bundle of E (resp. E) in S (resp. X). Then we have (1) N EIX ~ (9E ElH9E( - 2) if x E E and (C j " E) = 1 for j = 1 or n, (2) N EIX ~ (9E(-1)E9{9E(-1) ifx¢E.

Proof. In the proof of Theorem 3.2 in Morrison [13], we have only to replace the conormal bundle N;IS = (9£(2) by N £IS = (9£(1). Q.E.D.

The indeterminancy of the rational map ~(1): Xl ---_ p 3 can be resolved by the following way:

Let us consider the following sequence of blowing ups:

where (i) O"j+ 1: X j + 1 - Xj is the blowing up of Xj along C j ~ pl (1 ~ j ~ 5), (ii) Cj +1 is the negative section of the pl-bundle Cj = P(N2j lx) ~ lFi1 ~j ~ 4),

(iii) C6 is a section of Cs ~ pl X pl with (C6 "C6 ) = o. Then we have the morphism <1>: X6 _ p3 and a diagram:

Xl~X6

I', 41>(1) l-UI '- CI> , ,

'). X _____ p3

CI> '

where 0":= 0"2°0"3°0"4°0"5°0"6.

This is a desired resolution of the indeterminancy of the rational map

~(1): Xl --- _ p3 (or ~: X ____ p3).

4. We will prove the facts above.

Notations:

• Yj + 1 :

• E j + 1 :

• C j+l:

the proper transform of Yj in X j + 1.

the proper transform of Ej in X j + 1 •

a section of Cj = P(N CjlXj *).

(D-1)

• q/ the singularity of Yj of As _ rtype (Ao-type means the smoothness).

• Yo: • f(i+1):

• ly+ 1):

the contraction of the exceptional set Uf~s/; in Y. a fiber of the pl-bundle Cj ~ X j + 1 •

the proper transform of 1\1) in X j +1 (i = 1,2).

186 Mikio Furushima

• Jlj: Yo ~ Yj: a birational morphism with Jlj

Step 1. Let U 2 :X2 ~Xl be the blowing up of Xl along Cl ~ pl. Since (Kx, ·Cl) = (u!H - 2El ·Cl ) = 0, we have degNctlx, = -2. Since qlE Cl C

Y l is the singularity of Y l of A4-type and (C·/l)r = 1, by Lemma Mo, we have

(3.15)

Thus we have C'l ~ 1F2.1t is easy to see that Y2 has two rational double points q2 of A3-type and qo of D6-type with q2 E C2 C Y2, qo ¢ C2 · Since (KX2 • C2 ) = 0, by Lemma Mo, we have

In particular, we have

Jl21(q2) = 12 U 13 U 14'

Yo - (/2 U 13 U 14) ~ Y2 - {q2}'

Jl2(C) = C2 ,

Jl2(fl) = P 2),

Jl 2 (11) = W) (i = 1,2).

(3.16)

(3.17)

(Step k,2~k~5). Let Uk: Xk~Xk-l be the blowing up of X k- l along Ck - l ~ pl. Then Yk has two rational double points qk of As-k-type and qo of D6-type with qk E Ck C Y,., qo ¢ Ck (k ~ 5). Since (KXk • Ck) = 0, we have deg NCklxk = -2. By Lemma Mo, we have

N CklXk ~ (9Ck E9 (9Ck( - 2) (2 ~ k ~ 4)

N C,IX, ~ (9c,( -1) E9 (9c,( -1) (k = 5).

In particular,

Jl;;l(qk) = It u ... U 14

Yo - (It u ... u 14) ~ Yk - {qk}

Jlk(C) = Ck,

Jlk(h-d = Pk).

(3.18)

(3.19)

Complex analytic compactijications 01 C3 187

Step 6. Let cr6: X6 -+ Xs be the blowing up of Xs along Cs ~ pl. By (3.18), we have Cs = crs 1 (CS) ~ pl X pl. Then we have anisomorphismJ.t6: Yo ~ Y6. We identify Yo with Y6 (see Fig. 6). Thus we put

J.t6(C) =: C, J.t6(15) =: 15

J.t6(/;) =: /;, J.t6(IJ =: Ii·

(3.20)

Then C = Y6 ' Cs gives another ruling on Cs. Let Cj (1 ::::;; j::::;; 4) be the proper transform of Cj in X 6. Then we have Figure 4 (see also Pagoda (5.8) in Reid [20]).

Figure 4

Now, since

Y 6 = cr~ cr~ cr: cr~ cr! crr H - 3cr~ cr~ cr: cr~ cr! E 1

-5Cs - 4C4 - 3C3 - 2C2 - C1,

we have

(9y.(Y6) = (9y.(15 + 3Ky• - 5C - 4/4 - 3/3 - 2/2 -/1)

~ (9Yo(15 - 3Z - 5C - 4/4 - 3/3 - 2/2 -/1)

~ (9 Yo (2/),

(3.21)

(3.22)

where I is a general fiber of v: Y -+ pl(see Fig. 4). This shows that the linear

188 Mikio Furushima

C I I I

-11 I I

-1

fs

-2

-2

Figure 5

Figure 6

-2

\ \ \ \

\ \ \ \ \ \ \ B

o

f

Complex analytic compactijications oj (:,3 189

system I (Oy.(Y6 )1 has no fixed component and no base point Therefore it defines a morphism vo: Y6 = Yo -+ Q of Y6 onto a smooth conic Q ~ !p1 in !P2. Since H1(X 6' (Ox.) = 0 and Pic X ~ lL, the linear system I Y6 1 = I (OX.(Y6 ) I has no base locus. Therefore we have a morphism (1):= (l)IY.I: X 6 -+ W ~ !p3 defined by the linear system I Y6 1, and have the diagram (D-1), which is desired. It is easy to see that

<D(Y6 ) = <D(Y) = VO(Y6 ) = Q ~ 12 (3.23)

(l)CQ Cj U Cs) = (I)(j5) (a line in !p3 ).

5. Since N C,IX, ~ (Oc,( -1) $ (Oc,( -1), by Reid [20], Cs can be blown down along C, and then the blowing downs can be done step-by-step. Finally, we have a smooth projective 3-fold V with b2 (V) = 2, morphisms (1)1: X 6 -+ V, (1)2: V -+ W, and a birational map p: X 1 -+ V, called a flip, such that

(i) (I) = (1)2 0 (1)1'

(ii)X 1 - C1 !!:. (V - h, where h:= <D1(f3)' (see (D-2».

(D-2)

~x~ U1r -----;P\I ~2

X - - - - - - - - - --+ W c:..--.. 1P3. cp

Since -Kx, = Y 1 + E10 by (ii) above, we have -Ky = A + I:, where A:= (I)(Y6 ) and I::= (1)1 (E6). For a general fiber F of (1)2: V -+ W, since deg(KF) = (KyoF) = -(I: oF) ~ -1, we have F ~!p1 and (I:oF) = 2. Since (1)1 (11 ) is a smooth rational curve contained in I:, and since (1)2 0 (1)1 (11 ) = (1)(11) is a point, I: is a meromorphic double section of (1)2: V -+ W.

Let G be a scheme-theoric fiber. Then we have (GoI:) = 2. Since V(I: u A) ~ X 1 - (Y1 u E 1) ~ (:,3 by assumption, it contains no compact analytic curve. Thus (1)2: V -+ W is a conic bundle over W, and (1)2 is the contraction of an extremal rayon V. Thus, W is smooth by Mori [12]. Since deg W = 2, W ~ !p1 X !PI, hence, b2(V) = 3. This is a contradiction, since b2 (V) = b2(X 1) = 2. Therefore we have:

Conclusion

The case (a) of Proposition 2.5 can not occur.

190 Mikio Furushima

40 Non-existence of the case (b)

1. Assume that there is a compactification (X, Y) = (V22' H 22) of the case (b) in Proposition 2.5. Then we have Sing Y = {x,y}, where x is a minimally elliptic singularity of Cu-type (Table L-3, (7)), and y is a rational double point of Alo-type. Let n: Y --+ Y be the minimal resolution of the singularities of Yand put E:=n- l (x),n- l (y)=:U}2 1 Bj • Then E is an irreducible rational curve with a cusp, and Ky = - E, (E 0 E)y = - 2. We can easily see that Y can be obtained from [p>2 by succession of 11 blowing ups at a smooth point p on a cubic curve Co ~ [p>2 with a cusp (infinitely near points allowed). Let /1: Y --+ [p>2 be the projection. Then E is the proper transform of Co in Yand /1-l(p) = U}llB j ,

where Bll is the exceptional curve of the first kind (see Fig. 7). We take sufficiently general hyperplane section H such that D:= HoY does not

pass through the points x and y. Then D is a canonical curve of the genus 9 = 12 with deg D = 22 in Y. Let 15 be the proper transform of D in Y. Then we have

(15 0 15)1' = deg D = 22

15 = 3E + 2G,

(4.1)

(4.2)

where G is the proper transform of a line G ~ [p>2 with p 1: G in Y. In particular, (GoG)y = 1.

LEMMA 4.1. There is no line in X through the point x. Proof Since the multiplicity 1tl((()Y,x) is equal to two, any line through the point

x is contained in Y. Let 9 be such a line in X, a:nd g be the proper transform of Yin 9 ~ Y. Since (Dog)y = (Dog)y = (Hog)x = 1, by (4.2), we have

3(Eog) + 2(Gog) = 1. (4.3)

A,O·type

r~----------------~~~------------~' E

B8 Bll B3 -- - -- B6

B9 Bs B2

B7 B, B4

-y

Figure 7

Complex analytic compactijications of (:,3 191

This is a contradiction. Q.E.D.

2. Let 0": Xl -4 X be the blowing up of X at the point x, and put E:= O"-l(X) ~ /p z . Let Y 1 = 0"* H - 2E be the proper transform of Yin X. Then we have:

(i) Y1 • E = 21, where 1 is a line in E ~ /pz,

(ii) Sing Y 1 = 1, (iii) N/1x ~ (9/( -1) EB (9/(1).

Let r: X z -4 Xl be the blowing up of Xl along 1 ~ /pl. By (iii) above, we have I..::= r- 1(1) ~ !Fz. Let us denote the negative section (resp. a fiber) by s (resp. f). Let Yz be the proper transform of Y 1 in Xz' Then we have Yz = r*Y1 - 2I..:. Let Yo be the contraction of the exceptional curve UJ21Bj in Y. Then Yo has a rational double point of A10-type. By an easy computation, we have an isomorphism Yo ~ Yz. We identify Yz with Yo via v. For

. simplicity, we put C:= v( C), D := v(D), G:= v( G). Then we have

(see Fig. 8).

3. We will study the linear system I.P I := I r* Y 1 - L'I on X z· Let us consider the exact sequence

-2 s

Figure 8

(4.4)

(4.5)

(4.6)

192 Mikio Furushima

Since -.* Y1 - I: = -,*(1* H - 2£ - 3I:, we have

by (4.2), where £ ~ 1P2 is the proper transform of E ~ 1P2 in X 2' Since H i (y2 , lP y2 (2G)) = 0 for i > 0, by the Riemann-Roch theorem, we have HO(y2,lPy2(9')) ~ C6. Since HO(X2,lP(X2,lPY2(I:)) ~ C and H 1(X2,lPX2 (I:)) =

0, we have finally the following exact sequence:

0-+ HO(X 2, lPX2(I:)) -+ HO(X 2, lPX2(9')) -+ HO(Y2, lP y2(9')) -+ 0

J II J II J II (4.6)

Since diml9' I = 6, we have a rational map cD:= cD1.2'1: X 2 -+ 1P6 defined by the linear system 19' I.

Since the linear system I lP y2 (2G) I has no base locus on Y2 , neither does 19'1 by (4.6). Therefore cD: X2 -+ 1P6 is a morphism X 2 to 1P6 with

where IPIl'JY2(2G)1 is a morphism defined by IlPy2(2G)I. Thus we have the following:

LEMMA 4.2. cD: X 2 -+ cD(X 2) ~ 1P6 is a morphism of X 2 onto a 3-fold cD(X 2) of degree 4 in 1P6. Moreover, the restriction cD1Y2: Y2 -+ cD(Y2) = IP(Y2) ~ IPs gives an birational morphism of Y2 onto a surface IP(Y2) of degree 4 in Ifl1S.

Proof. Since (-'*Y1 - I:? = 4, we have degcD(X2) = 4. Q.E.D.

0-+ lP y2(-.* Y1 - 2L')) -+ lPX2(9') -+ lPd9') -+ 0

J II J II (4.7)

(4.8)

Since lPds + 3f)is very ample onI:, the morphism p := Pls+3fl: I: -+ p(I:) ~ IPs is an isomorphism of I: onto a smooth surface of degree 4 in IPs. Thus we have the following

Complex analytic compactifications of C3 193

LEMMA 4.3. The restriction (f)1L': [j --+ (f)([j) ~ 1P 5 is an isomorphism of [j onto a smooth surface (f)([j) of degree 4 in 1P5.

Finally, let us consider the exact sequence:

0--+ @x2{-r*(Yl - E)) --+ @X2(.P) --+ @E(.P) --+ ° f II f II (4.9)

@X2{-r*(a* H - 3E))

Then we have Hl(X 2'@X,(r*(a* H - 3E))) = 0, namely, we have a surjection

(4.10)

Thus we have the following

LEMMA 4.4. The restriction (f) IE: E --+ 1P2 gives an isomorphism of E onto 1P2.

5. Let y be an irreducible curve in X 2 such that ('t"* Y1 - [j 0 y) = O. Since 't"* Y1 - [j = Y2 + [j, we have (Y2 oy) + ([j oy) = O. By Lemma 4.2, Lemma 4.3, Lemma4.4,y c's;: Y2 u [j. Thus (Y2 oy) = ([joy) = O,namely, (Y1 0't"(y)) = O. Hence, we have Y 1 11 't"(y) = <p and Ell 't"(y) "# <p. This shows that there is no irreducible surface T in X 2 such that dim <D(T) :::; 1. There are a finite numbers of conics in X through the point x (see [8J). Let y be the proper transform of a conic in X through x. Then dim (f)(y) = O. In particular, there are a finite number of irreducible curves y' in X 2 such that dim (f)(y') = O. Therefore we have the following

LEMMA 4.5. (f): X 2 --+ W:= (f)(X 2) ~ 1P6 is a birational morphism of X 2 onto a 3-fold W of degree 4 in 1P6. In particular, b2 (X2 ) = b2 (W) = 3.

6. Since deg W = 4 in 1P6 , we have an equality

deg W = codim W + 1. (4.11)

Since there is a smooth rational curve y in X 2 such that dim (f)(y) = 0 and b2 (X 2) = b2 (W), the 3-fold W has a finite number of isolated singularities. Thus, W is a cone over a rational scroll or a cone over the Veronese surface. Hence, b2 (W) = 1. This is a contradiction, since b2 (W) = 3 by Lemma 4.5. Thus, we have:

Conclusion

The case (b) of Proposition 2.5 can not occur. We have proved in Section 3 that the case (a) of Proposition 2.5 can not occur.

194 Mikio Furushima

Therefore, in the case of the index r = 1, such a compactification of (:3 does not exist. Thus we have the Theorem (see the Introduction).

5. Remarks and an example

1. Let (X, Y) be an analytic compactification of (:3. Then we have (cf. [2], [3]):

Y has at most isolated singularities.

:::;. Y is normal. <\: Y is projective.

Y is irreducible. ¢>b2 (X) = 1

In the case where Y is normal, we have determined the complete structure of such a (X, Y) (see Theorem in the Introduction).

On the other hand, we know that there is a non-normal hyperplane section Es of the Fano 3-fold Vs such that Vs - Es ~ (:3([3]). This gives an example of a compactification (X, Y) of (:3 with a non-normal irreducible boundary Y.

Recently, Peternell-Schneider [18] and Peternell [19] proved the following

THEOREM 5.1. Let (X, Y) be a projective compactijication of (:3 with b2 (X) = 1. Assume that Y is non-normal. Then, X is a Fano 3-fold of the index r(1 ::::;; r::::;; 2), and (i) r = 2:::;. (X, Y) ~ (Vs,E s) (up to isomorphism).

(ii) r = 1:::;. X ~ V22 '+ jp>13(or V22 '+ jp>12) (Mukai-Umemura [14]).

2. Finally, we will prove that there is a non-normal hyperplane section H22 of V22 such that V22 - H22 ~ (:3. Let (ao :a1: ... :a12) be a homogeneous coordinate of jp>12. Then V 22 '+ jp>12 can be written as follow (see p. 506 in [14]):

aOa4 - 4a 1 a3 + 3a~ = 0

aoas - 3a1a4 + 2a2 a3 = 0

7aOa6 - 12a1 aS - 15a2 a4 + 20a~ = 0

aOa7 - 6a2 as + 5a3 a4 = 0

5aoaa + 12a1 a7 - 42a2 a6 - 20a3 aS + 45ai = 0

aOa9 - 6a 1 aa -6a2 a7 - 28a3 a6 + 28a4 aS = 0

aOa10 + 12a1 a9 + 12a2 aa - 76a3 a7 - 21a4 a6 + 72a; = 0

Complex analytic compactifications of (:3 195

aOall + 24a l al0 + 90a2a9 - 130a3as -405a4a7 + 420aSa6 = 0

aOa12 + 60a l all + 534a2alO + 380a3 a9 - 3195a4as - nOaS a7 + 2940a~ = 0 (*)

al a12 + 24a2all + 90a3 alO - 130a4 a9 - 405asas + 420a6a7 = 0

a2a12 + 12a3 all + 12a4al0 - 76a sa9 - 21a6as + na~ = 0

a3 a 12 - 6a4all - 6a S al0 - 28a6a9 + 28a7 aS = 0

5a4a12 + 12asall -42a6al0 - 20a7 a9 + 45a~ = 0

aS a12 - 6a7 a lO + 5aS a9 = 0

7a6a12 - 12a7 all - 15asal0 + 20a~ = 0

a7 a 12 - 3aS all + 2a9 al0 =0

aS a 12 - 4a9 all + 3aio = 0

In the affine part {ao = I} ~ (:12(a l , . .. ,a12 ), let us consider the following coordinate transformation:

X3 = a3

X 4 = a4 - 4a l a3 + 3a~ Xs = as - 3a l a4 + 2a2a3

X6 = 7a6 -12al aS -15a2a4 + 20a~ X 7 = a7 - 6a2aS + 5a3 a4

Xs = 5as + 12a l a7 - 42a2a6 - 20a3 aS + 45a~ X9 = a9 - 6a l aS - 6a2 a7 - 28a3 a6 + 28a4aS

X l0 = al0 + 12al a9 + 12a2as - 76a 3 a7 - 21a4a6 + 72a;

X ll = all + 24a l a lO + 90a2a9 - 130a3 as - 405a4 a7 + 420a Sa6

X 12 = a12 + 60a l all + 534a2a l0 + 380a3 a9 - 3150a4 as - 720a Sa7 + 2940a~

Then the Jacobian IO(Xl, .. " x12)/o(al,' •. , a12)1 = 35 -# 0, and further we have

We put HZ2 := V22 (l {aD = O}. Then HZ2 is non-normal. Therefore the pair (VZ2' HZ2 ) is a compactification of (:3 with a non-normal

boundary. One can easily see that the singular locus of HZ2 is a line in VZ2 '

Question 1. Is there a non-normal hyperplane section E22 in V22 ( -# V~2) such that V22 - E22 ~ (:3?

196 Mikio Furushima

Acknowledgements

The author would like to thank the Max-Planck-Institute fiir Mathematik in Bonn, especially Professor F. Hirzebruch for hospitality and encouragement, and would also like to thank Professor N. Nakayama for the stimulating conversations we had and for his valuable comments.

References

1. M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966) 129-136. 2. L. Brenton, Some algebraicity criteria for singular surfaces, Invent. Math. 41 (1977) 129-147. 3. M. Furushima, Singular del Pezzo surfaces and complex analytic compactifications of the

3-dimensional complex affine space C3, Nagoya Math. J. 104 (1986) 1-28. 4. M. Furushima, Singular K3 surfaces with hypersurface singularities, Pacific 1. Math. 125 (1986)

67-77. 5. M. Furushima and N. Nakayama, A new construction of a compactification of C3, Tohoku

Math. J. 41 (1989), 543-560. 6. M. Furushima and N. Nakayama, The family of lines in the Fano 3-fold Vs, to appear in

Nagoya Math. J. 116 (1989). 7. M. Furushima, Complex analytic compactifications of C3, Proc. Japan Akad. 64 Ser. A (1988)

25-26. 8. V.A. Iskovskih, Fano 3-fold I, Math. U.S.S.R. Izvestija 11 (1977) 485-527. 9. V.A. Iskovskih, Anticanonical models of three-dimensional algebraic varieties, J. Soviet Math.

13-14 (1980) 745-814. 10. Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, in

Algebraic Geometry, Sendai, Advanced Studies in Pure Math. 10, Kinokuniya, Tokyo and North Holland, Amsterdam (1987) 551-590.

11. H. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977) 1257-1295. 12. S. Mori, Threefolds whose canonical bundles are not numerical effective, Ann. Math. 116 (1982)

133-176. 13. D. Morrison, The birational geometry of surfaces with rational double points, Math. Ann. 271

(1985) 415-438. 14. S. Mukai and H. Umemura, Minimal rational threefolds, Lecture Notes in Mathematics 1016,

Springer Verlag, Berlin, Heidelberg, New York (1983) 490-518. 15. S. Mukai, Curves, K3 surfaces and Fano 3-folds of genus .::; 10, Algebraic Geometry and

Commutative Algebra in Honor of Masayoshi Nagata (1988), to appear. 16. S. Mukai, New classification of Fano 3-folds and Fano manifolds of coindex 3, Preprint (1980). 16a. S. Mukai: Private letter, May 1986. 17. D. Mumford, Topology of normal singularities of an algebraic surface and a criterion for

simplicity, Publ. Math. I.H.E.S. 9 (1961) 5-22. 18. T. Peternell and M. Schneider, Compactifications of C3 (I), Math. Ann. 280 (1988) 129-146. 19. T. Peternell, Compactifications of C3 (II), Math. Ann. 283 (1989), 121-137. 20. M. Reid, Minimal models of canonical 3-folds, in Algebraic Varieties and Analytic varieties,

Advanced studies in Pure Math. 1, Kinokuniya, Tokyo and North-Holland, Amsterdam (1983) 131-180.

21. J.P. Serre, Groupes algebriques et corps de classes, Actualites Sci. Indust. no. 1264. Hermann, Paris (1959).

22. Y. Umezu, On normal projective surfaces with trivial dualizing sheaf, Tokyo J. Math. 4 (1981) 343-354.

23. P. Wagreich, Elliptic singularities of surfaces, Amer. J. Math. 2 (1970) 419-454. 24. S.T. Yau, On maximally elliptic singularities, Trans. A.M.S. 257 (1980) 269-329.

Compositio Mathematica 76: 197-201, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

Analytic curves in power series rings

HERWIG HAUSER1 & GERD MULLER2

1 Institut fur Mathematik, Universitiit Innsbruck, 6020 Innsbruck, Austria; 2 Fachbereich Mathematik, Universitiit Mainz, 6500 Mainz, FRG

Received 3 December 1988; accepted 20 July 1989

Let us state a standard result on algebraic group actions:

PROPOSITION. For an analytic map germ y: S -+ (V, v), S a reduced analytic space germ and (V, v) the germ in v of a finite dimensional complex vector space V, together with an algebraic subgroup G of GL(V) the following holds:

(i) The germ T of points t in S for which y(t) lies in the orbit Go v of G through v is analytic.

(ii) There is an analytic map germ <fJ: T -+ (G, 1) such that y(t) = <fJ(t) 0 v for all t in T.

Indeed, the orbit Go v is a locally closed submanifold of V, isomorphic to the homogeneous manifold G/Gv via the orbit map G/Gv -+ Gov, and the natural map G -+ G/Gv admits local sections.

The second part of the Proposition asserts that every analytic curve in a G-orbit is locally induced from an analytic curve in G. The object of the present article is to establish this statement in the case where V = (!)~ is a finite free module over the I[>algebra (!)n of convergent power series in n variables and G = % =

GLp«(!)n) ><l Aut (!)n is the contact group acting naturally on (!)~. Recall that the orbits of % through f in (!)~ just correspond to the isomorphism classes of the analytic space germs in (C",O) defined by f Thus we shall obtain analytic trivializations of local families of space germs whose members lie in the same isomorphism class.

DEFINITION. Let S always denote a reduced analytic space germ. A map germ y: S -+ E with values in some subset E of (!)~ is called analytic if there is an analytic map germ G: (C", 0) x S -+ cP such that y(s)(x) = G(x, s).

198 Herwig Hauser and Gerd Muller

Choosing coordinates on (C", 0), the group Aut (9 n can be considered as a subset of (9~. The analyticity of a map germ with values in Aut (9 n does not depend on this choice (cf. [M, sec. 6]) and thus analytic map germs with values in % are defined. We then have:

THEOREM 1. Let y: S --+ (9~ be an analytic map germ, S reduced. (i) The germ T of points t in S for which y(t) lies in the orbit % " y(O) is analytic.

(ii) There is an analytic map germ <fJ: T --+ % with <fJ(0) = 1 such that y(t) = <fJ(t) " y(O) for all t in T.

Proof For mn C (9n the maximal ideal and kE N consider Ak = (9n/m~+1 and the algebraic group %k = GLp(Ak) ><I Aut Ak acting rationally on the finite dimensional vector space Vk = A~. The composition Yk: S --+ (Vk' Yk(O)) of Y with the natural map (9~ --+ Vk is analytic. By the Proposition the germ Tk of points t in S with Yk(t)E%k"Yk(O) is analytic. Clearly Tk+1 C Tk. As (9s is Noetherian the sequence becomes stationary, say Tk = T* for k» O. The Proposition gives analytic <fJk: T* --+ (%k' 1) such that Yk(t) = <fJk(t)"Yk(O) for tE T*. By Theorem 2 below there is an analytic <fJ: T* --+ % with <fJ(0) = 1 such that y(t) = <fJ(t) " y(O) for all t E T*. This implies T* c T. Obviously T c T* and Theorem 1 is proved.

THEOREM 2. For two analytic map germs y, 1]: S --+ (9~, S reduced, the following conditions are equivalent:

(i) There exists an analytic <fJ: S --+ % with <fJ(0) = 1 such that y(s) = <fJ(s) "I](s) for all SE S.

(ii) For any kE N there exist analytic <fJk: S --+ %k with <fJk(O) = 1 such that Yk(s) = <fJk(S) "l]k(S) for all s E S.

Proof Embed S in (C m, 0) and choose G,H:(cn+m,O)--+CP such that one has y(s)(x) = G(x, s) and I](s)(x) = H(x, s). We have to find u(x, s) E GLp((9n+m) and y(X,S)E(9:+m such that:

u(x,O) = 1, y(x, 0) = x, y(O,s) = 0,

and

H(y(x, s), s) == u(x, s)" G(x, s) mod 1(S)

where 1(S) is the ideal of (9 m defining S in (cm, 0). By condition (ii) this system of equations can be solved up to order k. A generalization of Artin's Approximation Theorem by Pfister and Popescu [P-P, Thm. 2.5] and Wavrik [W, Thm. 1] yields the solutions u(x, s) and y(x, s).

Let us now indicate some applications of Theorem 1. We first determine the tangent spaces to the orbits of the contact group % in (9~:

DEFINITION. (a) The tangent vector in y(O) of an analytic curve y: (C, 0) --+ (9~

Analytic curves in power series rings 199

given by G: (Cn x C,O) --+ CP is defined as:

dy _ aG P

d (0) - a 1 E(!Jn· S S .=0

(b) The tangent map at 0 of an analytic map germ y:(cm,O)--+(!J~ given by G:(C n x Cm,O)--+C P is defined as

m ay v --+ L Vi ·-(0)

i=l as;

where

ay aG -a (0) = -a 1 E (!J~.

Si Sj .=0

(c) For a subset E of (!J~ and gEE let

COROLLARY. Let gE (!J~ with f-orbit fog. (i) Tifog) = I(g)o(!J~ + mnoJ(g)

where I(g) is the ideal of (!In generated by the components of g and J(g) is the (!In-submodule of (!J~ generated by the partial derivatives of g.

(ii) Let y: (Cm, 0) --+ (!J~ be analytic, y(O) = g, restricting to Yls: S --+ f 0 gfor some reduced S c (Cm,O). Then Toy(ToS) c Tg(fog).

Proof. (i) If G(x, s) = u(x, s) 0 g(y(x, s» with u(x, 0) = 1, y(x, 0) = x, then

aG

as 1.=0 au ag ay og+_o_ . asl.=o ax asl.=o

Theorem 1 therefore implies" c". The other inclusion is obvious. (ii) Since any analytic S --+ f can be extended to an analytic (Cm,O) --+ f,

Theorem 1 gives an analytic map germ 15: (Cm, 0) --+ (!J~, 15(0) = g, such that l5(s) E fog for SE(Cm, 0) and l5(s)= y(s) for seS. Clearly Tol5(C m) c T,(f 0 g) and Tol5(v) = ToY(v) for VE ToS.

Next, let us interpret Theorem 1 geometrically. For an analytic y: S --+ (!J~ given by G: (Cn,O) x S --+ cP consider the space germ

X defined in (Cn, 0) x S by G. For fixed s E S the vector y(s) E (!J~ defines the germ

200 Herwig Hauser and Gerd M iiller

in a(s) = (0, s) of the fiber of n = prix: X -+ S over s. Conversely, a morphism n: X -+S of space germs with section a: S-+X has an embedding X c(Cn,O) x S over S with O"(S) = 0 X S, see [F,0.35]. Moreover an analytic <1>: S -+ Aut (9n given by (s)(x) = y(x, s) induces an automorphism 4> of (C n, 0) X S over S mapping 0 x S onto itself: 4>(x, s) = (y(x, s), s).

Combining these remarks we get:

THEOREM I'. For a morphism of analytic space germs n: X -+ S, S reduced, with section a: S -+ X denote by Xl> t E S, the germ in a(t) of the jiber of n over t.

(i) The germ T of points t in S with X t ~ X 0 is analytic. (ii) For any base change a: S' -+ S with S' reduced the induced morphism

n': X' = X x s S' -+ S' is trivial along the induced section a': S' -+ X' if and only if a maps into T. (We say that n' is trivial along a' if there is an isomorphism X' ~ X 0 X S' over S' mapping a'(S') onto 0 x S'.)

The universal property of(ii) applies in particular to the base change T c Sand then reads as follows: A local analytic family of analytic space germs with isomorphic members is trivia1. This is a local analogon of a result of Fischer and Grauert [F-G] and Schuster [Sch, Satz 4.9]: A flat analytic family of compact analytic spaces with isomorphic members is locally trivia1.

Theorem I' can be extended to the case where n does not come with a section a:

THEOREM 3. For a morphism of analytic space germs n: X -+ S with S reduced denote by X(a), a E X, the germ in a of the jiber of n through a.

(i) The germ Y of points a in X with X(a) ~ X(O) is analytic. (ii) The restriction ny: Y -+ S is a mersion (i.e., has smooth specialjiber Y(O) and

there is a germ T c S such that ny maps into T and Y ~ Y(O) x T over T). (iii) For any base change a: S' -+ S with S' reduced the induced morphism

n': X' = X x s S' -+ S' is trivial if and only if a maps into T. (iv) There is a germ Z with X(O) ~ Y(O) x z. Proof Choose embeddings X c (C n, 0) X S over Sand S c (Cm, 0) and let

F:(C n x Cm,O)-+iCP define X. Let y: (I[:" x cm,O)-+(9~ be given by y(a)(x) =

F(x + a1 , a2 ). For fixed aE X the germ y(a)E (9~ defines X(a). Hence Theorem 1 yields the analyticity of Y.

Let Y(a) be the germ in a of the fiber of ny through a. For a E Y fixed its reduction red Y(a) is the germ of those points bE X(a) with X(b) ~ X(a). As X(a) and X(O) are isomorphic, red Y(a) and red Y(O) are isomorphic. In particular, dim Y(a) = dim Y(O) for all a E Y.

By the Corollary we have for g = y(O) E (9~ and v E To Y(O) = To Y n (Cn X 0):

n 8g L V;o-E l(g) ° (9~ + mn ° J(g). ;=1 8x;

Analytic curves in power series rings 201

As g defines X(O) in (en, 0) this signifies that there are d = dim To Y(O) vectorfields ~ l' ..• , ~d on X(O) with ~ 1 (0), ... , ~d(O) linearly independent. A Theorem of Rossi [F, 2.12] implies X(O) ~ (Cd, 0) X Z for some Z. Hence, by definition, Y(O) must have dimension at least d and therefore Y(O) ~ (Cd,O). This gives (iv). Moreover 1!y is a mersion by [F,2.19, Cor. 2]. The universal property of (iii) is then a consequence of Theorem I'.

We conclude by some remarks: In the absolute case S = 0 we have recovered a result of Ephraim [E, Thm. 0.2]. Another Corollary is Teissier's economy of the semi-universal deformation: In the semi-universal deformation of an isolated singularity X(O) there are no fibers isomorphic to X(O), [T, Thm. 4.8.4].

Finally, it is possible to provide the germs Yand T of Theorem 3 with canonical non-reduced analytic structures. The universal property then holds for arbitrary base changes. A detailed exposition of this non-reduced case is given in [H-M]. We also refer to results of Flenner and Kosarew [F-K] and Greuel and Karras [G-K]. Using deformation theory and Banach-analytic methods they treat the case of flat morphisms.

Acknowledgements

We would like to thank G.-M. Greuel and H. Flenner for stimulating discussions and suggestions.

References

[E]

[F] [F-G]

[F-K]

[G-K]

[H-M] [M]

[P-P]

[Sch]

[T]

[W]

Ephraim, R.: Isosingular loci and the cartesian product structure of complex analytic singularities. Trans. Am. Math. Soc. 241 (1978) 357-371. Fischer, G.: Complex analytic geometry. Springer Lect. Notes 538, 1976. Fischer, W., Grauert, H.: Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten. Nachr. Akad. Wiss. Gottingen, Math. Phys. Kl. II 6 (1965) 89-94. Flenner, H., Kosarew, S.: On locally trivial deformations. Publ. Res. Inst. Math. Sci. 23 (1987) 627-665. Greuel, G.-M., Karras, U.: Families of varieties with prescribed singularities. Compos. Math. 69 (1989). Hauser, H., Miiller, G.: The trivial locus of an analytic map germ. To appear. Miiller, G.: Deformations of reductive group actions. Math. Proc. Camb. Phi/os. Soc. 106 (1989) 77-88. Pfister, G., Popescu, D.: Die strenge Approximationseigenschaft lokaler Ringe. Invent. Math. 30 (1975) 145-174. Schuster, H.W.: Zur Theorie der Deformationen kompakter komplexer Raume. Invent. Math. 9 (1970) 284-294.

Teissier, B.: The hunting of invariants in the geometry of discriminants. In: Real and complex singularities, Oslo 1976, 565-677. Holm, P., (ed.) Sijthoff & Noordhoff 1977. Wavrik, J.J.: A theorem on solutions of analytic equations with applications to deformations of complex structures. Math. Ann. 216 (1975) 127-142.


A Siegel Modular 3-fold that is a Picard Modular 3-fold*

BRUCE HUNT Universitiit Kaiserslautern, FE Mathematik, Postfach 3049,675 Kaiser/autern, FRG

Received 26 August 1988; accepted 4 October 1989

- 00 Introduction

Let ~ be a bounded symmetric domain. r c Aut(~) be a discrete, properly discontinuous group. If r is cocompact and acts freely, it has been known for several decades (Kodaira: [K], Hirzebruch: [Hi]) that r\~ is then an algebraic variety, and in fact of general type. The Hirzebruch proportionality theorem then tells us the (ratios of) Chern numbers of X = r\~, which allows us to recover ~ from the Chern numbers of X if we know only that X is of the form r\~ for some ~. The group r is then of course just the fundamental group 11:1 (X). So it can't happen, for example, that X = r\~ = r\~' for 2 non-isomorphic bounded symmetric domains ~ and ~'.

Arithmetically the condition r cocompact means that r has Q-rank zero. Although occasionally such groups occur in algebraic geometry (Shimura curves, for example), in most cases r will only be of finite covolume, and the space r\~ occur as moduli spaces of some sort, usually as moduli spaces of varieties with special properties (e.g. classes of abelian varieties, K3-surfaces, curves, etc.) More generally, the period domains according to Griffiths [GS] (classifying spaces of Hodge structures) are all of the form r\G/H, where G is a real simple non-compact Lie group and H is a compact subgroup, r a discrete subgroup. If one has a Torelli theorem for a corresponding class of varieties, then these spaces, locally homogenous complex manifolds are the moduli spaces for that class of varieties. These spaces are all fibre bundles over some symmetric spaces, for example

SO(a, b)/U(hO) x ... x U(hm-1) x SO(hm) --+ SO(a, b)/SO(a) x SO(b),

(a=ho +h2 + ... +h2m,b=h1 +h3 + ... +h2m-1)

Sp(2a, 1R)/U(hO) x ... x U(hm) --+ Sp(2a, IR)/U(a),

(a = hO + h1 + ... + hm).

*Research supported in part by NSF Grant 8500994A2

204 Bruce Hunt

Returning to the situation in which r\.@ is a moduli space, in general r\.@ will not be compact, and the compactification is instrumented by adding moduli of the degenerations at the boundary. This leads one to consider (smooth) algebraic varieties X which contain r\.@ as a Zariski-open subset. We call such X locally symmetric.

Locally symmetric varieties, together with a divisor D (usually assumed to be normal crossings) such that X - D = r\.@, still contain a lot of "symmetry", that is, structure determined by the group structure of Aut(.@), and so we can think of a correspondence of pairs (X, D) and pairs (.@,r), where now r = 1t 1(X -D). Thus it is now quite conceivable that given an algebraic variety X, there exist 2 different divisors Dl and D2 such that (X,Dl) corresponds to (.@I,r1) and (X, D 2 ) corresponds to (.@2,r2) in the above sense, but .@1 and .@2 are not isomorphic, nor even of the same \R-rank for that matter. This phenomena seemed worthy of study as soon as it occurred somewhere. For surfaces no such examples were, up to now, known.

Let X be the following modification (blow-up) of!p 3 (for notations and details see 2.5.1. below). Let D1 , ••• , D4 be the 4 coordinate planes (regular tetrahedron), Ds, ... ,D10 the 6 symmetry planes of this tetrahedron, and Dll , ••• ,D1S be the 5 exceptional!p2's gotten by blowing up !p3 at the corners and the center of the tetrahedron. Finally let E1 , .. • , E10 be the !pI x !pI'S gotten by blowing up along the 10 3-fold lines of the 10 planes (i.e. of the arrangement D = Dl U ... U D10 , see 2.5). The proof and study of the following result is the object of this paper:

THEOREM. (i) (X, D) corresponds to (§2' r(2)) in the above sense, where §2 = Siegel upper half space of degree 2, r(2) = principal congruence subgroup of level 2.

(ii) (X, E) corresponds to (1B3, r(1 - p)) in the above sense, 1B3 = complex hyperbolic 3-ball, r(1 - p)=principal congruence subgroup ofU((3,1),(9K),K = 1fJ(,J-3), p = e21ti/3 • Here D = 'LDi,i = 1, ... ,15, E = 'LE;.,A = 1, ... ,10.

There are several directions into which this result can be further investigated. On the group level, one can associate to the discrete group r a Tits building with scaffolding. Here we find that the corresponding Tits building with scaffoldings are dual to each other. It is not yet clear whether this is implied by the double structure as locally symmetric space or whether this is perhaps as additional coincidence of this particular example.

It is not difficult to determine the ring of modular forms of r(2), using theta constants (Igusa [11], [12]). As for the ring corresponding to the group r(1 - p), the structure can be determined by utilizing results of Holzapfel [Hol], together with results of Deligne-Mostow [DM]. Using in addition a result of Shimura [S] which characterises (X, E) as a moduli space of Picard curves, one can even describe the ring in terms of theta constants. Viewing things this way, the duality

A Siegel Modular 3101d that is a Picard Modular 3101d 205

above turns into projective duality of varieties (the singular Baily-Borel embeddings), and this duality is in fact classical (cf. [B]).

It turns out that both (X, D) and (X, E) have moduli interpretations, and it is easthetically pleasing to see this duality in terms of degenerations. X - D - E parametrises on the one hand hyperelliptic curves y2 = P6(X), genus 2 curves, and on the other hand Picard curves y3 = P6(X), genus 4 curves, the correspondence being given by the zeros of P6(X). But where zeroes of P6(X) coincide (x ED u E), the types of degenerations seem to be dual to each other. For the hyperelliptic curves D corresponds to curves aquiring doublepoints, E to curves splitting into 2 components. For the Picard curves it is the other way around. This is summarised in the table 5.4.

For each such family of curves, one can consider the Picard-Fuchs differential equation corresponding to the dependency of the periods on the moduli. Both our families of curves are associated with the famous hypergeometric differential equation, but with different parameters. Here we use results of [DM] on the one hand, and of Sasaki and Yoshida [KSY] on the other.

The variety we are studying in this paper is probably one of the most thoroughly studied algebraic 3-folds, so we don't claim to be deriving new results on this variety. Rather, our object is to study in detail the 2 structures of locally symmetric spaces and their interrelations. The Siegel picture (§1) is very well known whence we only sketch the necessary statements and facts; we refer to [V] for general results and to [L W] for combinatorial and cohomological results. The Picard picture (§2) is, to the best of our knowledge, essentially new, so we give this side of the picture in much greater detail. However, this Picard picture is an almost straightforward 3-dimensional generalisation of the Picard picture of a surface for which extremely detailed results are available, namely Holzapfel's monograph [HoI]. Hence here there is also little which is original.

As this subject also has an interesting history, as well as being a model for our efforts in dimension 3, we now recall some of the background and results of the surface case. As long ago as 1769, L. Euler considered the following partial differential equation in connection with acoustics:

02 Z a OZ a OZ 0 ------+---= . oxoy x - y ox x - y oy

About a century later Riemann constructed solutions of this equation by an inversion process, during which hypergeometric functions occurred. In 1880 Appell gave the generalisation to several variables of the hypergeometric function which also occurs in recent work of Mostow and Deligne. In 1881 E. Picard studied these and found the famous integral representation for Appell's

206 Bruce Hunt

hypergeometric series:

He in particular studied the integral

and discovered that this function (of t1 and t 2) is a special solution of the Euler equation (E 1/3 )! In fact, 3 of these integrals form a fundamental system of solutions for a system of 3 partial differential equations.

One recognises immediately the integrals above as forming a base of the (l,O)-differentials on the trigonal curve

which is a genus 3 curve with Galois action by 71/371. These are the so-called Picard curves, of which we will be studying a suitable generalisation (we also call our curves, genus 4 curves, Picard curves).

We now recall for the reader's benefit some of Holzapfel's results in the study of the family of Picard curves. Consider the following subgroups of Aut(1B2) = PSU(2,1):

Eisenstein lattice f:= PU(2, 1; (OK)' K = iQ(~ - 3) Special Eisenstein lattice r:= PSU(2, 1; (OK)'

(0 K = ring of integers in K.

Letting p = e21ti/3 be a primitive cube root of unity, (1 - p) is an ideal in (OK (note

-1 + J=3 that iQ(~ - 3) = iQ(p) since p = 2 . One considers also the con-

gruence subgroups r' = r(1 - p), and r' = r(l - p) (here we are using Holzapfel's notation). The following results are proved in [HoI]:

I. The monodromy group of the system of partial differential equations alluded to above is r'.

II. The rings of automorphic forms for f' and rare:

00 00

EB [f', m]x = 1C[~1' ~2' ~3]' EB [r, m]x = IC[G2, G3 , G4 ], m=O m=O

where the ~i have weight 1 and the Gj have weight j.

A Siegel Modular 310ld that is a Picard Modular 3101d 207

III. f'\1B2 ~ jp2_{4 points}. IV. If F l' F 2 and F 3 are 3 fundamental solutions of the system alluded to above,

then the automorphic forms of weight 1, ('1 :'2 :'3) give (up to coordinate transformations) the inverse to the many-valued function (F l:F 2:F 3)' in other words, ('1:'2:'3)o(F1:F2:F3) is 1 -1, and

1B2

(F,:F,:FW ~,:,,:,,) jp2 _{ 4 points} -+ jp2 -{ 4 points}

commutes. V. There is also a commutative diagram

(1B2)* _ {Picard curves}/projective equivalence

~ C(P).~ 7 G,(P)x' + G,(p)x + G.(P)}

f\(1B2)*

where (1B2)* = 1B2u {f-rational cusps}, f\(1B2)* is the Baily-Borel compactification.

VI. The (image of) f-fixed points of 1B2 are the set of Picard curves with automorphism group larger than 7L/37L.

In our work below we are able to give reasonable generalisations ofl, II, III and IV to dimension 3. It would be challenging and extremely interesting to also get a nice equation in terms of modular forms (V) and to generalise VI also to dimension 3.

Finally we would like to remark that during the last year it has started to emerge that this example is the first in some finite list of examples with similar properties. An upcoming paper with Weintraub will give more details on the other known examples related to this one.

Acknowledgements

The author would like to thank the organisers of the conference at the Humboldt Universitat for the invitation and the opportunity to speak on this topic there. Also conversations with R.-P. Holzapfel and J.-M. Feustel were helpful. Finally

208 Bruce Hunt

I am greatly indebted to S. Weintraub for many explanations and conversations on this and related topics, in particular for the technicalities of 2.1.

O. Notations and conventions

All varieties considered are over the complex number field C. We use freely the standard notions of hemitian symmetric spaces. For a non-compact locally hermitian symmetric space X, X* usually denotes the Baily-Borel compactification, X A some desingularisation.

Throughout, we use the following notations:

projective space (over C) the complex n-ball symmetric group on n letters alternating group on n letters some arithmetic group, particular cases of which are:

r(2), r(4) principal congruence subgroups (Siegel case) r(1 - p), sr(1 - p), r(1 - p)2 lattices in Picard groups

Sp(n, IR), U(p, q), SU(p, q) K (!)K

Mn(C) 1'C 1 (X)

1. A Siegel Modular 3-fold

field with p elements (p prime), which is sometimes also denoted Z/pZ the usual classical groups number field, ring of integers in K n x n matrices with C-coefficients fundamental group.

1.1. Let §2 = Sp(2,1R)/U(2) = {ZEM2(C)!Z = tZ. Im(Z) > o} be the Siegel upper half-space of degree 2, a 3-dimensional, IR-rank 2 bounded symmetric domain. Sp(2, Z) is a lattice in Sp(2, IR) which has Q-rank 2. The action of r = Sp(2, Z) on §2 is Z r-+ (AZ + B)(CZ + D)-l. We are particularly interested in the principal congruence subgroup of level 2, defined by the following exact sequence:

1 -+ r(2) -+ Sp(2, Z) -+ Sp(2, Z/U) -+ 1. (1.1.1)

r(2) is thus a normal subgroup, of index equal to ! Sp(2, Z/2Z)! = 720, since

A Siegel Modular 310ld that is a Picard Modular 310ld 209

Sp(2,7L/27L) = L6' the symmetric group on 6 letters, as is well known. r(2) does not act freely, but the quotient is smooth [11], [13], [e]. Let X(2) denote the non-compact quotient r(2)\§z.

1.2. A compactification of X(2) is constructed in the standard way, i.e. Baily-Borel. Adjoin to §z the rational (with respect to r) boundary components, which are copies of§l in dimension 1 (rank 1), and points in dimension 0 (rank 0). The action of r(2) extends to the rational boundary of §z, and the quotient r(2)\§! is a compact Hausdorff space. The action of r(2) on one of the §1 's on the boundary is via the principal congruence subgroup of level 2 of Sp(l, 7L) =

Sl(2, 7L), which has 3 inequivalent cusps. Since r(2) c Sp(2, 7L) has 15 inequivalent I-dimensional cusps as well as 15 inequivalent O-dimensional ones, one gets the following configuration on X(2)* = r(2)\§!:

15 curves Ci = r(2)\§t 15 points Pij = Ci n Cj = cusp of Ci and Cj •

3 different Ci meet at each Pij. Each Ci contains 3 cusps Pij

1.3. In order to describe the boundary components precisely it is convenient to work in (7L/27L)4. Since any two cusps are equivalent under Sp(2,7L) the exact sequence 1.1.1 implies the natural action of Sp(2.7L/27L) on (7L/27L)4 gives exactly the action of r(2) on the boundary components (see also [L W]):

. . {I E (7L/27L)4, 1 =f. 0 1-dlmenslOnal cusps of X(2)* - I _ ( )_

- e1> ez, e3' e4' ej - 0 or 1.

. . {h = 11 1\ Iz, 2-dimensional O-dlmenslOnal cusps of X(2)* -. . b

IsotropIc su spaces.

In this scheme the curves C i are numbered by 4-tuples (el' ... ,e4)' ej = 0, 1. Then Ci n Cj is the 2-plane spanned by

and the third curve Ck meeting Ci n Cj is (e~), e~ = e~ + et.

1.4. A desingularisation of X(2)* was constructed by Igusa in [13] by blowing up along the sheaf of ideals defining the boundary. In [V] van der Geer explains how to obtain the desingularisation directly by means of toroidal embeddings of X(2). The result is the same, and is as follows: there are 15 divisors D 1 , ..• , D 15 , each itself an algebraic fibre space Di -+ Ci , whose generic fibre is a Kummer

210 Bruce Hunt

curve = P 1 ( = elliptic curve/involution) and whose 3 special fibres are degenerate conics in p2, consisting of 2 copies of pl meeting at a point

D. _-\:---___ +-----\""--= -I

1 (1.4.1)

The fibre space Dj -+ Cj has 4 sections Sl" .. , S4' Another way to view the Dj is as p2 blown up in the 4 3-fold points of the line arrangement in p2:

D· -" ------------------~~p~

The fibering Dj -+ Cj is given by the pencil of conics passing through the 4 points, and the singular fibres are the 6 lines of the arrangement, 2 of them at a time forming a degenerate conic, and the sections Sj are the exceptionalpl's of the blow-up. Therefore on each Dj one can blow down the 4 sections, the result being p2. The Dj intersect 2 at a time along the singular fibres and 3 at a time at the double points of those fibres. This describes the structure of the normal crossings divisor D = ~ Dj • For more details on the intersection behavior see 2.5. We denote the Igusa desingularisation by X(2) 1\ •

1.5. We now describe another important set of divisors, the Humbert surfaces of discriminant 1. For each natural number A == 0 or 1 mod(4) there is such a Humbert surface H 4 ([V], §2), but we will only describe H 1 here. The diagonal §1 x §1 C §2 has 10 inequivalent transforms under r(2), and the action of r(2) restricted to each copy is by r 1(2) x r 1(2) c Sl(2, Z) x SI(2, Z). (Here r 1 (N) denotes the principle congruence subgroup of level Nand degree 1, i.e. in SL(2,Z).) Let E 1 , ... ,ElO be the images in X(2)* of these diagonals. Then each Ej =r1(2)§! x r 1(2)\§! is a copy of pl x pl. The EJ

are disjoint, but intersect the D i in X(2) A at the sections of each. Each E .. intersects 6 Di , 3 in each direction:

I 0.

1 e~ e~ e~ e~

0· e{ ~ e{ ei J

e~ e~ e~ el

Ok

°1 Om On

1.6. To describe the incidences E .. n Di =1= 0, we follow [LW, §2J. Let L\ =

{15, 151- }, an unordered pair of 15, a non-singular plane and 15\ its orthogonal complement. Such L\ are in 1 - 1 correspondence with the EM and E .. n Di =1= (/)

iff(e~)E 15 or E 151-. Hence the E .. can be numbered by {(e~) A (e!),(e~) A (em, for example. We just give one example of this. Say E1 will be numbered by (1,0,0,0) A (0,0,1,0) and (0,1,0,0) A (0,0,0,1), and the other D/s meetingE1 are (1,0,1,0) and (0,1,0,1) which, plugging into the above scheme, describes all intersections quite explicitly. The action of Sp(2, 7l./271.) induces an action of the E .. , i.e. gE .. = E /J' E .. associated to L\ and E /J associated to L\g, for g E Sp(2,71./271.).

1.7. Finite covers We just state a result here which follows from Theorem 3.3.2. and the proof of

Theorem 2.7.1, but whose statement belongs here. Let r(4) denote the principle congruence subgroup of level 4. This is a normal subgroup of r(2) and Pr(2)/Pr(4) = (71./271.)9 (it is the projective groups that are acting effectively). This is the Galois group of the Fermat cover of degree 2 branched along the arrangement H (see 2.7. and [HuJ for Fermat covers), and in fact

THEOREM 1.7.1. The (smooth) Fermat cover Y(2, H) branched along H,

is the (Jgusa compactijication of the) Siegel modular 310ld of level 4.

2. A Picard Modular 3-fold

2.1. Let 183 := SU(3, 1)/S(U(3) x U(1» = {ZE C3 1 ~lzd2 < 1} be the complex hyperbolic 3-ball, a 3-dimensional, ~-rank 1 bounded symmetric domain. The

212 Bruce Hunt

best-known lattices in SU(3,1) are the Picard modular groups. For each square-free integer dIet K = Q(J="d) be the imaginary quadratic field associated with d and (!) K = the ring of integers in K. Then (!) K C C is a lattice, and SU(3, 1; (!)K) is the Picard Modular group of discriminant D (D the discriminant of K) which is a lattice in SU(3,l) (note that whereas Sp(2, IR) is a group with real coefficients, so integer coefficients give a lattice, SU(3, 1) is a group of complex matrices, so we need coefficients in a lattice in C). We shall be concerned in this paper with the field of Eisenstein numbers K = Q(..j=3), and the corresponding Picard modular group. Actually, the group more basic to our applications is the lattice U(3, 1; (!)K). These lattices are related as follows: SU <J U, and U /(SU = 7L/37L, since an element E U(3, 1; (!) K) may have determinant = p or p2, as well as 1. We will refer to U as the Picard lattice, and to SU as the special Picard (or Eisenstein) lattice.

The action of Aut(1B3) = PU(3, 1) (PU since the center acts trivially) is by fractional linear transformations. For yE Aut(1B3), Z = (Zl,Z2,Z3)E 1B3, this action is described as follows:

1 ~ i,j ~ 4. (2.1.1)

The jacobian of the action is (~ai4zi) - 1 at z = (zJ When considering lattices r in U(3, 1) or SU(3, 1) we will, without mentioning it, take their images in PU(3, 1). However, as this is a potential source of confusion, we describe this in some detail. Since the result is quite different in dimensions 2 and 3 we describe both. Let r, f, r', f' be as in the introduction (Holzapfel's notation) and let a P in front of one of the groups denote the projectivised group. Let Z(G) denote the center of a group G. Then:

z(f) = {±1, ±p, ±p2}, Z(r) = {1,p,p2}

Z(f') = {1, p, p2}, Z(r') = {1, p, p2}.

Thus we have the diagrams:

r 1 -6 .f r' 1-3 • f'

13 -1 16 -1 13 - 1 13 -1 pr 1 - 3 . pf pr' 1 -3 . pf'

and then a diagram relating the congruence subgroups:

1-pf' - pf - PU(2, 1; IF 3) = ~4 --1

J J <II 1-Pr' -pr -PSU(2, 1; 1F3) = ~4-1

the bottom sequence of which Holzapfel proves is exact [Ho1,1.3.1]. Both inclusions Pr' c pf' and pr c pf are of index 3.

We now give the corresponding diagrams in dimension 3, and fix, for the rest of this paper, the following notations:

r:= U(3, 1; (!JK)' r(l - p) congruence subgroup,

sr:= SU(3, 1; (!JK)' Sr(l - p) congruence subgroup.

Here we have

Z(r) = {±1, ±p, ±p2}, Z(Sr) = {±1}

Z(r(l - p» = {1,p,p2}, Z(Sr(l - p» = {1},

giving the following diagrams:

Sr(l - p)

11 - 1

1-3 ) r(1 - p)

13 - 1

psr 1- 2 ) pr PSr(l - p) 1 - 1 )pr(1 - p).

Furthermore, from the Atlas of finite simple groups we have

U(3, 1; 1F3) = ~6 x 2

PU(3, 1; 1F3) = ~6

SU(3, 1; 1F3) = A6 x 2

PSU(3, 1; 1F3) = A 6 •

We have the following sequence

1- pr(1 - p) - pr-

<II J

214 Bruce Hunt

(where A6 is the alternating group on 6 letters), and PSr(1 - p) c pr(l - p) is an isomorphism, whereas psr c pr is a subgroup of index 2.

Now just like r(2) in Section 1, r(1 - p) does not act on 1B3 freely but it turns out that the quotient is nonetheless smooth. The subgroup r(l - pf does act freely, and the singularities ofr(1 - p)\1B3 can be analysed by studying the cover corresponding to r(l - p)2 c r(1 - pl. We will describe this in 2.8 below. Let Y(1 - p) denote the non-compact quotient r(1 - p)\1B3 .

2.2. We now discuss the boundary components of 1B3 with respect to r. To do this, think of IB 3 c V4, a 4-dimensional vector space over C with hermitian form

Let BB c V be the positive cone, i.e. rJB = {v c V (v, v) > o}. This looks as follows:

Then 1B3 = p(BB), p: V -+ IP(V) the natural projection. From this one sees that aIB3 = p(aBB) = p(Iff), Iff c V the set of isotropic vectors, Iff = {v E V I (v, v) = o}. With this picture in mind it is obvious that the K-boundary components (or the boundary components with respect to r) are: (here we should fix an embedding KcC),

these are the isotropic lines in IP(K4 ). Hence the number of r-cusps is the order

A Siegel Modular 3-fold that is a Picard Modular 3-fold 215

of r\oKIB3. This question is discussed in [Z,I] for any discriminant and corresponding Picard group. The answer is: rd has f.i(d) cusps, where f.i(d) =

# 1 equivalence classes of hermitian unimodular lattices in K 4 1. This number has been calculated by Hashimoto [HK] and for d = 3 there is a unique equivalence class (1 cusp).

2.3. Hence, to determine the number of cusps of r(1 - p), just consider the exact sequence 2.1.2, and the ensuing action of r /r(1 - p) = PU(3, 1; 1'/3) = L6 on (£:/3)4. The number of cusps is just the number of totally isotropic lines {v E 1'/34 1 (v, v) = o}/ ± 1 (note that IF~ = ± 1), just reducing the form mod 3. This works as follows. The form on (;4 is ZlZl + Z2Z2 + Z3Z3 - Z4Z4. The involution Z -4 Z in @ (where it is given by p -4 p2) descends to the trivial involution on 1'/3 (as it must, £:/3 having no non-trivial automorphisms) which can be seen by the fact that p == 1 mod(1 - p) so p2 == 12 = 1 == p mod(1 - p). It is easily checked that there are, up to sign, 10 isotropic vectors: (1,0,0,1), (1,0,0, -1), (0,1,0,1), (0,1,0, -1), (0, 0,1,1), (0, 0,1, -1), (1,1,1,0), (1,1, -1,0), (1, -1, 1, 0) and (-1, 1, 1, 0). I am indebted to Steve Weintraub for this nice exposition.

2.4. A desingularisation of Y(1 - p)* is constructed by blowing up at the cusps. The resolving divisor is A/I, A = E x E an abelian surface which is a product of

2 copies of the elliptic curve with complex multiplication by .j=3 and I is the involution (z l' Z2) ~ ( - Z l' - Z2)' in other words the resolving divisors are pi x pi'S. This can be extracted from the standard construction ([He], [HoI]). This is of course the same as a desingularisation by means of toroidal embeddings which in the case of Q-rank 1 yields isolated resolving divisors. Let Y(1 - p)" denote this resolution. We will see below that Y(1 - p)" is actually already smooth.

2.5. We now describe another important set of subvarieties, which we call modular subvarieties. Let D = 1B2 C 1B3 be a totally geodesic embedding such that r D act properly discontinously, that is r D := {y E r 1 yD = D} C Aut(1B2) is properly discontinous. That is to say the diagram

commutes. This is what is usually called a modular embedding (r D' D) c (r, 1B3). We will be interested in the r D such that r D = U(2, 1; @K(1 - p)) is the Picard (surface) group. We can then draw on the very detailed results of Holzapfel for

216 Bruce Hunt

these surfaces (i.e. the whole book [HoI]). As we will see below there are 15 modular subvarieties of this kind on Y(l - p). These 15 meet in a union ofspecial curves on each copy, the r-reftection discs of [Hol, 1.3.3]. To describe their intersections (which will be identified as such below) we use the figure alluded to in the introduction. Let Hi" .. , H 10 be the 10 planes consisting of the 4 facet planes and 6 symmetry planes of a regular tetrahedron in jp>3:

2.5.1.

This arrangement has the following singularities (i.e. is not a normal crossings divisor because of):

56-fold points 10 3-fold lines

(4 corners and the center) (6 edges and 4 diagonals).

The divisor H = I;.Hi is turned into a normal crossings divisor by blowing up jp>3 at the 5 points, then along the 10 lines just mentioned. Let 1fl>3 denote this blow up, [HJ =: Di the proper transforms, Du ,"" D15 the exceptional jp>2,S

and E1, ... ,E10 the exceptional jp>1 x jp>l'S. Then under the isomorphism Y(l - py = 1fl>3 (see 2.7) D1 , ••• , D 15 are the modular varieties just introduced. Notice that after the blow-up each of the Di , i = 1, ... ,15, is identical. Each is the blow up of jp>2 at 4 points in general position, the 3-fold points of the linear arrangement 1.4.2, hence in each Di there are 10 jp>l'S, all of which have self-intersection (-1) in each Di • These are of course the same surfaces occurring in 1.4.

~5 acts in a natural way permuting the 10 jp>l'S; in fact Di - P, (P = ~Ph Pi the exceptional divisors under the modification Di ~ jp>2) is a GIT-quotient, arising as follows: Let (Xi' Yi)' i = 0, ... , 4 be a set of homogenous coordinates on

(iJ=Di)5. Let X c (iJ=Di)5 be the Zariski open subset consisting of those (Xi' Yi) such that no 3 of the 5 are identical. PGL(2, IC) acts freely on X, and the quotient can be compactified to iJ=D2 blown up in 4 points ([Y], p. 140). The action of~5 is then just permutation of the factors on X.

On the other hand we have the natural action of ~4 on the Di as described in [Ho1, l.3.ff]. This comes about as soon as you have chosen a subset of 4 (disjoint) out of the 10 iJ=D1>s to be blown down, i.e. identified a set of cusps.

2.6. We now give the combinatorial description of the Di in 71-/3Z4 • Obviously, in K4 each such Di(K) (K-valued points) is given by the intersection of the cone of 2.1 with a hyperplane, fixed by r as in 2.5. However, in Zj3Z4 there is no distinction between signature (3,0) and (2, 1), so we must find the hyperplanes He K4 such that tll restricted to H n lEE has signature (2,1), then take their images in Zj3Z4 . Note that we can find representatives of the cusps (cf. 2.3) in Z4 c K4:

(1,0,0,1),(0,1,0,1),(0,0,1,1),( -1,0,0,1),(0, -1,0,1),

(0,0, -1,1), (1, - 2, - 2,3), (1, 2, 2, 3), (2,1,2,3), (2, 2,1,3).

Letting the cusps (now in Z4) be denoted by Vi (i = 1, ... , 10), there are e30) = 120 sets of 3 of them. For each such triple, say (Vi' Vi' vk ), we can find an orthogonal base of the 3-space they span:

Wi = Vi + Vj

w2 = Vi - Vj

W3 = -(Vj' Vk)V i - (Vi' Vk)Vj + (Vi' V)Vk·

(Here (,) denotes the form tll for notational simplicity). Using this base we can calculate the signature: (note (Vi' Vj) < 0 for any i, j)

(Wi' Wi) = (Vi + Vj'Vi + Vj) = 2(vi,v) < 0

(W 2 , W 2 ) = (Vi - Vj'V i - V) = -2(vi , Vj) > 0

(W 3 , W 3 ) = - 2(vi , Vj)(Vj , Vk)(Vi , Vk ) > 0,

so on any such 3-space, the form tll has signature (2,1). Of these 120, there are exactly 15 which contain a 4th cusp, and the images of these 15 subspaces of Z4 in Zj3Z4 give the combinatorial description of the modular subvarieties. This amounts then, a postiori, to a linear combination, in Zj3Z, of the cusps given in 2.3. For example, the 3-plane spanned by (Vi' V2 ' v8 ) also contains V9: -Vi + v2 + V8 == v9(mod3).

2.7. We now come to the proof of

THEOREM 2.7.1. Y(l - p)" = IfD3 , the Di are the modular subvarieties of 2.5.

218 Bruce Hunt

i = 1, ... ,15 and the EJ. are the compactijication divisors of2.4. A. = 1, ... ,10. Proof Let Y(3, H) be the Fermat cover of degree 3 associated to the

arrangement H (see [Hu] for details on this construction), i.e. the variety whose function field is

where {Ii = O} = Hi' In [Hu] I constructed a desingularisation and calculated the Chern numbers of Y(3, H), as well as the logarithmic Chern classes of (Y(3,H),E), where E = 7t- 1(l:EJ.)' It turned out that C1 3 (Y,E) = 3C1 C2 (Y,E) (logarithmic Chern numbers) so by Kobayashi's generalisation ofYau's theorem quoted in [Hu], (also proved by Yau), Y - E is a smooth, non-compact ball quotient, Y its compactification. The desingularisation described in [Hu] is affected by blowing up 1?3 in exactly the same manner as above, so the smooth cover Y --+ iP 3 is a branched cover of iP 3 , or put differently, iP3 is a ball quotient; we just have to identify the group. Let r y be the group such that r y \1B3 = Y - E. Then r => r y, r = 7tl (1Jl3 - E - D), as r y \1B3 is a cover of iP3 which is unramified over iP3 - E - D. The quotient r/r y = (7../37..)9 is the Galois group ofY--+iP3 .

Utilising known results on the hypergeometric differential equation, r is the monodromy group of Appell's equation, number 1 in the [DM] list in dimension 3. Later we will identify this differential equation with the Picard Fuchs equation of the periods of Picard curves, whose monodromy group is easily identified with r(1 - py = U(3, 1; (9K(1 - p» (see §5-§6). The identification of this particular group is thus by means of the scheme:

(Fermat cover) .4 ([DM]- #) ~ (Picard curves) ~ (monodromy group).

Step 1 was done in [Hu]. Step 2 will be done in Sections 5-6, Step 3 in Section 6. The statements about Di and EJ. follow from [H01] (identification of the Di) and direct calculations (showing EJ. is an abelian variety as in 2.4).

Let us just mention that much of this is more or less well-known.

2.8. Finite covers In this section we clarify a few questions which were left untouched up till now.

We consider the following coverings:

A Siegel Modular 3{old that is a Picard Modular 3{old 219

We explain now the inclusion Y(2) c iP3, which will be given an easy proof in the next section. Delete uD; from iP3. Then the inclusion Y(2) c iP3 is such that the Humbert surfaces are the divisors E l , • .• , E lo ' As mentioned above, each D; is a Kummer modular surface (compactification divisor), and iP3 is the Igusa desingularisation of Y(2)*.

Taking that for granted, one has two natural covers,

Y(4)A --+ Y(2)A and

Y(1 - pfA --+ Y(1 _ p)A

(using obvious notations). We claim these are in fact both Fermat covers, of degrees 2 and 3, respectively. To see this, first note that both Pr(4}/Pr(2) and Pr(1 - p)/Pr(I- p)2 are abelian. In fact, it is true that rcn)/r(!)2) is abelian for any ideal !). (Steve Weintraub pointed this out to me. Just calculate (A + B)2 mod(!)2).) The coefficients are in lL/4lL/lL/2lL = lL/2lL and (!!K(1 - p)/(!!K(1 -p)2 = lL/3lL, respectively, and a matrix in PSp(2, lL) and PU(3, 1; (!!K), respectively, will have 9 independent entries.

Once we know the groups are correct, we just have to note that the fixed point set under these Galois groups which consists of the union uD; u Ej of 2.5, are hermitian symmetric, and in fact identical in the Fermat covers as well as in Y(4Y--+ Y(2Y and Y(1 - p)2A--+ Y(1 - p)A, respectively. This also allows us to count modular subvarieties and compactification divisors:

Siegel: 10'24 = 160 modular subvarieties, 15'23 = 120 compactification divisors,

Picard: 15'33 = 135 modular subvarieties, 10'34 = 270 compactification divisors.

We remark that this discussion finishes, modulo the proof of 3.3.3. below, the proof of 1.7.1. above.

3. Modular forms

In this paragraph we prove the theorem stated in the introduction, utilising for the proof modular forms. First we recall the structure of R(r(2)), a result due to Igusa [11]. We then deduce the structure of R(r(1 - p)). It turns out that these rings are dual to each other, i.e. the projective varieties Proj(R(r(2))) and Proj(R(r(1 - p))) are dual. This implies they are birational, and our theorem follows.

220 Bruce Hunt

3.1. Theta Constants In this section we review the work ofIgusa [11]-[12]. For later use we will need

theta constants of genus 2 and 4, so in this section we give definitions and results for any g. Let. E §g, z E ICg, and m = (m', m") E (p29.

DEFINITION 3.1.1. The theta function of degree g and characteristic m is

Om(., z) = L exp(!t(n + m').(n + m') + t(n + m')(z + m")). nelg

The corresponding theta constant is

Igusa has studied these theta constants. Some of his results are:

LEMMA 3.1.2. Om(.) == 0~mmod(1) satisfies exp(4nit(m')(m")) = -1.

The Siegel modular group r g(1) := Sp(g, Z) acts on the arguments (., z) by:

M(.,z) = «A. + B)(C. + D)-\(C. + D)-lz)

and on the characteristic itself by

M:m=(m',m")f-+(_DB -C) . !(diag(CtD)) A m + 2 diag(AtB) .

The behavior of the thetas under M is given by

LEMMA 3.1.3. (Igusa's transformation law), [11], p. 226

OMm(M(., z)) = K(M) exp(2ni<pm(M)) det(C. + D)1/2 x x exp(nitz( C. + D)-lCz)Om(., z),

where K(M) is some 8th root of unity,

(3.1.2a)

(3.1.2b)

<Pm(M) = -ttm'BDm' + tm"'ACm" - 2tm"BCm" - tdiag(AtB)(Dm' - Cm").

In particular, for the theta constants the formula becomes

3.2. The ring of modular forms for r(2). We review the results of Igusa describing R(r(2)), the ring of modular forms for

r(2). For the rest of this section we take the following particular case of 3.1: g = 2,

A Siegel Modular 3-fold that is a Picard Modular 310ld 221

m = (m', m") E t Z4. There are 16 such characteristics, 6 of which yield odd theta functions (Om(-r, z) = - Om(-r, - z)) and hence zero theta constants (this is a special case of 3.1.2). There are 10 even characteristics, and among their fourth powers there are 5 linear relations (Riemann theta formula). In fact,

THEOREM 3.2. [11, p. 397] Let

Yo = 0(0110><,r), Y1 = 0('o100><,r), Y2 = O('oooo><,r), Y3 = 0(1000><",r) - O('oooo><,r), Y4 = -0(1100><,r) - O('oooo><,r).

X10 = no;.

Then,

R(r(2)) = C[yo," . 'Y4' X]/,r, C the ideal generated by

This theorem implies in particular, that the (singular) quartic defined by R1 is the Baily-Borel embedding of X(2). The Yi are modular forms of weight 2 with respect to r(2), as follows from the transformation law 3.1.3. A more symmetric description of the same variety is given by van der Geer in [V, §5]. This is done by taking all thetas O!(-r), and the map into i!=D9,

(3.2.1)

The image being onto the i!=D4 which is cut out by the 5 linear relations. The equation for X(2)* is then ([V], 5.2)

(3.2.2)

In this description the action of Sp(2, Z/2Z) on X(2)* is the action on the characteristics 3.1.2b. It is known that each theta vanishes along exactly one of the Humbert surfaces, so we can identify this action with the action of 1:6 on H 1

described in 1.6. This replaces the rather artificial action 3.1.2b by the much more natural one ofl:6 on (Z/2Z)4 described in 1.6. This observation is due to van der Geer [V] and Lee and Weintraub [LW].

3.3. The ring of modular forms on Y(l - p)* We now proceed in the following manner: first we recall the (well-known)

identification of Y(l - p)* with the Segre Cubic. Then, since we know the

222 Bruce Hunt

coordinate ring of the Segre cubic, we know the coordinate ring of Y(1 - p)*. Finally it is not difficult to see that the natural coordinates used are indeed modular forms.

The Segre Cubic

There is a unique cubic 3-fold S in iP'4 with 10 ordinary doublepoints. S is given most symmetrically by the following 2 equations in the homogenous variables xo, ... ,Xs on iP's:

LX;=O

LX? = o.

The double points are (1,1,1, -1, -1, -1) and its permutations under ~6' which acts naturally on S by permuting coordinates. There are 15 iP'2,s lying on S:

Xa(l) + Xa(4) = Xa(2) + xa(S) = Xa(3) + X a(6) = 0, (fE ~6'

The GIT -Quotient

It was already mentioned above that the following was proved in [L W], see also [KLW]:

LEMMA 3.3.1. S = IfD3 = GIT-quotient # 1 in [DM]. (S denotes the (big) resolution of S).

From this and 2.7.1 it follows first that S = Y(1 - p)* and from this that the coordinate ring of Y(1 - p)* coincides with that of S, i.e.

R(Y(1 - p)*) = C[xo, ... , xs]/t&"

t&" = {Rl = ~x; R2 = ~x?

(3.3.2)

An important additional bit of information we get from the argument of2.7. is the following: The D;, being in the branch locus ofthe cover Y --+ iP3 , are pointwise fixed under the Galois group, hence subball quotients. (The only submanifolds in the universal cover 1B3 fixed under automorphisms are subballs.) This implies on the one hand the fact (that we already know) that the D; are modular subvarieties, and on the other implies that the divisors D; are the zero-loci of modular forms (since roots of their quotients give the field extension of the finite cover Y --+ iP 3

described in 2.5). Hence there are 15 modular forms (j;, i = 1, ... ,15, with the


property that each vanishes identically along precisely one of the Di and does not vanish identically at any other. Taking all of these forms as coordinates we get a map

ljJ: Y(1 - p) -+ 1P 5 C 1P14 t -+ bi(t)E S

onto the Baily-Borel compactification (the Xi above will be linear combinations of the b;) much in the spirit of van der Geer's (3.2.1.-3.2.2. above).

THEOREM 3.3.3. p3 is the common resolution of singularities of X(2)* and

Y(1 - p)*, i.e. we have a diagram

Siegel X(2)* Y(1 - p)* Picard.

Using the notation in the introduction, we have

(p3, D) corresponds to (§2, r(2))

(p3, E) corresponds to (IB\ r(1 - p)).

This follows from the above results as follows: we know X(2)* is the quartic described in 3.2, Y(1 - p)* is the Scgre cubic, and classically (1880's !) it is known that these varieties are dual to each other. This implies they are birational, and (11

and (12 have been explicitly described above. (11 blows down the Di to 1P 1's, (12

blows down the E j to ordinary double points.

3.4. Theta constants of degree 4 Assume for the moment we have a modular embedding (1B3, r(1 - p)) C

(§4, r(?)), where r(?) is a not further specified level subgroup. We will see later in Section 5 that this exists via lacobians of Picard curves. Consider the diagram

1 1

It follows from Igusa's results ([12, Cor., p. 235]) that theta constants can be used to embed X4(?) in projective space (ala Satake-Baily-Borel). Restricting these

224 Bruce Hunt

thetas to r(1 - p)\1EB3 yields Picard modular forms. This is the idea behind Feustel's proof (in dimension 2) of

THEOREM 3.4.1 [F, II]. The modular forms ~i mentioned below in 6.1.8 for U(2, 1; lDK (l - p» can be written asfollows in terms of theta constants:

This gives at least partial information on r(?): it contains the group r(6, 36). Since we have Di given by the equations, say

we can use Xl> X2 and X3 as coordinates on r(1 - p)\1EB2, so by 3.4.1, the Xi can be written in terms of genus 3 thetas when restricted to the Di • Hence in principle at least, we can take the modular forms in 3.3.2 to be genus 4 theta constants which restrict on the modular subvarieties Di to the theta constants in 3.4.1. It would be very interesting to get some explicit results in this direction. This is, however, a highly non-trivial task. One would first have to find an explicit embedding (1EB3, r) C (§4, r(?» together with 15 compatible sub-modular embeddings

(1EB3, r) C (§4' r(?» u u

(1EB2, r 2,1) c (§3, r i?».

The 15 subballs will give the intersection of (1EB3, r) with the zero locus of the sought for theta constants. Let us briefly describe what this looks like in local coordinates. Say

1EB2 = {(Zl>Z2)E IC IIzil2 < 1},

1EB2 = {Z3 = O} C 1EB3 = {(Zl>Z2,Z3)EIC3Irlz;l2 < 1},

and

[" 't'l 't'2

X'] [" 't'l

:} §4~ :: §3 = 't'l

't'3 't'4 X2 't'3

't'4 't's X3 't'2 't'4

Xl X2 X3 't'6


The modular embeddings are:

These 4 spaces parameterise:

§4 = principally polarised abelian 4-folds u :=: = Shottky divisor = locus of jacobians = genus 4 curves u

1EB3 = Jacobians with complex multiplication = Picard curves §4 ::::> §3 = genus 3 curves ( = degenerating abelian 4-folds)

1EB2 = Picard curves (Jacobians with complex multiplication)

II 1EB3 n §3 = Degenerate Jacobians with complex multiplication.

Next one would have to find the right level groups acting on §3 and §4 and show the equivariance of the diagram above with respect to the groups involved. Finally, one would have to describe the genus 3 thetas as special values of genus 4 theta constants on §4. As the embeddings involved are rather complicated (see for example those given by Resnikoff-Tai), this would seem to be a formidable task.

4. Tits buildings with scaffoldings

4.1. Let r c Go be an arithmetic subgroup, and P 1> ••• ,Pk a complete set of r-inequivalent maximal parabolics. One can construct a simplicial complex associated to r, the Tits Building, as follows:

vertices: Vi +-+ Pi

I-edges: vij +-+ Pi C Pj inclusions 2-faces: Vijk +-+ Pi c P j c P k flags

First, let r = Sp(2, Z). Then there are 2 maximal parabolics PI, P 2, corresponding to the O-dimensional and I-dimensional boundary components, respectively. There is one inclusion PI c P2 • Hence the Tits building is:

Sp(2, Z): VI 0-0 V2.

The building for r R = SU(3, 1; (JIK) is even more boring, being just one point:

SU(3, 1; (JIK) V10

226 Bruce Hunt

Now we consider the congruence subgroups r(2) and r(1- p). The building of r(1 - p) is still not too interesting, consisting of 10 disjoint verticies:

The interesting Tits building is that of r(2). There are 15 verticies Vb •. . , V15

corresponding to the I-dimensional components, and 15 verticies Wb ... , W15

corresponding to the O-dimensional components. Each Wi has 3 inclusions into v/s, and Vj contains 3 Wi:

(4.1.1.)

There are therefore 3(15 + IS).! = 45 edges, see Figure 4.1.2. We now "blow-up" this Tits building, i.e. replace each Wi by a 2-simplex:

~jly~b ·W·

- 1

v· -22

and we note that the blown-up Tits building is the dual complex of the normal crossings divisor D. This means we have 1 vertex for each component, 1 edge for each double intersection and 1 2-simplex for each triple point:

Vi ~ Di divisor (15)

ViVj ~ Di (") Dj (45)

~ViVjVk = W;. ~ Di (") Dj (") Dk (15)


4

Figure 4.1.2.

4.2. Modular Scaffoldings We consider a fixed locally symmetric space (X,~) = (f», r) (we recall that this

notational convenience explained in the introduction means X - ~ = r\f»), and modular subvarieties (Di' ~ n D;) = (f»i' r) for iE Index set.

DEFINITION 4.2.1. A finite set (f»i' r i) c (f», r), i = 1, ... ,N, is called a Modular scaffolding of (f», r), if:

(i) Each (f»;, r i) c (f», r) is a modular embedding. (ii) Each intersection (Di n Dj , ~ n (Di n Dj )) = (f»ij' rij) is a modular embed

ding in both (f»i' r i) and (f»j' rJ (iii) 3).,./ljEO Cl (X) = ~AiDi + ~Jlj~j, ~ = ~~j. Now it is an easy matter to see that the modular subvarieties discussed in 1.5 and

228 Bruce Hunt

2.5 form modular scaffoldings of X(2) and Y(1 - p), respectively. The necessary Chern class calculations can be found in [V, §3], for example.

4.3. Scaffoldings on buildings We now introduce a notion due to Lee-Weintraub ([LW1]), in a somewhat

different fashion. According to 4.2 a modular scaffolding on (X, L1) will be a normal crossings divisor.

DEFINITION 4.3.1. A scaffolding on the Tits building of (X, L1) is the dual complex of a modular scaffolding.

This dual complex is a simplicial complex consisting of 1 vertex for each component, 1 edge for each simple intersection, and so on. Now just looking at the example above, we have:

PROPOSITION 4.3.2. The scaffolding ofY(1 - p) is the blown-up Tits building of X(2). The scaffolding of X(2) is the Tits building of Y(1 - pl.

We now would like to combine the Tits building with the scaffolding of (X, L1). The easy way to do this is to take the graph of the normal crossings divisor L1 + D, L1 the compactification divisor and D the modular scaffolding as above. We call this complex the Tits building with scaffolding, Tbws.

PROPOSITION 4.3.3. The Tits building with scaffolding of r(2) and ofr(1 - p) are the same, and the scaffolding of r(2) is isomorphic to the Tits building of r(1 - p) and vice versa.

This is what we mean by saying that r(2) and r(1 - p) have dual Tits buildings with scaffolding. Now going back to the diagramm in 3.3.3 we recall the GIT-quotient had both L1 and D as compactification divisor. Consider its (blown-up) Tits building.

PROPOSITION 4.3.4. The GIT-quotient's (blown-up) Tits building coincides with the Tbws mentioned in 4.3.3. The scaffolding of this GIT -quotient is trivial.

Hence we see the Tits building of the GIT -quotient specialising to a Tbws in 2 different manners (dual to each other).

5. Modular interpretation

5.1. Families of marked lines Let S = {I, 2, 3, 4, 5, 6} and {XdieS a set of6 points on Pi, i.e. a map tjJ: S -+ pl.

The set of all such maps is naturally isomorphic to (Pl)6. The "diagonals" L1 are the divisors {Xi = Xi' i =F j = 1, ... , 6} and their intersections. PGL(2) acts on X.= (Pl)6 - L1 and the quotient, which is the GIT quotient discussed in 3.3, is isomorphic to p 3 - H. H the arrangement consisting of the 10 planes of 2.5. In

A Siegel Modular 3-fold that is a Picard Modular 3jold 229

other words we have a fibre space

with fibre PGL(2). The explicit form of this map is given in [LW, §6], as mentioned above in Section 3. Here we are using the identification of p 3 - H with S - D, S the Segre cubic, derived in 3.3. We let fFo denote this family of marked pl'S over p3 - H.

5.2. Hyperelliptic curves Let the {Xi};eS be as in 5.1, and consider the hyperelliptic curve

6

y2 = O(t - Xi) (5.2.1)

which is a branched cover of Pi, branched at each of the Xi. We let vi( 0 -+ fFo be the double cover of fFo branched at the {CPi}. To be precise, this object only exists in the category of algebraic stacks, i.e., the universal curve only exists locally, and cannot be globally constructed as -1 E r(2). However this is sufficient for our purposes. Hence, for each {x;} mod PGL(2) E p3 - H, the fibre (vi( O){Xi) is the double cover of pl branched at the {Xi} mod PGL(2). Thus we get a "fibre space"

vi( 0 -+ p3 - H = X(2) - E,

of genus 2 curves. We now describe the degenerations of 5.2.1 corresponding to the divisors

D (compactification divisor, 1.4) and E (Humbert surfaces, 1.5). We shall employ the following notation:

D;, i = 1, ... , 15: Dij=DinD/ D ijk = Di n Dj n Dk :

EA., A. = 1, ... , 10: DinEA.: Dij n EA.:

2 of the Xn coincide 2 pairs of the Xn coincide 3 pair of the Xn coincide 3 of the Xn coincide 1 pair and 1 triple of the Xn coincide 2 triples of the Xn coincide

Di : If 2 of the Xn coincide, then we have a double cover with 5 branch points; this is a genus one curve (elliptic) with one double point.

Dij: Reasoning as above we get here a rational curve with 2 double points.

230 Bruce Hunt

Dijk : Now we have only 3 branching points and at each branch point the degeneration is:

so that the double cover splits into 2 curves which are permuted by the Galois group, and each branch point is now a double point of the covering.

E).: when 3 ofthe Xn coincide, the curve is y2 = IIi(t - x n ), an elliptic curve. To see precisely what is going on, let

be the original equation, and write it as

and the degeneration is then given by letting A -+ 00. The limit curve is

(5.2.2)

with 4th branch point at infinity, and changing variable to t = At we get

which for A -+ 00 becomes

6

2 TI-y = (t - Xi), (5.2.3.) 4

with 4th branch point at infinity. Therefore, over pEE). the corresponding degeneration consists of 2 elliptic curves 5.2.2 and 5.2.3, meeting at their common branch point.

REMARK. The degeneration just described of course depends on the birational model of X(2)" used. That described here corresponds to the big resolution described in 2.4. There is also a small resolution of the ordinary double point, and in that case the degeneration is just an elliptic curve.


It is now obvious what the remaining degenerations are. A summary is given below in table 5.4.

5.3. Picard curves Let {Xn}nES be again as in 5.1 and consider the trigonal curve

(5.3.1)

This curve is called a Picard curve, being studied by Picard more than a century ago (actually it was the genus 3 curve he studied [P]). It is a genus 4 curve, whose Jacobian has complex multiplication by a cube root of unity, coming from the Galois action on the curve. As above we construct the triple cover ceo --+ :#'0 whose fibre (ceO){Xn) is the curve 5.3.1. We get a fibering

ceo --+ 1P3 - H = Y(l - p) - D,

of genus 4 curves. This result was first proven by Shimura [S]. We now describe the degenerations of 5.3.1. corresponding to the divisors

D (modular subvarieties, 2.5) and E (compactification divisors, 2.4). We use the same notations as in 5.2.

Di : If one pair of the x. coincide, we get a 3-fold cover, branched at 5 points, which is a smooth, genus 3 Picard curve (see the introduction and 2.5.)

Dij: Now the equation becomes y3 = IIt(t - xn ), and each branch point induces:

I ~:-;z: z 2 I '---1 3 . I

so after 2 branch points we see that the cover splits into one elliptic and one rational component.

Dijk : the picture now becomes:

so the cover splits into 3 rational curves. E .. : At a point where 3 of the Xn coincide, the action of the Galois group is ~--~

at that branch point, so it is a double point of the curve, and checking euler numbers it has genus 2.

232 Bruce Hunt

E .. n Dj : As above, where the pair of {xn} coincide the cover is still smooth, with the double point of E .. (from the triple of the {xn} which coincide), and an euler number calculation shows it is elliptic.

E .. n Dij: Here one gets a rational curve with 2 double points.

We can extend the family ~o --+ Y(l - p) - D to all of Y(l - py by adding in the degenerations just described. We denote this by ~A --+ Y(l - py. We note that this is not a semi-stable family of curves (the arithmetic genus changes).

5.4. Totally degenerate stable curves of genus 3. There is (at least) yet another moduli interpretation of p3, described in detail in

[GHv]. I am indebted to F. Herrlich and L. Gerritzen for this.

DEFINITION 5.4.1. A connected projective curve C, of arithmetic genus g, is called a totally degenerate stable curve of genus g if: (a) every irreducible component of C is a rational curve. (b) every singular point of C is an ordinary doublepoint, and (c) every non-singular component Cj of C meets C - L in at least 3 points.

Naturally associated with such a degenerate curve is a more combinatorial object, a tree;

DEFINITION 5.4.2. A connected set of mutually intersecting pl,S is called an n-pointed tree of projective lines if;

(i) the components intersect in ordinary double points (ii) The intersection graph is a tree, and

(iii) given is a set {Pi>' .. , Pn} of distinct ("marked") points on the components. It is called stable if, in addition,

(iv) on every component there are at least 3 points which are either singular or marked.

Totally degenerate curves are also parameterised by the (XdE p3. The generic curve is as follows:

I )'

stable 6-pointed tree ~ 3 II 5 C. ~ » ~ ( K

totally degenerate curve

~ In the following we describe the 6-pointed trees corresponding to the loci Dj and F .. as above:

A Siegel Modular 310ld that is a Picard Modular 3-fold 233

The corresponding degenerate curves depend in addition on the identification of the double points, for example the tree:

I Z. ~ l.f F gives both ~ X X )( ]I

~ (z.

4

We have included these in the following table.

6. Differential equations

6.1. Picard-Fuchs equations and monodromy Let n: Cfj -t S be a family of smooth curves over a parameter space S, i.e. Cfj, S are

algebraic varieties, and n is a hoI om orphic map. We consider the sheaf R 1n*C, which is a locally constant sheaf whose stalk at XES is just Hl(F x, q, where F x is the fibre at x. Since F x is a Riemann surface this stalk splits according to the Hodge decomposition:

H 1 ,O(F x) is the vector space of holomorphic 1-forms, and the decomposition is such that the position of H 1'°(Fx) in Hl(Fx, q = Hl(F, q depends holomorphically on x (here F is a typical fibre). Fix a base Wl (x), ... , Wg(x) of H 1,0(F x), and let <5 b ... , <5 2g be 1 cycles forming a basis for H 1 (F, Z). The period matrix of S is the g x 2g matrix

Q(x) =

which can be put in a normalised form Q(x) = (1g, Z(x)), where Z(x) E §g.

234 Bruce Hunt

'< o

-.: .,.; ..!!l "'" ~

c . ."

+ " '"

o 0 __ 0 fI II

~ ~ ~

A' :~

~ Q

cS.~ ~ ..... c:? uj c: <::

ltJ'" ll.J ...;


Associated with the family C(j -+ S one then gets a holomorphic period map:

ro: S -+ §g

x- Z(x).

(6.1.1)

Furthermore the Torelli theorem tells us that from Z E §g (mod Sp(g, Z)) we can recover the curve F x such that Z = Z(x). This is the general picture, and well-known.

Now fix a section of the (1, 0) part of R 11t.C, in other words, fix a holomorphic I-form ro(x) on each fibre, depending smoothly on x. Then the periods

are functions of the parameters, i.e. of x. It was first observed by Fuchs that, given ro, the 2g periods f~iro are all solutions of a linear differential equation of degree 2g, the Picard-Fuchs equation. See Katz [Ka] for general remarks and higher-dimensional analogues.

Getting back to our family C(j -+ S it is intuitively clear that if the fibres are "special" this leads to "special" Picard-Fuchs equations. For example, the special case of cycloelliptic curves

(6.1.2.)

is discussed in detail in part II of Holzapfel's book [HoI], yielding equations he calls of Euler-Picard type. Both of our families are in fact special cases of these curves (§5.2: hyperelliptic, §5.3.:trigonal). Our special Picard-Fuchs equations will turn out to be hypergeometric differential equations.

We now just sketch how the monodromy of an algebraic differential equation relates to the geometry of the base space. First ofall, the Picard-Fuchs equations have regular singular points (see e.g. [Gr] for precise definitions and statement of results). The singular locus ~ c S is a divisor, which may be taken as normal crossings. For any XE S - ~, local solutions cp(x) will be single-valued. If the space of solutions of the differential equation has dimension d, let CPt. ... , CPd be a local base at some fixed point .ES -~. For yE 1tl(S - ~,.) we can consider analytic continuation of the CPi' and the continued solution, upon returning to .' can be written as a linear combination of the cP;'s one started with, yielding a representation

(6.1.3)

called the monodromy representation. Of course, interesting things might happen

236 Bruce Hunt

if the image of p lies in some particular subgroup. For example, suppose C(J -+ Sis a family of elliptic curves over a curve, i.e. C(J is an elliptic surface. The Picard-Fuchs equation here is a second-order linear differential equation (see CSt])

(D2 - P(x)D + Q(x))F = ° (6.1.4)

with P and Q being singular at the discriminant ofthe family C(J -+ S. One then has the theorem CSt, §3]:

The differential equation 6.1.4 coming from a family C(J -+ S of elliptic curves is characterised as follows: (1) The dimension of the solution space is 2. (2) If Wl and W2 are 2 linearly independent solutions, then Wl (X)/W2 (x) E § 1 for

XES -~. (3) The monodromy lies in SL(2, £'). If S = IP\, ~ = {a, 1, oo}, the resulting equation 6.1.4 is the hypergeometric differential equation (in 1 variable). In that case the curve C(J x is given by the equation

y2 = (t - 1)t(t - x), (6.1.5)

i.e. a double cover of p 1 branched at {a, 1, 00 and x}. The differential in this case is

dx dx W=-=--;=====

y J(t - 1)t(t - x)'

and the solutions to the HGDE are Wi = LiW, Yh Y2 a basis of H l(C(Jx, £'). Taking their quotient we get a many-valued function

which takes values in the upper half plane (for XES - ~), yielding a diagram:

w(x) J (6.1.6)

The composition of these two maps is now a well defined (single-valued) function from S - ~ to Pi, since the many-valuedness of w(x) is precisely offset by the

A Siegel Modular 3101d that is a Picard Modular 3101d 237

SL(2, Z)-invariance of the modular function J. For S = !PI, ~ = {a, 1, oo}, this map is an isomorphism onto !p l _ {a, 1, oo}. For a complete list of examples in this case, i.e. elliptic surfaces over !pI with 3 singular fibres see [SH].

In the upcoming sections we will find diagrams analogous to 6.1.6 for the families.A and ~ considered in Section 5. Before we go into some of the details, let us begin by describing some of the very detailed results of Holzapfel for the analogous case where S is 2-dimensional (in our cases it will be 3-dimensional). One might start by considering the simplest 2-dimensional analogue of 6.1.5.

y2 = (t - l)t(t - xd(t - X2),

however this of course doesn't define a family of smooth curves. Instead, consider

(6.1.7)

branched also at 00. This is a genus 3 curve. Let S = !p2, ~ = 11 U ... U 16 , the six lines forming the arrangement 1.4.2. Then 6.1.7. defines a smooth family over S - ~, and its period matrix has the form (1'/1 = dx/y, '12 = dx/y2, '13 = X dx/y2)

C' A2 -P2A l A3 p2A2

pA, ) TI = Bl B2 -pBl B3 pB2 P2B3

Cl C2 -pCl C3 pC2 P2C3

where

Ai = f 'II, Bi= f '12, Ci = f '13, i = 1,2,3. a, a, a,

from which it follows that II is already determined by the first row: x f-+

(Sal '11 : Sa2'1l: Sa3'1l) is a many valued map from S - ~ into !p2, whose image lands in 1EB2 c !P2 • Hence, in analogy to 6.1.6 above, we get a diagram

w(x) (6.1.8.)

where w(x) = (Sal'll(X):Sa2'1l(X):Sa3'1l(X)), and the (~l: ~2: ~3) are automorphic forms of weight 1 ([Ho1, pp. 20-21], see also 3.4.1 above).

238 Bruce Hunt

6.2. The Hypergeometric Differential Equation (HDE) A general reference for this section is Terada [T]. Let iP'n be given synthetic

coordinates {(xo, ... , Xn+ 1) 11: Xi = O}. The natural way to get these coordinates is to consider the en + 1 C en + 2:

and we take iP'n = iP'(en + 1) with the induced homogenous coordinates. Since ~n + 2 acts naturally on en + 2 by permuting the coordinates and en + 1 is an invariant subspace, we get the so-called projective symmetric representation of ~n+2 (cf. [H02]). In different language ~n+2 is a unitary reflection group, and as such defines an arrangement H of planes in iP'n (cf. [OS]). The first 3 such arrangements are:

n = 1. H = {XOXl(XO - Xl) = O},

n = 2. H = {XOXIX2(XO - Xl)(Xl - X2)(X2 - xo) = O} = 1.4.2.

n = 3. H = {XOXIX2X3(XO - Xl) . .. (X3 - XO) = O} = 2.5.1.

The hypergeometric differential equation is:

(Xi - xj)8iojF + (Aj - 1)OiF - (Ai - 1)ojF = 0, 1 ~ i <j ~ n,

Xi(Xi - 1)o~ F + [Xi (Xi - 1) L (1 - Aa)/(Xi - xa) + Ao + Ai - 2 + 1 ~1X::S:;n, a#-i

+ (4 - 2Ai - Ao - An+ l)Xi]OiF + (Ai - 1) L Xa(Xa - 1)oaF/ 1 ~a:::;;n,lX#=n

(Xi - Xa) + Aoo (1 - Ai)F = 0 1 ~ i ~ n. (6.2.1)

where the Ai are rational numbers with ~Ai = n + 1. This is an algebraic differential equation on iP'n with regular singular points, non-singular off the arrangement He iP'n defined by ~n+2. A solution of6.2.1 is the period of a holomorphic 1-form on the curve

for parameters I1dv = 1 - Ai. This is part of more general results of [DM]. The exact conditions on the parameters l1i are known for the differentials Wi =

L,y-l dx to give a uniformisation into the balllEBn. If this is the case, the monodromy group is a discrete subgroup in PU(n,1), and the parameter space therefore is locally symmetric. These conditions are:

INT

A Siegel Modular 3jold that is a Picard Modular 3-fold 239

There are a finite number of parameter values {Jli} satisfying INT, listed in [T] and [DM, §14].

6.3. Picard Curves In this section we consider Picard curves as in Section 5,

y3 = t(t - 1)(t - xd(t - X2)(t - X3), (6.3.1)

which are curves as considered in 6.2 with parameter values

(Jli)=(1,··.,t)

yielding a particular HDE, the one listed as # 1 in [DM]. Let p: 1[1 (S - 1:)--. GL(3, C) be the monodromy representation.

THEOREM 6.3.2. The monodromy group of the HDE associated with thefamily 6.3.1. is the Picard lattice r(1 - p).

Proof. There are several ways to prove this. The most straightforward is analogous to Holzapfels proof of the surface case. [Ho pp. 12~125]. We sketch this argument, in a series of lemmas.

LEMMA 6.3.3. Let r be the monodromy group of the Picard family 6.3.1.,

1[l(S -1:) thefundamental group, Ylo ... , Yl0E 1[1(8 -1:) generators. Then P(Yi)E U(3, 1; (PK(1 - p».

COROLLARY 6.3.4. Y c U(3,1; (PK(1 - p» has finite index.

This follows from [DM]: ris an arithmetic lattice in PU(3, 1). From this one gets a diagram

and X is a finite branched cover. 6.3.2 then follows from

LEMMA 6.3.5. X is unbranched.

In fact, since (r(1 - p)\1B3 )* is simply connected, it has no unbranched covers, and 6.3.5 implies X is an isomorphism. 6.3.5 could be proved in a manner similar to [HoI], (his 6.3.11), which is a detailed analysis of X near the cusps.

There is a somewhat easier proof of 6.3.2, which we now sketch. The idea is simple: we will identify fundamental domains of r(1- p) and r in IB 3• First of all, both groups have 10 cusps. For r(1 - p) this was shown in 2.2, and for 1: it can be proved using the cover Y --. p3 described in 2.7. Secondly, both have a scaffolding

240 Bruce Hunt

consisting of 15 modular subvarieties. Proof of this is the same. Now utilising the map

from above, we see both groups have the same fundamental domainin 1B3: the fundamental domain of Y would consist of deg x-copies of the fundamental domain of r(1 - p) if r(1 - p) acted freely. But since r(1 - p) and Y have the same elliptic points (see the discussion in 2.8) this is sufficient.

Putting all this together, we get the following diagram:

(6.3.6)

where w(x) is the map given by the periods giving solutions of 6.2.1, and (~;) are the modular forms discussed in 3.3.

6.4. Hyperelliptic curves In this section we consider the family of Section 5.1.

(6.4.1)

Here, once again, we have exponents, this time

(J.l;) = (t, ... ,-!-)

and a corresponding HDE. However this set of (J.li) does not fulfill the condition INT, and so the situation here is different than that considered in 6.3 (as it should be). Sasaki and Yoshida [HSY] have succeeded in figuring this example out, yielding a diagram analogous to 6.3.6 in this case. This goes as follows:

THEOREM 6.4.2. Let E(-!-) be the HDE in 3 variables with all parameters = -!-. Let E /\ E be the wedge product, a 6-dimensional system. Then for these parameter values, the solution space of E /\ E splits into an irreducible 5-dimensional space + complement. This 5-dimensional space is spanned by the 2 x 2 minors of


the periods:

of the family 6.4.1.

Let us denote this solution by n = {O"l""'O"s}, Then we get a diagram

(6.4.3)

analogous to 6.3.6. Here we identify §2 with the non-compact dual of the hyperquadric iQl 3 in 1P'4, and n maps onto projective coordinates.

There is the notion of dual differential equation, [HSY, §4], and one of the things proven in [HSY] is that Em is the unique HDE which is self-dual. This raises

PROBLEM. Is the self-duality of E(t) related to the double structure of IFi>3 as locally symmetric space?

References

[B]

[C] [DM]

[F]

[GHv]

[Gr] [GS]

[HK]

Baker, H.F., Principles of Geometry, Vol. IV, Ch. 5, Cambridge University Press, London, 1940. Christian, u., Dber die Modulgruppe zweiten grades I, Math. Zeit. 85, (1964) 1-28. Deligne, P. and Mostow, G., Monodromy of hypergeometric functions and non-integral lattice monodromy, Pup/. Math. IHES, 63 (1986) 113-140. Feustel, I.M., Ringe automorpher Formen auf der komplexen EinheitskugeI und ihre Erzeugung durch Theta-Konstanten. Preprint P-Math-13/86, Akademie der Wissenschaften, Berlin. Gerritzen, L., Herrlich, F. and van der Put, M., Stable n-pointed trees of projective lines. To appear in Indagationes. Griffiths, P., Periods of integrals on algebraic manifolds. Bull. AMS 76, (1970) 228-296. Griffiths, P. and Schmid, W., Locally homogenous complex manifolds. Acta Math. 123 (1969) 253-302. Hashimoto, K. and Koseki, H., On class numbers of positive definite binary and ternary unimodular hermitian forms. Math. Gottingensis (1987) 21-22.

242 Bruce Hunt

[He] Hemperly, Ie., The parabolic contribution to the number of linearly independent automorphic forms on a certain bounded domain. Amer. J. Math., 94 (1972) 1078-1100.

[Hi] Hirzebruch, F., Automorphe Formen und der Satz von Riemann Roch. in: Symp. Inter. de Top. Alg., Unesco 1958.

[Ho1] Holzapfel, R.-P., Geometry and Arithmetic around Euler partial differential equations. D. Reidel Publishing Company: Boston 1986.

[Ho2] Holzapfel, R.-P., On the nebentypus of Picard modular forms. To appear in Math. Nach. [HSY] Hara, M., Sasaki, T. and Yoshida, M., Tensor products of Linear Differential Equations.

Preprint Kyushu University. [Hu] Hunt, B., Coverings and Ball Quotients. Bonn. Math. Schrif. 174, Bonn, 1986. [11] Igusa, J., On Siegel modular forms of genus two. Am. J. Math. 84 (1962) 175-200 and 86

(1964) 392-412. [12] Igusa, J., On the graded ring of Theta Constants. Am. J. Math. 86 (1964) 219-246 and 88

(1966) 221-236. [13] Igusa, J., On a desingularisation problem in the theory of Siegel modular forms. Math. Ann.

168 (1967) 228-260. [Ka] Katz, N., On the differential equations satisfied by period matrices. Pub!. Math. IHES 35,

71-106 (1968) [KL W] Kirwan, F., Lee, R. and Weintraub, S., Quotients of the complex ball by discrete groups. Pac.

J. Math. 130, (1987) 115-141. [K] Kodaira, K., On Kahler varieties of restricted type. Ann. of Math. 60, (1954) 28-48. [L W] Lee, R. and Weintraub, S., Cohomology ofSP4(Z) and related groups and spaces. Topology,

24, (1985) 391-410. [LW1] Lee, R. and Weintraub, S., Moduli spaces of Riemann surfaces of genus two with level

structures. I. Math. Gottingensis. 66 (1986), to appear in Trans. AMS, 1988. [OS] Orlik, P. and Solomon, L.: Arrangements defined by unitary reflection groups., Math. Ann.

261, (1982) 339-357. [P] Picard, E., Sur les fonctions hyperfuchsiennes provenant des series hypergeometriques de

deux variables. Ann. Ecole Norm. Sup. 3" ser. 62, (1885) 357-384. [SH] Schmickler-Hirzebruch, U., Elliptische Flachen fiber pi mit drei Ausnahmefasem und die

hypergeometrische Differentialgleichung. Diplomarbeit, Bonn, 1978. [S] Shimura, G., On purely transcendental fields of automorphic functions of several variables.

Osaka J. Math. 1, (1964) 1-14. ESt] Stiller, P., The differential equation associated with elliptic surfaces. J. Math. Soc. Japan 24,

(1972) 20-59. [T] Terada, T., Fonctions hypergeometriques F 1 et fonctions automorphes I. J. Math. Soc.

Japan 35, (1983) 451-475. [V] van der Geer, G., On the geometry of a Siegel modular threefold. Math. Ann. 260, (1982)

317-350. [Y] Yoshida, M., Fuchsian differential equations. Viehweg, BraunschweigfWiesbaden 1987 [Z] Zeitinger, H., Volumina und Spitzenanzahlen Picardscher Modulvarietaeten. Bonn. Math.

Schrif. 136, (1981).


Linearizing some 7L/27L actions on affine space

JERZY JURKIEWICZ Institute of Mathematics, University of Warsaw, P.K.i. N. 9p., 00-901, Warsaw, Poland


Let V be the affine space k" over an algebraically closed field k, G a linearly reductive group and A: G x V -4 V a group action with a fixed point, say the origin. Then for all g e G let me denote by A(g) the corresponding automorphism of V. We have

A(g) = L(g) + D(g)

where L(g), D(g)e End V, L(g) linear and D(g) the sum of terms of higher degrees. Let me recall the well known linearization problem: is the action A linearizable, i.e. conjugated to the linear action L: G x V -4 V (see e.g. [B] and [K])? Recently counter-examples have been found, see [S] and [K + S], so it is reasonable to study additional assumptions on the action A. One of them is considered in the present paper.

First I want to define some morphism (J A: V -4 V which turns out to be a conjugating automorphism for A, provided (J A is invertible. It will be done using the Reynolds operator i.e. the equivariant projection p: (J'J(G) -4 k. For a finite dimensional k-space W we have the unique linear map JG: Mor(G, W) -4 W such that for all linear maps f: W -4 k the induced diagram

f : Mor(G, W)- W

G 1f* 1f p:(J'J(G) I k

is commutative. Now let 4J: G -4 End(V) be such a map that the induced map G x V -4 Vis an algebraic morphism. Then W:= lin hull (4J(G» is finite dimensional, hence JG4J is a well-defined element of End(V). Let us apply the above to the map 4J: G 3 g 1-+ L(g-l )A(g) e End(V) and set (J = (J A = J 4J (compare [J]). We have

L(h)(J = r L(h)L(g-l)A(g) = (r L(hg-1)A(gh-1»)A(h) = (JA(h) JgeG JgeG

for all heG.

244 Jerzy Jurkiewicz

So (J invertible implies that A(h) = (J-1 L(h)(J. In particular the action A is linearizable. Later we will give an example of an action A which can be linearized but for which (J A is not invertible.

As mentioned in [J], the morphism (J A can be interpreted as an average deviation of A from being linear.

CONJECTURE (Kraft, Procesi). Assume for some d ~ 2

A(g) = L(g) + HAg) + Hd+1(g) + ... + H 2d - 2(g), for all g,

where Hm(g) is a homogeneous endomorphism of V of degree m. Then (JA is invertible. In particular the action A is linearizable.

THEOREM. The above conjecture is true in the following cases

1. G linearly reductive, d = 2 and char k -# 2, 2. G diagonalizable, d = 2 and char k arbitrary, 3. G = 71./271., d arbitrary and char k = o. Cases 1 and 2 are the objects of [1]. Proof for the case 3. Let I denote the identity map of V. We can write:

G = {I, L + D}, where Land Dare endomorphisms of V, L linear and D = Hd + ... + H 2d - 2. We have L2 = (L + D)2 = I. It follows that

LD + D(L + D) = o. (1)

Let me denote by Hd the d-linear symmetric map from Vd to V corresponding to Hd • Then we have

D(L + D) = DL + dHAL, ... , L, Hd ) + ...

where the first summand consists of terms of degrees d, ... , 2d - 2, the second is of degree 2d - 1 and all further summands have higher degrees. Considering the possible cancellations in (1) we obtain:

- LD = DL = D(L + D). (2)

By defnition (J = 1(1 + (I + LD)) = I -1DL. We will prove that I + 1DL is the inverse of (J.

LEMMA. D(1 + mDL) = D for m = 0, 1,2, ... Proof Suppose the above holds for some m - 1, m > O. By (2), D = D(1 + DL).

Therefore

D = D(1 + (m - 1)DL)(1 + DL) = D(1 + DL + (m - 1)DL(I + DL)).

Linearizing some 7L/27L actions on affine space 245

On the other hand DL(/ + DL) = - LD(/ + DL) = - LD = DL, and we are done.

Since char(k) = 0 the Lemma implies that D(/ + rDL) = D for all r E k. Then taking r = t we have

(/ - tDL)(/ + tDL) = / + tDL + tLD(/ + tDL) = /,

and the same applies if we interchange the order of factors at the left hand side. Q.E.D.

EXAMPLE OF A NON INVERTIBLE (1. Let the linear endomorphism L of P be given by L(x, y) = (x, - y) and an automorphism -r by -r(x, y) = (x - (x + y)2, Y + (x + y)2) so that -r-1(x, y) = (x + (x + y)2, y - (x + y)2). The automorphism -r - 1 L-r has order two, so it defines an action of the group of order two on k2• The corresponding endomorphism (1 = t(1 + L-r- 1 L-r) takes (x, y) to (x - u + v, y + u + v), where u = t(x + y)2, v = t(x - y - 2(x + y)2)2. Direct computation shows that the Jacobian determinant of (1 is

Therefore the endomorphism (1 is not invertible, while the considered group action can obviously be linearized.

References

[B] Bass, H., Algebraic group actions on affine spaces. Group actions on rings. Contemporary Mathematics, Vol. 43 (1985), AMS, 1-24.

[J] Jurkiewicz, J., On the linearization of actions of linearly reductive groups; Commentari Mathematici Helvetici 64 (1989) 508-513.

[K] Kraft, H., Algebraic group actions on affine spaces. Geometry today, Progress in Mathematics. Birkhauser 1984, 251-266.

[S] G.V. Schwarz, Exotic algebraic group actions, preprint. [K + F] Kraft, H. and Schwarz, G. V., Reductive group actions on affine spaces with one

dimensional quotient. To appear in Contemporary Mathematics, Proceedings of the Conference on Group Actions and Invariant Theory, Montreal 1988.

Normal forms and moduli spaces of curve singularities with semigroup (2p, 2q, 2pq + d)

IGNACIO LUENG01 & GERHARD PFISTER 2

lDto. de Algebra, Fac. de Matematicas, Univ. Complutense, 28040-Madrid, Spain; 2SektionMathematik, Humboldt-Universitaet zu Berlin, 1086 Berlin, Unter der Linden 6, DDR

Received 23 September 1988; accepted in revised form 20 September 1989

This work has been possible thanks to a Scientific Agreement between the Universidad Complutense and the Humboldt Universitaet. This cooperation agreement supported our stay in the Bereich Algebra and Departamento de Algebra respectively.

The aim of this paper is to classify map germs (C2, 0) ~ C and germs of curve singularities in C2 given by an equation of the type f = (x P + yq)2 + ~iq+jp>2pqaijxiyj = 0 with a fixed Milnor number JI.(f) = dimeC[[x,y]]/ (of/ox, of/oy). Here we always suppose p < q and gcd(p, q) = 1.

The moduli space M p,q,1l of the map germs described above is an affine Zariski-open subset ofC2(p-l)(q-l)-p-q+2+[q/p] devided by a suitable action of Jl.2pq (the group of 2pq-roots of unity) depending on JI.(f).

The moduli space Tp,q,1l of all plane curve singularities described above (which is the moduli space of all plane curve singularities with the semigroup (2p,2q,JI.-2(p-1)(q-1)+ 1) if JI. is even) is C(p-2)(q-2)+[q/p]-1 devided by a suitable action of Jl.d, d = JI. - (2p - 1)(2q - 1).

In both cases we also get an algebraic universal family. It turns out that the Tjurina-number -r:(f) = dime C[[x, y]]/(f, of/ox, of/oy) = JI.(f) - (p - 1)(q - 1) depends only on JI.(f) and p and q.

Constructing the moduli spaces we use the graduation of C[[x, y]] defined by p, q: deg xiyj = iq + jp.

We use the following idea to construct the moduli spaces: Let JI. = (2p - 1) (2q - 1) + d. We prove that for all f of the above type we can choose the same monomial base of C[[x, y]]/(of/ox, of/oy) (Lemma 2). We choose IX, p and that IX < p, IXq + pp = 2pq + d. Hence J.l{fo) = JI. with fo = (x P + yq)2 + xl1.yp. Then we consider a universal JI.-constant unfolding of fo as a "global" family (Lemma 3). The parameter space U of that unfolding is an affine open subset of C2(p-l)(q-l)-p-q+2+[q/p] The group II acts on U and M = U/II . t"'2pq p,q,1l t"'2pq'

248 Ignacio Luengo and Gerhard Pfister

To construct Tp,q,Il we consider the Kodaira-Spencer map of the universal Jl-constant unfolding. The Kernel of the Kodaira-Spencer map is a Lie-algebra acting on U. The integral manifolds ofthat Lie-algebra are the analytically trivial subfamilies of the unfolding.

We choose a suitable section transversal to those integral manifolds, which turns out to be isomorphic to C(p-2)(q-2)+[q/p]-1. The group Jld acts on the corresponding family and we prove that Tp,q,Il = C(P-2)(q-2)+[q/P]-1/Jld'

1. A normal form for map germs (C2,0) -+ C with initial term (xP + yq)2

LEMMA 1. Let

then Jl(f) ~ (2p - 1)(2q - 1) + d, and Jl(f) = (2p - 1)(2q - 1) + d iff

fd'= L (-1)[i/ p ]Wij oF O. iq+jp=2pq+d

Proof. Either f is irreducible or the components of f have the same tangent direction. This implies that

Jl(]) = Jl(f) - 2p(2p - 1),

where J is the blowing up

( )2

= x P + yq-p + L hi,j_i+pxiyj + i(q-p)+ jp>(q-p)p

+ L Wi,j_i+2pXiyj. i(q-p)+ jp;;.2(q- p)p+d

Using induction we may assume that

Jl(]) ~ (2p - 1)(2(q - p) - 1) + d

Normal forms and moduli spaces of curve singularities 249

and

J1(J) = (2p - 1)(2(q - p) - 1) + d iff 0 -# iq+jp=2pq+d

L (-1)[i/ p)Wi,j_i+2P' i(q- p)+ jp= 2(p-q)p+d

This yields the if part of the result. Now if f is as above and J1(f) > (2p - 1) (2q -1) + d, the condition h = 0 says that (x P + yq) divides ~iq+ jp=2pq+dWijXiyj, and adding -t~iq+jp=2pq+dWijXiyj to the first part of f one gets f= (x P + yq + .. y + terms of degree greater than 2pq + d. Continuing this way, we get the result.

LEMMA 2. Let f = (x P + yq)2 + ~iq+jp>2pqhijxiyj and J1(f) = (2p - 1) (2q -1) + d. Let y, () such that yq + ()P = 3pq - q - p + d, y < p. Let B = {(i,j)E N 2/i < 2p -1,j < q -1} u {(i, j)EN2/i < p, j < q} u {(i, j), (i, j)EN2/i < y, j < () + q}. Then {Xiyj}(i,j)EB is a base ofC[[x, y]]/(of/ox,of/oy).

Proof. We use the algorithm of Mora (cf. [3]) to compute a Groebner base of the ideal (of/ox, of/oy). We consider C[[x, y]] as a graded ring with deg x = q, deg y = p. Let f1 = 1/2p(of/ox) and f2 = 1/2q(of/oy).

Consider S(f1' f2) = yq-1 f1 - x p- 1 f2 and let f3 be the reduction of S(fb f2) = yq-1 f1 - x p- 1 f2 with respect to the initial terms X2p - 1 resp. x Pyq-1 of f1

resp. f2' i.e.

f3 = L lixYili Yi pq - p resp. > pq - q. f3 -# 0 because of J1(f) < 00. Let k be the minimal such that lk -# 0, i.e, IkxYk/k is the initial term of f3' Consider now

S(f2' f3) = Iklk-q+1f2 - xv- Ykf3

= lklk+q + terms of degree> P()k + pq

=:f4'

It is not difficult to see that the reductions of S(f1' f3) and S(fi, f4) i = 1,2,3 with respect to the initial terms of fb f2' f3, f4 are zero, i.e. f1' f2' f3, f4 is a

Groebner base of (aJlax, aJlay). This implies that

{Xi yi}(i,i)EB" B' = {(i, j), i < 2p - 1, j < q - 1} u {(i, j),

i < p, j < 15k} u {(i, j), i < Yk, j < 15k + q},

is a base of C[[x, y]]/(aJlax, aJlay).

q -1 B'

p 2p -1

This implies /1(f) = (p - l)(q - 1) + qYk + pc5k and therefore Y = Yk and 15 = 15k and B = B'. D

LEMMA 3. Let J = (x P + yQ)2 + ~iQ+ip>2pQaijxiyi and /1(f) = (2p -1) (2q -1) + d.

Let y,c5 be defined by

Y pq, i ~ p - 2, j ~ q - 2};

Bl = {(i, j), iq + jp ~ 2pq + d, i < p, j < c5}

U {(i, j), iq + jp ~ 2pq + d, i < y, j < 15 + q}.

There is an automorphism <p: C[[x, y]] --+ C[[x, y]] such that

Jor suitable hij, wij E C. Proof Using Lemma 1 we may assume that

Assume that there is an automorphism cp(k) such that

f(cp(k») = (xp + yq + L hl~)Xiyj + L bl~)Xiyj)2 + (i,j)eBo iq+ jp<>pq+k

+ L Wl~)Xiyj + L Cl~)Xiyj, (i,j)eB, iq+ jp<>2pq+k

cp(l) = identity. Now

L Cl~)Xiyj = (x P + yq)H + L (-I)[i/Plcl~)xioyjo iq+jp=2pq+k iq+jp=2pq+k

for a suitable homogeneous H of degree pq + k and io < p, ioq + joP = 2pq + k. If ~iq+jP>2pq+k(-I)[i/Plc\~);60 and (io,jo)¢;B 1 then k~pq-q-p+d

(Lemma 1) and jo ~ 15, io ~ Y or jo ~ 15 + q.

Let IX, {3 be defined by qlX + p{3 = 2pq + d, IX < p, then wap ;6 O. Notice that IX - 1 == Y mod p and {3 - 1 == 15 mod q.

Let

g:= x P + yq + L hlYxiyj + L bl~)Xiyj (i,j)eBo iq+ jp<>pq+k

and

( Og 0 og 0) ill:=e'x~y~ oy ox - ox oy ,~q+l1P=k-pq+p+q-d

e:= 1 . L (_I)[i/pl-[a-l+~/P1+lc!~) (lXq + {3p)waP ip+jq=2pq+k

. h (i" ) _ {(io - y, jo - 15) if jo ~ 15, io ~ Y WIt <,,11 - .

(io - Y + p, jo - 15 - q) If jo ~ 15 + q.

Let ljI: C[[x, y]] -+ C[[x, y]] the automorphisms corresponding to the vector field ill, then g( ljI) = g.

Hence,

f(ljI 0 cp(k») = g2 + L Wl~)Xiyj + L Cl~)Xiyj (i,j)eBo iq+ jp<>2pq+k


and

L (-l)[i/Plcl~) iq+jp=2pq+k

L (-l)[i/Plcl~) + (_l)[a-1+~/pl(O(q + {3p)wap • e.

iq+ jp=2pq+k

If (io,jo)rJ;B 1 we may assume now that ~iq+jP=2pq+k(-1)[i/Plcl~) = O. Let gl :=g +!H and

" b(k)Xi j + .1H + Ogl + Ogl - " d(,.kJ}xiyj. L... ij Y 2 mkax nkay - L... iq+jp=pq+k (i,j)EBo

The degree of the initial part of mk resp. nk is q + k resp. p + k. We define cp(k+ 1) by

cp(k+ l)(X) = cp(k)(X) + mk

cp(k+ 1)(y) = cp(k)(y) + nk

and

wlr 1) = wl~) if (i, j) #- (io, jo)

W~k+.1)=W~k). + " (-1)[i/ P1 c (,.kJ) if (io,j·o)EB 1 . lO , }O lO ,}o l..J

iq+jp=2pq+k

Then

f(cp(k+ 1)) = (xp + yq + L hl~+ l)Xiyj + L bl~+ 1)Xiyj)2 + (i,j)EBo iq+ jp;.pq+k+ 1

+ L wl~+ l)Xiyj + L cl~+ l)Xiyj (i,j)EB, iq+ jp;.2pq+k+ 1

for suitable M~+ 1), d~+ 1). 'J 'J

LEMMA 4. Let

iq+jp>2pq

and J1(ft) = (2p - 1)(2q - 1) + d for t E C.

D


Let y, b, Bo, Bl be as in Lemma 3. There is a C[t]-automorphism CPt: C[t][[x, y]] -+ CCt][[x, y]] such that

!r(cpt) = (xp + yq + L hij(t)Xiyj)2 + L Wij(t)xiyj (i,j)eBo (i,j)eBl

for suitable hiit), Wij(t)EC[t].

The proof is similar to that of Lemma 3. o Let us consider the family

F(x, y, H, W) = (xp + yq + L HijXiyi)2 + L Wijxiyi (i,j)eBo (i,j)eBl

depending on the parameters H = (HiJ(i,j)eBo' W = (Wij)(i,j)eBl and define N = #Bo + #Bb then J.1(F) = (2p -1)(2q -1) + d on the open set V defined by WaP 'I- 0, aq + pp = 2pq + d, in CN = Spec C[H, W]. Notice that N = 2(p -1) (q - 1) - p - q + 2 + [q/p] is not depending on d!

The group of 2pq-roots of unity acts on V:

o THEOREM 1. V / J.12pq is the moduli space of all functions

f = (x P + yq)2 + L aijXiyj iq+ jp> 2pq

with J.1(f) = (2p - 1)(2q - 1) + d and F is the universal family. Proof Using Lemma 3 we have to prove the following

LEMMA 5. Let cP be an automorphism cp: C[[x, y]] -+ C[[x, y]] such that

F(cp(x), cp(y), Ii, w) = F(x, y, h, w)

for (Ii, w), (h, w) E V S;; CN then A· (Ii, w) =(h, w) for a suitable A E J.12pq' Proof. Let x:=cp(x), y'=cp(y) then grouping the squared part of(*) one

gets:

(xp + yq + x P + yq + LnijXiyj + Lhijxiyj) x

x (xp + yq - x P - yq - IJiijxiyj + L hijxiyj)

= L WijXiyj - L WijXiyj.

This equation implies obviously that the degree of the initial term of </J(x) is ~ q

and

- 1P ( + " (2) i j) 1 Y = JI. Y L. aij x Y ,Jl.EJl2pq· iq+ jp> p

We may assume that A = 1 and prove (Il, w) = (h, w). Let the degree of the leading parts of both sides of the above equation be

2pq + m and let r be the degree of </J, i.e. aD) = 0 if iq + jp < q + r, aU) = 0 if iq + jp < p + r and aD) i= 0 or al;) i= 0 for suitable i, j with iq + jp = q + r resp. iq + j p = p + r.

1. Step. We prove that (a) r ~ pq - p - q

L aWxiyj=~yq-l.k, iq+ jp=q+r P

L a!~)xiyj = - ~XP-l. k iq+jp=p+r 'J q

(b) h = nand wij = wij if iq + jp < 3pq - p - q + d. First of all m ~ d + r because the leading part of the left side of the equation is divisible by x P + yq and m < d + r would imply that the leading part of the right side is a monomial. This implies Wij = Wij if iq + jp < 2pq + d + r. Now hij = nij

if iq + jp < pq + r. Otherwise the leading part of the left side of the equation would be 2(xP + yq)(hij - nij)xiyj for some i, j with iq + jp < pq + r and therefore of degree 2pq + r < 2pq + m.

Now suppose r < pq - p - q. Then there is at most one monomial of degree p + r resp. q + r.

If iq + jp = pq + r for some (i, j) E Bo and

qio + pjo = q + r

qi 1 + pjl = P + r

then

otherwise the leading part of the left side of the equation would have degree 2pq + r < 2pq + m.

But (i, j)E Bo, i.e. i < P - 1 and j < q - 1. This implies hij = nij, al~}o = al;}, = 0 (because of r < pq - p - q we have il < P - 1). This is a contradiction since al~}o i= 0 or al;}, i= 0 by the definition of r.

Similarly one gets a contradiction if there is no (i, j) E Bo with qi + pj = p + r, resp. no io,jo with qio + pjo = q + r resp. no ibjl with qi1 + pj1 = P + r.

This proves that r ~ pq - p - q. With the same method we obtain

L iq+jp=p+r

and L aljlxiyj = ~ yq-1k. iq+ jp=q+r p

(b) is clear now by the choice of Bo and the fact that r ~ pq - p - q.

2. Step. We prove that r ~ 2pq - p - q. Assume that r < 2pq - p - q. Then deg k < pq, i.e., k is a monomia1. The leading part of the left side of the above equation is divisible by x P + yq. The leading part L of the right side is

if iq + jp = 2pq + m for some (i,j)EB1 or

- k(1X a-1 P+q-l P a+p-1 P-1) wap • - X Y - -x y p q

if iq + jp =f. 2pq + m for (i, j) E B 1 .

Let k = K' x~y~. If IX + ~ - 1 < p, then i = IX + ~ - 1 and j = P + 1'/ + q - 1. If IX + ~ - 1 ~ p, then i = IX + ~ - 1 - p and j = P + 1'/ + 2q - 1. But (IX + ~ - 1, P + I'/+q-1)¢Bl and (1X+~-1-p,P+I'/+2q-1)¢Bl. This implies L= waP'kxa-1yP-l«IX/p)yq - (P/p)x P ) which is not divisible by x P + yq. This is a contradiction and therefore r ~ 2pq - p - q.

Now iq + jp ~ 4pq - 2p - 2q + d for (i,j)EBl then wij = wij for all (i,j)EB l .

D

2. The construction of the moduli space

We will construct the moduli space of all plane curve singularities given by an equation (x P + yq)2 + Liq+jp>2pq aijx iyj = 0 with fixed Milnor number Jl.

For Jl being even we get especially the moduli space for all irreducible plane curve singularities with the semigroup r = (2p, 2q, Jl- 2(p - 1)(q - 1) + 1).

We use the family

V(F) ~ U x C2 ~ U

constructed in Theorem 1.

U admits a C*-action defined by

We get

If J.l = (2p - 1)(2q - 1) + d and rxq + pp = 2pq + d, rx < p, then U £: CN was defined by W",fJ oF O.

For the construction of the moduli space it is enough to consider the restriction of our family to the transversal section to the orbits of the C*-action defined by

W",fJ =1. Let W' be defined by W = (W",fJ, W') and G(x, y, H, W') = F(x, y, H, 1, W').

The parameter space of Gis CN - 1 = SpecC[H, W']. The group J.ld of dth roots of unity acts on the family

induced by the above C*-action

G(AqX, APy, h, w') = A 2pqG(X, y, A 0 (h, w)) A e J.ld'

LEMMA 6. Let cp: C[[x, y]] -+ C[[x,y]] be an automorphism and ueC[[x, y]] a unit such that

u· G(cp(x), cp(y), h, w') = G(x, y, Ii, w')

then there is a A e J.ld such that (h, w') and A 0 (Ii, w') are contained in an analytically trivial subfamily of V(G)-+CN - 1.

Proof. Let

cp(x) = Lal})xiyi and cp(y) = LaWxiyi, u = LUiixiyi.

U' G(cp(x), cp(y), h, w') = G(x, y, Ii, w')

implies

(1) aU> = 0 if iq + jp < q

(2) a(1)2p = a(2)2q = a(1)p a(2)q = u- 1 1,0 0,1 1,0 0,1 0,0'

Let a~~~ = Aq and a~~)l = AP•

We will prove later that Ad = 1.

Now we may assume that A = 1 and prove that (h, Wi) and (Ti, Wi) are contained in an analytically trivial subfamily of V( G) -+ CN - 1.

We choose (1) u(t) E C[t] [[x, y]] with the following properties u(O) = 1, u(1) = u and u is

a unity for all t E IC. (2) ({>t: C[t] [[x, y]] -+ C[t] [[x, y]] with the following properties ({>o = identity,

({> 1 = ({> and ({>t is an automorphism of positive degree for all t E IC. Let H(t):= u(t)G({>t(x), ((>t(Y), h, Wi) and apply Lemma 4. There is an C[tJ-auto

morphism <l>t: C[t] [[x, y]] -+ C[t] [[x, y]] such that

H(t) = F(x, y, h(t), w(t))

for suitable hij(t), wij(t) E C[t] with the property

h(O) = h

w(O) =(1, Wi).

H(t) has a constant Milnor number, i.e. wa,p(t) has to be constant. This implies

H(t) = G(x, y, h(t), w'(t)).

But,

G(x, y, h(1), w'(1)) = H(l) = G(l (x), <1>1 (y), Ti, Wi).

Using Lemma 5 and the fact that <1>1 has positive degree we get

11 = h(1)

Wi = w' (1),

i.e. (h, Wi) and (Ti, Wi) are in the trivial family

G(x, y, h(t), w'(t)) = u(t)G(t({>t, h, Wi).

To finish the proof of Lemma 6 we have to prove

LEMMA 7. Let

oc < p, ocq + fJp = 2pq + d.

Let <p: lC[[x, yJJ --+ C[[x, yJJ be an automorphism with the property

<p(x) = AqX + terms of degree> q

<p(y) = APy + terms of degree> p

and u a unit, such that

then Ad = 1. Proof. u = A 2pq + terms of higher degree.

u'f2 = A2Pq(XP + yq + L a:~)xiyj)2 + A2pqx~yP + L ~~)Xiyj iq+jp>pq iq+jp>2pq+d

+ A2pq+dx~yP + L ~;)Xiyj iq+ jp;;,2pq+d

f . bl -(k) v<,k) or sUlta e aij , ij' This implies

( 2XP + 2yq + L (a:;) + a:~»)xiyj), L (a:;) - a:~»)xiyj iq + jp > pq iq + jp > pq

= (1- Ad)X~YP + L A - 2Pq(~~) - 5:;»)xiyj. iq+ jp > 2pq+d

Because the leading term of the left side of the equation is divisible by xP + yq, we get Ad = 1. 0

We consider now the Kodaira-Spencer map of the family

V( G) --+ CN - 1 :

p: DerclC[H, W'J --+ C[H, W'J[[x, YJJ/( G, ~~, ~~)

defined by

p(b) = class(bG).


The kernel of the Kodaira-Spencer map is a Lie-algebra L and along the integral manifolds of L the family is analytically trivial. We will choose a transversal section to the integral manifolds of L and divide by the action of Jld to get the moduli space. To describe this transversal section we choose a suitable subset of B 1 :

fJ + q ';;-// ,////'" -::./~&

fJ B2

fJ-q+1

y + 1

y p 2p -1

Let M.= #Bo + #B2 = N - (p -1)(q -1) = (p - 2)(q - 2) + [q/p] - 1. Let W".= (Wij)(ii)eB2 and eM = Spec e[H, W"}

Gu(x, y, H, W"):= (xp + yq + L HijXiyi)2 + xlZyP + L W;jXiyi (i,j)eBo (i,i)eB2

As before Jld acts on the family V(Gu) £; e2 x eM -+ eM.

THEOREM 2. eM / Jld is the moduli space of all plane curve Singularities defined by an equation

(xp + yq)2 + L ai]xiyi = 0 iq+ ip > 2pq

with Milnor numbers Jl = (2p - 1)(2q - 1) + d and Gu is the corresponding universal family.

Especially the Tjurina number 'r = Jl - (p - 1)(q - 1) only depends on Jl for these singularities.

COROLLARY. Let r = (2p, 2q, 2pq + d), d odd, a semigroup. Then e(p-2)(q-2)+lq/p]-1/Jld is the moduli space of all irreducible plane curve

singularities with the semigroup r. Gu is the corresponding universal family. Proof. To prove the theorem we compute generators of the kernel of the


Kodaira-Spencer map. Let

G(O) = x p + yq + L HijXiyj

(i,JJeBo

(i,j)eBI iq+jp>2pq+d

w: i j . ijX y, I.e.

Let () E Dere C[H, W'] be a vector field which belongs to the kernel of the Kodaira-Spencer map, i.e.

( oG OG) {)G E G, ox 'oy .

Now

()G = 2G(O)

(i,j)eBo (i,j)eBI iq+ jp>2pq+d

for a suitable SE C[H, W'] [[x,y]].

. . oG oG {)W;·x·yJ = S· G mod - -

'J oX ' oy

We will associate to any monomial xayh, (a, b) oF (0,0), a vector field {)a,bE Dere[H, W'] such that

a b (OG OG) {)a,b G = X Y Gmod ox' oy .

Obviously {{)a,b} generate the kernel ofthe Kodaira-Spencer map as C[H, W'} module.

Now consider

for suitable Eft E C[H, W'], L10 L2 E C[H, W'] [[x, y]],

L1 = ! xa + 1 yh + terms of higher degree p

then

oG(l) (OG OG) -1L 2 By mod OX' oy .

The leading term of

oG(l) OG(l) XaybG(l) - tL -- - 1L --

1 ox 2 oy

Using Lemma 2 we get

xaybG(l) _ tL oG(l) -tL oG(1) 1 ox 2 oy

L Dfjxiyimod(~~, ~Gy) (i,J)eB,

iq+ jp;;.2pq+d+aq+bp

for suitable Dfj E IC[H, W']. This implies

xaybG = G(O) L Efjxiyi + (i,J)eBo

L Dfjxiyimod(~~, ~Gy) (i,j)eB,

iq+ jp;;.2pq+d+aq+bp

We define for (a, b) i= (0,0)

(j 1 "Eab 0 "Dab 0 a,b = -2 £..., ij oH .. + £..., ij ow..

lJ lJ

The vector fields (ja,b have the following properties: (1) (ja,b is zero if aq + bp > 2pq - 2p - 2q (2) (ja,b(Wij) = 0 if iq + jp < 2pq + d + aq + bp (3) (ja,b(Wij) = - d/2pq if (i,j) = (oc + a, f3 + b) or (i,j) = (oc + a - p, f3 + b + q)

(in this case iq + jp = 2pq + d + aq + bp).

(4) (ja,b(Hij) = 0 for all (i, j)E Bo if aq + bp ~ pq - 2p - 2q (5) For iq + jp ~ 3pq + d - q and (i,j)E Bl there is (a',b') such that

(i,j) = (IX + a', p + b') or (i,J) = (IX + a' - p, p + b' + q),

i.e. (ja',b,(Wij) = - d/2pq.

(6) For any (a, b), aq + bp < pq - q, always (IX + a, p + b) or (IX + a - p, p + b + q) E B l' i.e. (ja,b(Wij) = - d/2pq for the corresponding (i, j) E B l'

(1) and (4) hold because of the fact that

iq + jp ~ 2pq - 2p - 2q if (i,J)E Bo

iq + jp ~ 4pq + d - 2p - 2q if (i, J) E B1 •

(2) and (3) hold because of the fact that the leading term of G(l) has degree 2pq + d and because of Lemma 2. To prove (5) we consider two cases

1. Case i ~ IX - 1 In this case (i,j)E Bl implies i ~ P - 1. But iq + jp ~ 3pq + d - q implies

j ~ p. Then a' = i-IX, b' = j - p have the required properties. Notice that i < IX - 1 and iq + jp ~ 3pq + d - q implies (i,j)rtB 1

2 Case i < IX - 1 Now iq + jp ~ 3pq + d - q implies j ~ p + q then a' = p + i-IX, b' =

j - p - q have the required properties. (6) is similar to (5):

We may assume that (IX + a, p + b) rt B1 • This implies 2p - 3 ~ IX + a ~ p and IX ~ 2 because b ~ q - 2, a ~ p - 2. Suppose (IX + a - p, p + b + q)rt Bl then p + b + q ~ p + 2q - 1, i.e. b ~ q - 1, or IX + a - p ~ IX - 1, i.e. a ~ p - 1, but this is not possible. 0

For the coefficients to the vectorfields (ja,b we get, because of (1)-(6), the following matrix:

This implies that the kernel of the Kodaira-Spencer map is generated (as C[H, W']-module) by the vector fields

and

~' _ _ pq ~ EfJ~ _0_ + _0_ + I,m - d L. ~H ~W

(i,j)eBo U ij U I,m

+ L (- 2Pq) Di!~ _0_ (i,j)eB, d 'J oWij

2pq+d+aq+bp<iq+ jp< 3pq+d-q

x (I, m) E B1 \(B2 U {(IX, P)}, Iq + mp < 3pq + d - q,

. h (I ) {(IX + a, P + b) if I ~ IX Wlt ,m =

(IX + a - p, P + b + q) else,

The vectorfields ~;,m act nilpotentely on C[H, W']. Namely, if we consider CEH, W'] as a graded algebra defined by deg Hij = pq - iq - jp < 0, deg Wij = 2pq - iq - jp < 0 then the Efj resp. Dfj are polynomials in C[H, W'] of degree ~ aq + bp + pq - iq - jp resp. ~ aq + bp + 2pq - iq - jp. Notice that their degree is always ~O. Let A E CEH, W'] be any polynomial of degree o ~ deg A = s (deg A = minimum of the degrees of the monomials in A). Then the degree of ~;m(A) > s. Therefore there is some n with ~;::'(A) = O.

LEMMA 8. Let A be a ring offinite type over afield k. L ~ Derk(A) a Lie-Algebra. Let ~ l' ... '~r vector fields with the following properties:

(1) ~1'"'' ~r ELand L ~ ~~iA (2) [~i' ~j] E ~k>max{i.j} ~kA (3) There are x l' ... , Xr E A such that

(4) ~1"'" ~r act nilpotentely on A.

Then AL [X 1 , ••• , xr ] = A.

The Lemma is not difficult to prove. A similar lemma was used in the construction of the moduli space for curve singularities with the semi-group <p, q) (cf. [1], [2]).

Obviously AL is the ring of all elements of A being invariant under ~1"'" ~r'


Now A"r[x,] = A and the conditions (2)-(4) of the lemma are satisfied for c51, ... , c5,-1 acting on A"r.

Now we may apply the Lemma 8 to the kernel ofthe Kodaira-Spencer map and its generators {c5;m}.

Because of the lemma the geometric quotient ofCN - 1 = Spec C[H, W'] by the action of the kernel ofthe Kodaira-Spencer map exist and is isomorphic to the transversal section to the maximal integral manifolds (which intersect therefore each of these integral manifolds exactly in one point) defined by

W/,m = 0, (l,m)e B1 \(B2 u {(a, pm.

Now we use Lemma 6 and get Theorem 2. Notice that

is a base of the free C[H, W'J-module C[H, W'] [[x, y]]/(G, fJG/fJx, fJG/fJy). This implies Jl. - 'r = #(B1 \B2 ) = (p - 1)(q - 1). 0

References

[1] Laudal, O.A., Martin, B., and Pfister, G., Moduli of irreducible plane curve singularities with the semi-group (a,b), Proc. Con! Algebraic Geometry Berlin. Teubner-Texte 92 (1986).

[2] Laudal, O.A., and Pfister, G., Local moduli and singularities, Springer, Lecture Notes 1310 (1988). '

[3] Mora, F., A constructive characterization of standard bases, Boll. U.M.I. sez. D. 2 (1983) 41-50. [4] Zariski, 0., Le probleme des modules pour les branches planes, Ed. Hermann, Paris 1986.


On Castelnuovo's regularity and Hilbert functions

UWENAGEL Martin-Luther-University, Dep. of Mathematics, DDR-4010 Halle, German Democratic Republic


Abstract. New bounds for Castelnuovo's regularity are established. As a consequence, we show that a property of Hilbert functions stated by J. Harris and D. Eisenbud in [7], p. 82 is only true for curves and false for higher-dimensional subschemes. The letter ofW. Vogel [25] gives rise to study this property again.

1. Introduction

Castelnuovo's regularity was first defined by D. Mumford [10], who attributes the idea to G. Castelnuovo, for coherent sheaves on projective spaces. In a more algebraic setting it was defined by D. Eisenbud and S. Goto [3] and A. Ooishi [13] (see Section 2). It comes out that Castelnuovo's regularity gives an upper bound for the maximal degrees of the syzygies in a minimal free resolution [3]. D. Bayer and M. Stillman [2] showed that an estimate of the regularity of an ideal gives a bound on the complexity of algorithms for computing syzygies. In [3], p. 93 D. Eisenbud and S. Goto stated the following well-known conjecture on such an estimation.

Let X s;;: IP~ (K an algebraically closed field) be a nondegenerate, that is, X is not contained in a hyperplane of lPn, irreducible, reduced subvariety then holds

reg(X) ~ degree(X) - codim(X) + 1.

So far, this conjecture has been proved for curves [6] and, if char(K) = 0, for smooth surfaces [9] (see also [5]), for a large class of smooth threefolds in 1Pi: [18], Theorem 3.3, and if X is arithmetically Buchsbaum or degree(X) ~ codim(X) + 2 [21]. In the other cases only weaker results are known by using certain correction terms (see, e.g., [1], [22], [23], [5], [11], [18]). We also note that [11] and [12] describe applications ofCastelnuovo's regularity. The aim of this paper is to describe a new approach for providing Castelnuovo bounds as presented in [23]. This provides new bounds for Castelnuovo's regularity which improve bounds of [23] in some cases. Moreover, we prove special cases of the conjectures ofD. Eisenbud and S. Goto and ofD. Bayer and M. Stillman [2] (see Theorem 2). In Section 4 we will apply these results in order to show that an assertion of J. Harris and D. Eisenbud [7], p. 82 on the equality of the abstract

266 U we Nagel

Hilbert function and the Hilbert polynomial is not true in general (see Theorem 3). Providing our Theorem 3 we construct our counterexamples.

Acknowledgement

I thank W. Vogel for several helpful discussions and suggestions, especially for showing me the letter [25].

2. Notations and preliminary results

We work over an algebraically closed field K. Let S = K[xo, . .. , xn] be a polynomial ring and 0 ~ S be a homogeneous ideal. We set A = S/o., that is, A is a graded K-algebra. We denote by P A:= EBn>oAn the irrelevant ideal of A. When there is no possibility of confusion we will denote P A simply by P. Let M = EBnEZMn be a graded A-module. The i-th local cohomology module of M with support in P, denoted by H~(M), is also a graded A-module. Let [M]i denote the i-th graded part of M for i E Z, i.e. [M]; = Mi' Let j be an integer then let M(j) denote the graded A-module whose underlying module is the same as that of

M and whose grading is given by [MU)]; = [M];+j for all iE Z. We set for an arbitrary A-module M: e(M):= sup{ t E Z: [M]t =I O}

( ) { 1, if Mn=lO sgn M :=

n 0, otherwise and

rk(M):= sup{i + e(H~(M)): i ~ k}.

For a finitely generated graded A-module M we define Castelnuovo's regularity, denoted by reg(M), by reg(M):= r o(M) = r dePth(M)(M).

Let {Xl"'" Xm} be a part of a system of parameters for M. It is said to be a filter-regular sequence if

e((xl,,,,,Xi-l)M: MX;/(Xl, ... ,x;-dM) < r:fJ for i = 1, ... ,m

(see, e.g., [20], appendix for further informations). We set Mi = M/(x l , ... , x;)M (i = O, ... ,m) for a filter-regular sequence {xl, ... ,xm } for M. We have the following result.

LEMMA 1. Let M be a Noetherian graded A-module of dimension d > O. Then there is afilter-regular sequence {li''''' ld} offorms EA 1•

Proof. It is sufficient to show that there is a filter-regular element I E A 1 for M.

On Castelnuovo's regularity and Hilbert functions 267

Let {Pl' ... ' P.} be the set of prime ideals P of Ass .. (M) with Krull-dim(A/p) ~ d. Using, for example [16], Theorem 2.3 we can find an element IE Al \(P·A l U Pl U ... UP.) since P ¢ Pl U ... uP.. It follows from [20], Theorem 7 of the appendix that I is a filter-regular element for M. Q.E.D.

Let M be a Noetherian graded A-module of dimension d ~ 1. We denote by ho(M) the (d - l)!-fold of the leading coefficient of the Hilbert polynomial PM(t). We recall that PM(t) = rankK[M]t for all t »0. If M = A = S/a we define

d () { ho(s/a), if Krull-dim(S/a) > 0 egree a :=

length(S/a), if Krull-dim(S/a) = o.

Further, we set a: (P) = {XES: there is an integer m ~ 0 with pm·x s;;; a}. Let X be a subscheme of P'lc. Then we denote by I(X) the defining ideal of X in

S = K[xo, . .. ,xnl If a s;;; S is a homogeneous ideal let V(a) be the corresponding subscheme of 1P'lc. The ideal a is said to be regular if V(a) is smooth. Note that degree (X) = degree(I(X)). We set rk(X):= rk+ 1 (I(X)) = rk(S/I(X)) + 1 (k ~ 0).

For a set B we write card(B) for its cardinality. Finally, we set for integers a, b ~ 0: {a/b}:= inf{tE Z: a ~ tb}. If a> b we define a sum ~t=a ... to be zero and a condition, say Bi , for i = a, a + 1, ... , b to be empty.

3. Castelnuovo bounds

Studying our integers rk we will prove a generalization of a theorem of D. Mumford [10], p. 99 and A. Ooishi [13], Theorem 2.

THEOREM 1. Let M be afinitely generated graded A-module of dimension d and let m and k ~ 1 be integers. Suppose that [H~(M)]m-i = 0 for all i ~ k. Then rk(M) ~ m - 1. Moreover, reg(M) ~ m - 1 provides AiMj = Mi+ j for all i ~ 0 andj~m-1.

Proof. We induct on d. In case of d = 0 the assertions are trivial since H~(M) = M and H~(M) = 0 for all i > O. Let d > O. According to Lemma 1 we can choose a filter-regular element lEAl for M. Then we get H~(M/O: I) ~ H~(M) for i > 0 from the long exact cohomology sequence of 0 --+ 0: MI --+ M --+

M/O: Ml--+ o. The exact sequence 0 --+ M/O: l( -1) 4 M --+ M/IM --+ 0 gives rise to the cohomology sequence

because we have 0: MI s;;; H~(M) by the choice of I. Considering the following

268 Uwe Nagel

sequences of (*)

[H~(M)]m-i -+ [H~(M/IM)]m-i -+ [Hi+1(M)]m_i_1 (i ~ 0)

we get [H~(M/IM)]m-1 = 0 for all i ~ k by assumption. Therefore the induction hypothesis provides [H~(M/IM)]j = 0 for all i andj with i ~ k and i + j ~ m. The following sequences of (*)

[H~(M)]m-i -+ [H~(M)]m-i+1 -+ [H~(M/IM)]m+1-i (i ~ 0)

gives us [H~(M)]m+ 1-i = 0 for all i ~ k, By induction we therefore have the first assertion. Proving the second assertion we first note that reg(M) ~ m - 1 and (*) yield reg(M/IM) ~ m - 1. If we set M' = MIIM and A' = AliA we have A;Mj =

Mi+ j for i ~ 0 and j ~ m - 1 by induction hypothesis. Hence AiMj + IMi+j- 1 = M i+j. It follows from this by induction on i that M i+j- 1 = Ai- 1M j. We obtain M i+j = AiMj + IMi+j- 1 = AiMj. Q.E.D.

Moreover we get from the proof of Theorem 1. (see also [24], Lemma 2.3):

LEMMA 2. Let M be a finitely generated graded A-module and let IE A1 be a filter-regular element for M then

e(H~+l(M)) < e(H~(M/lM)) ~ max{e(H~(M)), 1 + e(H~+l(M))} (i ~ 0).

Proof. The assertions follows from the exact sequence (*) of the proof of Theorem 1. Q.E.D.

LEMMA 3. Let k ~ 1 and C be integers. Then we have for aI/finitely generated graded A -modules M of dimension d

d

rk(M) ~ C - 1 + L card{tE 7L: t ~ c - i and [H~(M)]t # O}. i=k

Proof. The assertion is trivial for k > d. Let 1 ~ k ~ d. We have 'Lt=k sgn[H~(M)]m-i ~ 1 for all m ~ rk(M) by Theorem 1. Thus it follows from all CE 7L:

rk(M) d

rk(M) ~ C - 1 + L L sgn[H~(M)]m-i m=c i=k

d rk(M)-i

= C - 1 + L L sgn[H~(M)]t i=k t=c-i

d

= C - 1 + L card{tE 7L: t ~ C - i and [H~(M)]t # O} i=k

where the last equality follows from the definition of rk(M). Q.E.D.

In case of M = A we obtain something more.

MAIN LEMMA: Let k ~ 0 and c be integers. Let A be a graded K-algebra of

Krulldimension d. Then:

d

rk(A) ~ c - 1 + L card{tE Z: t ~ c - i and [HMA)]t ~ a}. ;=k

Proof. We consider only the case k = 0 according to Lemma 3. Since ro(A) = max{e(H~A), rl(A)} the assertion follows from Lemma 3 with k = 1 in assuming e(H~(A)) ~ r I (A). We therefore suppose that e(H~(A» > r I (A). We set A = S/a n q where S = K[xo, ... , xn ] and a, q !;;; S are homogeneous ideals such that a: (P) = a, a r;f:. q and q is a primary ideal belonging to P. If we set A' = S/a we get depth(A') ~ 1. Hence reg(A') = r I (A') = r 1 (A), consequently reg(a) = 1 + r 1 (A). It follows from the second assertion of Theorem 1. that a is generated by forms of degree ~ 1 + r 1 (A). Since H~(A) = a/a n q we can deduce [H~(A)]t =1= 0 for all t with r 1 (A) < t ~ e(H~(A». This gives us

ro(A) = e(H~(A)) = r 1 (A) + card{tE Z: t > r1 (A) and [H~(A)]t =1= a}.

Therefore the assertion follows again from Lemma 3 with k = 1. Q.E.D.

COROLLARY 1. Let A be a graded K-algebra of Krull dimension d. Letj and k be

integers such thatj + k ~ d and {il"'" Ii} be afilter-regular sequencefor A. We set Ai.= A/(ll" .. , lilA. Then:

k+ j-l

rk(A) ~ c - 1 + L card{tEZ: t ~ c - i and [H~(A)]t =1= o} ;=k

Proof. Lemma 2 gives us for i = k, ... , d - j

Therefore the assertion follows from the Main Lemma. Q.E.D.

REMARKS (i) If we suppose k ~ 1 in Corollary 1 the above result remains true even for finitely generated graded A-modules because we can apply Lemma 3.

(ii) If we set k = 1 and j = d - 1 in Corollary 1 we obtain the main lemma of [23]. Hence we could deduce the Castelnuovo bounds of [23]. Here we want to state some new bounds.

270 U we Nagel

THEOREM 2. Let a = 1'1 n ... n Pm C S = K[xO' ••• ' xn ] be an intersection of m equidimensional (homogeneous) prime ideals. Let d be the Krulldimension of A = S/a. Then we have:

(i) reg(a):::;; c + r.t,:-lcard{tE 7L: t ~ c - i and [H~(A)]t"# O} for all c ~ degree(a).

(ii) In case of m = 1 we get: (ii.l) reg(a):::;; c + r.t':-lcard{tE 7L: t ~ c - i and [H~(A)]t "# O} for all c>

{degree(a)-l/rankK[A]l-d} ifchar(K)=O (ii.2) reg(a):::;; c + r.t':-lcard{tE 7L: t ~ c - i and [H~(A)]t "# O} for all c>

degree(a) + d - rankK[A]l (ii.3) reg(a):::;; c + r.t,:-tcard{tE 7L: t ~ c - i and [H~(A)]t "# O} for all c>

degree(a) + d - rankK[A]l if a is regular and char(K) = o.

Proof. First we show (i) and (ii.2).1f d = 0 or d = 1 then A is Cohen-Macaulay and we get reg(a) :::;; degree(a) + d - rankK[A]l + 1 by [13], Proposition 13. This proves (i) and even (ii.2) since rankK[A] 1 ~ d, where equality holds if and only if A is isomorphic to a polynomial ring over K. But in this case we have degree(a) = reg(a) = 1. Let d ~ 2. According to H. Flenner [4] there are generic linear forms 11, ... , Id - 2 E S such that a + (/1, ... , li)S is an intersection of m prime ideals of dimension d - i up to a primary component belonging to P for i = 0, ... , d - 2. Therefore {11> ... , Id-z} is a filter-regular sequence for A. If we set Ad - 2 = A/(/l> ... , Id - 2 )A and a' = a + (/1' ... ' Id - 2 )S: (P) we obtain r1(Ad - 2 ) = r1(S/a') = reg(S/a'). Thus we get r1(Ad - 2 :::;;degree(a') + 2-rankK[S/a']l by Theorem 1.1. of [6] for m = 1 and r 1 (Ad - 2 ):::;; degree(a') - 1 according to the remark after the proof of Theorem 1.1. in [6] for m ~ 1. Therefore Corollary 1 with k = 1 and j = d - 2 proves (i) since degree(a) = degree(a') by Bezout's theorem.

Proving (ii.2) we will show that 2 - rankK[S/a']1 :::;; d - rankK[A]l. Then we can apply Corollary 1. It follows from [22], Lemma 3 that

rankK [a']l = rankK[a + (/1> ... , Id - 3 )S]1 + 1

:::;; rankK[a + (/1> ... , Id - 3 )S: (P)]l + 1

= rankK[a + (/1, ... , Id - 4 )S]1 + 2

:::;; ... :::;; rankK[a] 1 + d - 2.

The proof of (ii.3) is analogous to the proof of (ii.2). For this we note that we can choose the linear forms Ii according to [4] such that a':= a + (11, ... , Id - 3)S: (P) is regular if a is regular. Therefore the assertion follows from Corollary 1 with

k = 1 and j = d - 3 by using

r 1 (Aj(ll' ... , Id- 3)A) = reg(Sja')

::::; degree(a') + 3 - rankK[Sja']] by [9] (or [5])

::::; degree(a') +d -rankK[A]1 by [22],

Lemma 3 and Bezout's theorem. Now we show (ii.l). The assertion is trivialfor d = 0 and d = 1. Let d ;:::: 2. Take

the linear forms 11 , ••• , Id _ 2 constructed in our proof of (ii.2). Consider a general linear form Id-1 and set a' = a + (11) .. . , Id- dS: <P). Then we get

, {degree( a') - 1 } r 1(Aj(l1> ... , Id- dA) = reg(Sja) ::::; k [ j '] 1

ran K Sal -

(see, e.g., [22], Lemma 1). We have again degree(a) = degree(a') by Bezout's theorem and rankK[a'] 1 ::::; rankK[a]l + d - 1. On the other hand Lemma 3 of [22] gives us

rankK[a']l = rankK[a + (lb···, Id-2)S]1 +

+ 1 ;:::: rankK[a]l + d - 1.

Putting all together we obtain

( d-1) {degree(a) - 1 }. r1 A ::::;

rankKCA]l - d (+ )

Consequently Corollary 1 with k = 1 and j = d - 1 proves the assertion (ii.l). This completes the proof of Theorem 2. Q.E.D.

COROLLARY 2. Let X be a nondegenerate, irreducible and reduced subscheme of [pn of dimension d. Then we have:

d+1 {degree(X) - I} d + 1 + e(Hp (Sj/(X)))::::; codim(X) .

Proof. The assertion follows from (+) of the proof of Theorem 2(ii.l) and Lemma 2. Q.E.D.

REMARKS. Theorem 2(ii.l) is Theorem 2(ii) of [23]. Theorem 2(i), (ii.2) and (ii.3) improve Theorem 2(i) of [23] in some special cases.

272 U we Nagel

The assumption char(K) = 0 is necessary in Theorem 2(ii.1) because the general position lemma does not remain true if char(K) > 0 (see [17], Example 1.2).

The conjecture of D. Bayer and M. Stillman [2] gives reg(a) ~ degree(a). Therefore Theorem 2(i) and (ii.2) (see also [22], corollary) prove the conjectures of Bayer and Stillman and of Eisenbud and Goto in case of depth(Sja) ~ d - 1. This means, for example, that the latter conjecture is true for surfaces in iP'4 if the homogeneous coordinate ring has depth ~ 2. Note that (see the introduction) singular surfaces in iP'4 are the simplest varieties such that the conjecture of D. Eisenbud and S. Goto is open.

4. Counterexamples to an assertion of J. Harris and D. Eisenbud

In this section we will apply Theorem 2 in order to study the equality between Hilbert functions and Hilbert polynomials. Let X be a subscheme of pn and A = SjI(X) be its homogeneous coordinate ring. We recall that the Hilbert function of X is defined by hx(t):= rankK[A]t for t ~ O. The so-called Hilbert polynomial, denoted by Px(t), is given by hx(t) for t » O. It is well-known that Px(t) = ~i"'O( _1)ihi(X, (l)x(t)) where hi(X, Ox(t)) is the dimension of Hi(X, (l)x(t)). Following [7] the function h'x(t):=hO(X, (l)x(t)) is said to be the abstract Hilbert function of X. The index of regularity of X, denoted by r(X), is defined as r(X) :=

min{tE N: hx(i) = Px(i) for all i ~ t}. Moreover, we set

r'(X):= min{ tEN: h'x(i) = Px(i) for all i ~ t}, reg(X) = reg(I(X)).

LEMMA 4. (i) ([14], Corollary 2.2) r(X) ~ reg(X) - depth(SjI(X)), (ii) r'(X) ~ r2(X) - max{2, depth(SjI(X))}.

Proof We have hx(t) - Px(t) = ~i"'O( -1)irankKCH~(SjI(X))]t according to [19]. This proves (i). We obtain (ii) from the characterization of Px(t) as an Euler-Poincare characteristic and the isomorphisms Hi(X, (l)x(t)) ~ [H~+ l(SjI(X))]t for i > O. Q.E.D.

In [7], p. 82 J. Harris and D. Eisenbud assert for reduced and irreducible sub schemes X of iP'1 (char(K) = 0):

'( ) ~ {degree(X)} r X "" d. ( ) . co ImX

The letter [25] gives rise to study this claim again. In this connection, we will prove the following theorem.


THEOREM 3. (i) We have for nondegenerate, irreducible and reduced curves X:

'(X) ~ {degree(X) - I} _ 1 r ..... codim(X) ,

that is, the assertion (* *) is true for such curves. (ii) There are nondegenerate, irreducible and reduced subschemes X of lPn, n ~ 4,

of dimension d such that the assertion (**) is not true for all d ~ 2. Proof. (i) Corollary 2 gives us for d = 1

2 {degree(X) - I} r 2 (X) - 1 = 2 + e(Hp(S/I(X») ~ codim(X) .

Hence (i) follows from Lemma 4(ii). (ii) We consider the following class of examples: Let m ~ 3 be an integer. Let

Xm s;;; 1P4 be the surface given parametrically by {um, Um-1V, Um- 2VW, uwm- l , Wm}. It follows from [8], Proposition 3 that degree(X m) = m + 1. Moreover, Corollary 3.4(ii) of [24] shows H}(S/"Pm) = 0, that is depth(S/"pm) ~ 2 where "Pm C S = K[xo, ... , x 4 ] denotes the defining prime ideal of Xm. That is why we can apply Theorem 2(ii.2) and obtain reg(X m) ~ degree(X m) - codim(X m) + 1 = m. Hence "Pm is generated by forms of degree ~ m according to Theorem 1. Thus we can compute a minimal basis of "Pm from its parametrization and obtain

Since "Pm needs a generator of degree m we get reg(Xm) = m. Since PxJt) - h'x~(t) = 1:i>o(-l)irankK[H~+l(S/"Pm)]t we obtain from Corollary 2 that

Applying Theorem 1 we get from reg(X m) = m and Corollary 2 that [H;(S/"Pm)]t -:j; 0 for

{m} ,{m + I} {degree(Xm)} 2 -2~t~m-3.Hencer(Xm)=m-2> -2- = codim(Xm)

for m ~ 7. This shows (ii) in case d = 2. Letj ~ 0 be an integer. We denote by Ym the projective cone over Xm in lPi+4.

Then we get d:= dim(Ym) = dim(Xm) + j = 2 + j, degree(Ym) = degree(Xm) and

274 Uwe Nagel

depth(S/I(Ym)) = j + 2 where S = K[xo, .. . , Xj+4]. Moreover, Lemma 2 gives us reg(Ym) = reg(Xm) = m. Therefore we obtain from Corollary 2 as above:

'( ) = _ 2 _. {m + 1} = {degree(Ym)} r Ym m ] > 2 d· ( ) co 1m Ym

for m ~ 2j + 7 and for all d = 2 + j. Q.E.D.

REMARKS. (i) Using results of [6] the sub schemes Ym show that the conjecture of D. Eisenbud and S. Goto is sharp in the sense that there are nondegenerate, irreducible and reduced varieties X with reg(X) = degree(X) - codim(X) + 1 in any dimension ~ 1 and of any degree ~ 4.

(ii) If the assertion (**) were true we could deduce r 2 (X) ~ {degree(X)/ codim(X)} + 2. But this is also not true in general as the varieties Ym show.

(iii) (**) is true in assuming, for example, that the subschemes X are arithmetically Buchsbaum, i.e., that the homogeneous coordinate ring S/I(X) is a graded Buchsbaum K-algebra. In this case we obtain from [21], Theorem 1. reg(X) ~ {degree(X) - l/codim(X)} + 1. Therefore Lemma 4(ii) yields

'() {degree(X) - 1} r X < codim(X) .

(iv) The varieties Ym are not arithmetically Buchsbaum for m ~ 3 due to [24], Lemma 4.11 and Corollary 4.7 and even not locally Buchsbaum for m ~ 4 because Ym C IPj+4 has a singularity in the point l' = (Xl' . .. , Xj+4) which is not Buchsbaum for m ~ 4. Otherwise (S/I(Ym))p and consequently also (S/I(Ym) + X3S)p would be Buchsbaum. Since (I(Ym) + x 3S)p = (x3, Xl X4, xi- l , Xi- 2X4, ... ,

m-2 m-l) (m-l ) ( m-l m-2 m-l) h X2X4 ,X4 p = X2 ,X3, X4 p (l Xl> X3' X2 ,X2 X4, . .. , X4 p we ave x 2(xi-l,X3,x4)p cj;. (I(Ym) + x 3S)p for m ~ 4. We immediately obtain a contradiction to a Buchsbaum ring property. (For the facts on Buchsbaum rings used here we refer to [20].)

(v) We can not apply the results of [9] or [5] in order to obtain the bound reg(Xm) ~ degree(Xm) - codim(Xm) + 1 since the varieties Xm are singular.

(vi) Since our counterexamples are singular varieties it is an open problem if (**) is true in the case of smooth varieties, see [26].

(vii) In [15], p. 370 P. Philippon considers the ideal Q:= (Xl X4 - X2X3' XoX~ -xrX3, XoX~ - x~). He asserts that Q is a prime ideal used in his computations. But it follows from X4(XOX2X4 - X1X~)E Q that Q is even not a primary ideal because no power of X4 and XOX2X4 - Xl X~ is contained in Q. (Note that l'3 = Q + (XOX2X4 - X1X~)S.)


References

[1] E. Ballico, On the defining equations of subvarieties in p •. Boll. Un. Mat. Ital. A(6) 5 (1986) 243-246.

[2] D. Bayer and M. Stillman, On the complexity of computing syzygies. J. Symbolic Comput. 6 (1988) 135-147.

[3] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity. J. Algebra 88 (1984) 89-133.

[4] H. Flenner, Die Siitze von Bertini fiir lokale Ringe. Math. Ann. 229 (1977) 97-111. [5] V.A. Greenberg, A Castelnuovo bound for projective varieties admitting a stable linear

projection onto a hypersurface. Thesis, Columbia university, 1987. [6] L. Gruson, R. Lazarsfe1d and C. Peskine, On a theorem of Castelnuovo, and the equations

defining space curves. Invent. Math. 72 (1983) 491-506. [7] J. Harris (with the collaboration of D. Eisenbud), Curves in projective space. Les presses de

I'Universite, Montreal 1982. [8] L.T. Hoa, A note on projective monomial surfaces. Math. Nachr. (to appear). [9] R. Lazarsfeld, A sharp Caste1nuovo bound for smooth surfaces. Duke Math. J. 55 (1987)

423-429. [to] D. Mumford, Lectures on curves on an algebraic surface. Ann of Math. Studies 59, Princeton

Univ. Press, Princeton, N.J., 1966. [11] U. Nagel, Castelnuovo-Regularitiit und Hilbertreihen. Math. Nachr. 142 (1989) 27-43. [12] U. Nagel and W. Vogel, Bounds for Castelnuovo's regularity and the genus of projective

varieties. In: Topics in Algebra. Banach Center Publications, Volume 26. PWN-Polish Scientific Publishers, Warsaw 1989 (to appear).

[13] A. Ooishi, Castelnuovo's regularity of graded rings and modules. Hiroshima Math. J. 12 (1982) 627-644.

[14] A. Ooishi, Genera and arithmetic genera of commutative rings. Hiroshima Math. J. 17 (1987) 47-66.

[15] P. Philippon, Lemmes de zeros dans les groupes algebriques commutatifs. Bull. Soc. math. France 114 (1986) 355-383.

[16] P. Quartararo Jr. and H.S. Butts, Finite unions of ideals and modules. Proc. Amer. Math. Soc. 52 (1975) 91-96.

[17] J. Rathmann, The uniform position principle for curves in characteristic p. Math. Ann. 276 (1987) 565-579.

[18] J. Rathmann, On the completeness of linear series cut out by hypersurfaces. Preprint, M.l.T., June 1988.

[19] J.-P. Serre, Faisceaus algebriques coberents. Ann. of Math. 61 (1955) 197-278. [20] J. Stiickrad and W. Vogel, Buchsbaum rings and applications. Springer, Berlin 1986. [21] J. Stiickrad and W. Vogel, Castelnuovo bounds for certain subvarieties in p •. Math. Ann. 276

(1987), 341-352. [22] J. Stiickrad and W. Vogel, Castelnuovo bounds for locally Cohen-Macaulay schemes. Math.

Nachr. 136 (1988), 307-320. [23] J. Stiickrad and W. Vogel, Caste1nuovo's regularity and cohomological properties of sets of

points in po. Math. Ann. 284 (1989) 487-501. [24] N.V. Trung and L.T. Hoa, Affine semigroups and Cohen-Macaulay rings generated by

monomials. Trans. Amer. Math. Soc. 298 (1986) 145-167. [25] W. Vogel, Letter to D. Eisenbud dated 5 October, 1987. [26] U. Nagel and W. Vogel, Castelnuovo's regularity of graded k-algebras and applications. Stud.

Cerc. Mat. (in preparation).


Indecomposable Cohen-Macaulay modules and irreducible maps

DORIN POPESCU1 & MARKO ROCZEN2

lINCREST, Dept. of Math., Bd Padi 220, Bucharest 79622, Romania; 2Humboldt-Universitiit, Sekt. Mathematik, Unter d. Linden 6, Berlin 1086, DDR

Received 22 November 1988; accepted 20 July 1989

Introduction

Let (R, m) be a local CM-ring and M a finitely generated (shortly f.g.) R-module. M is a maximal CM module (shortly MCM R-modules. The isomorphism classes of indecomposable MCM R-modules form the vertices of the Auslander-.. Reiter quiver r(R) of R. Section 3 studies the behaviour of r(R) under base change; best results (cf. (3.10), (3.14)) being partial answers to the conjectures from [Sc] (7.3). Unfortunately, the proofs use the difficult theory of Artin approximation (cf. [Ar], or [Pol]).

A different easier method is to use the so-called CM-reduction ideals as we did in [P02] (4.9) or have in (3.2), (3.3). This procedure is very powerful in proving results describing how large is the set of those positive integers which are multiplicities of the vertices of r(R), in fact the first Brauer-Thrall conjecture (cf. [Di], [Yo], [P02] or here (4.2), (4.3)). However the Corollary (3.3) obtained by this method is much weaker than (3.10) which uses Artin approximation theory. The reason is that the conditions under which we know the existence of CM-reduction ideals are still too complicated. Trying to simplify them we see that the difficulty is just to prove some bound properties on MCM modules (cf. Section 2) which we hope to hold for every excellent henselian local CM-ring. Our Theorem (4.4) and Corollary (4.6) give sufficient conditions when the second Brauer-Thrall conjecture holds, and our Corollary (4.7) is a nice application to rational double points (inspirated by [Yo] (4.1)).

We would like to thank L. Badescu and J. Herzog for helpful conversations on (4.7), (3.10), (3.16) respectively.

1. Bound properties on MCM modules

(1.1) Let (R, m) be a local CM-ring, k:= Rim, p:= char k and Reg R = {qE Spec RIRq regular}. Suppose that RegR is open (this happens for instance

278 Dorin Popescu and Marko Roczen

when R is quasi-excellent). Then I.(R) = nq¢RegR q defines the singular locus of R. We say that R has bound properties on MCM modules if the following conditions hold:

(i) there exists a positive integer r such that I.(R)' Exti(M, N) = 0 for every MCM R-module M and for every f.g. R-module N, i.e. I.(R) is in the radical of the Dieterich Ext-annihilating ideal of R (cf. [Di] §2),

(ii) for every ideal a c R and every element y E I.(R) there exists a positive integer e such that (aM: ye)M:= {zEM I yezEaM} = (aM: ye+I)M for every MCM R-module M.

Clearly it is enough to consider in (i), (ii) only indecomposable M. Let M be a MCM R-module and AM:= {X E R I x Exti(M, N) = 0 for every f.g. Rmodule N}

(1.2) LEMMA. I.(R) c ~. Proof Let x EI.(R). Then Rx is a regular ring and so Mx is projective over Rx.

Indeed, if q SpecR with x¢q then Mq is free over Rq by [He](1.1), Mq being still MCM by [Ma2] (17.3) and Rq is regular. Thus

Rx ® R Exti(M, N) = ExtiJMx, N x) = 0

and so a certain power of x kills Ext ijM, N), i.e. x E ~.

(1.3) REMARK. The above Lemma shows that for each MCM R-module M we are able to find a positive integer rM such that I.(R)'M Exti(M, N) = 0 for every f.g. R-module N. Thus the trouble in (1.1)(i) is just to show that rMcan be bounded when M runs in CM(R). Also for each f.g. R-module M by Noetherianity we can find in (1.1)(ii) a positive integer eM such that (aM: yeM)M = (aM: yeM+I)M' Again the trouble is to show that eM can be bounded when M runs in CM(R). However, if R has finite CM-type (i.e. r(R) has just a finite set of vertices) then R has bound properties on MCM modules (compare with [Di] Proposition 8).

(1.4) LEMMA. Suppose that (R, m) is reduced complete with k perfect and

Reg(R/pR) = {q/pR I qEReg R, q :::> pR} if pR =F 0 (i.e. if p =F char R). Then R has bound properties on MCM modules.

Proof Clearly either R contains k or R is a flat algebra over a Cohen ring of residue field k, i.e. a complete DVR (T, t) which in an unramified extension of Z(P) , p > 0, t = p' 1 E T. Let x = (Xl,' .. , xn) be a system of elements from m such that (t, x) forms a system of parameters in R. By Cohen's structure Theorems the canonical map j: T[[X]] ~R,X = (XI, ... ,Xn)f-+X is finite. As R is CM we obtain R flat (thus free) over the image Sx of j.

Let I x be the kernel of the multiplication map R ® Sn R ~ Rand ..¥x:=

Indecomposable Cohen-Macaulay modules and irreducible maps 279

Il(AnnRl8isnRIx) the Noether different of Rover Sn. Then

(1)

where the sum is taken over all systems of elements x such that (t, x) forms a system of parameters of R (see [P02] (2.8), (2.10), the ideas come in fact from [Yo]).

Using the Hochschild cohomology we get a surjective map

for every MCM R-module M and every f.g. R-module N (see e.g. in [Di] Lemma 5). By (1) follows .Hx Ext1{M, N) = 0 and so there is re N such that

Is(R)' Ext1{M, N) = 0 (2)

for every MCM R-module M and every f.g. R-module N, i.e. (1.1)(i) holds. Now, let a c R be an ideal and yeIs(R). We show that there exists a positive

integer e such that (aM:ye)M = (aM: ye+1)M for every MCM R-module M. If there exists x as above such that y e.Hx then it is enough to apply [P02] (3.2) for Sx cR. Otherwise choose in Is(R) a system of elements (nih <;;i<;;s such that I s(R) = J}2iniR and for every i there exists x(i) as above such that ni e .Hx(i). Then yni e .Hx (;) and so there exists a positive integer ei such that

for every MCM R-module M. Choose a positive integer e' such that Is(R)e' c }2f=lnf;R. We claim that e = e' + maxl<;;i<;;sei works. Indeed, let M be a MCM R-module, if yVzeaM for a certain zeM and veN then (yni)V zeaM and so (yni)e; zeaM. Thus

(1.5) LEMMA. Suppose that

(i) (R, m) is quasi-excellent and reduced, (ii) Reg(RlpR) = {qlpR I q e Reg R, q ::::> pR} if pR #- 0,

(iii) there exists a fiat Noetherian complete local R-algebra (R', m') such that (iii1 ) R' is formally smooth over R, (iii2) k':= R'lm' is perfect.


ThenR has bound properties on MCM modules. Proof. By Andre-Radu's Theorem the structural morphismj: R --+ R' is regular

(see [An] or [BR1], [BR2]). Then

RegR' = {qESpecR'lr1qERegR}

by [Mal] (33.B). Hence

I.(R') = JI.(R)R' and Reg(R'/pR') = {q/pR'lqERegR',q::::> pR'}.

Thus R' has bound properties on MCM modules by Lemma (1.4) and it is enough to show the following

(1.6) Lemma. Let j: R --+ A be aflat local morphism of local CM-rings such that I.(A) ::::> I.(R)A. If A has bound properties on MCM modules then R has too.

Proof. Let r be the positive integer given for A as in (l.1)(i). We claim that r works also for R. Let M be a MCM R-module and N a f.g. R-module. By flatness we have

A ®R Ext1(M, N) = Ext~(A ®R M, A ®R N).

Since

depthA A ®R M = depthRM + depthA A/mA = depth R + depthA A/mA = depth A,

the A-module A ®RM is a MCM. Thus I.(A)' (so I.(R)') kills Ext~(A ®RM, A ®R N) and it follows I.(R)'(A ®R Ext1{M, N)) = O. By faithful flatness we get I.(R),Ext1(M, N) = O.

Now let a c R be an ideal and YEI.(R). Thenj(Y)EI.(A), and by hypothesis there exists an e E N such that

(aP :j(ynp = (aP :j(y)e+ l)p

for every MCM A-module P. Let M be a MCM R-module. As above, M' := A ®R M is a MCM A-module

and by flatness we obtain

(aM' : (y)')M' ~ A ®R (aM: y')M

for every sEN. Thus the inclusion

(1)

goes by base change into an equality. Then (1) itself is an equality, j being faithfully flat.

(1.7) PROPOSITION: Suppose that

(i) (R, m) is a quasi-excellent reduced and [k: kP] < 00 if p > 0, (ii) Reg(R/pR) = {q/pR I q e Reg R, q ::::> pR} if pR =F O.

Then R has bound properties on MCM modules. Proof If k is perfect then apply Lemmas (1.5), (1.6), where A is the completion

of (R, m). If k is not perfect, let K := k 1/poo • We have rankK r K/k ~ rankk Qk/kP =

[k: kP ] (see e.g. in [P02] (4.4)) and so there exists a formally smooth Noetherian complete local R-algebra (R', m') such that R'/m' ~ K and dim R' = dim R + rank r K /k (see EGA (22.2.6), or [NP] Corollary (3.6)). Now apply Lemmas (1.5), (1.6).

2. eM-reduction ideals

(2.1) LEMMA. (inspirated by [P02] (3.1)). Let A be a Noetherian ring, M, N two f.g. A-modules, xeA an element such that xExt~(M,P) = 0 for every factor A-module P of N, e a positive integer such that (0: Xe)N = (0: x e+ l)N and seN. Then for every A-linear map <p: M -+ N /xe+s+ 1 there exists an A-linear map

"': M -+ N which makes commutative the following diagram:

I 'PI

I .lN -----+N/xe+sN

Proof Let N':= (0: Xe)N. We have the following commutative diagram

O-+N/N' xe +s + 1

IN/N' I N/N' + xe+s+1 N 10

1 II 1 0-+ N/N'

xe +s IN/N' I N/N' + xe+sN I 0

in which the lines are exact. This follows from the elementary

(2.1.1) LEMMA. xeN 11 N' = O.

(1)

Indeed, if xe+szeN' for a certain zeN then xe+sz = 0 by the above Lemma and so zeN'. Applying the functor HomA(M, -) to (1) we obtain the following com-


mutative diagram

HomA(M, NjN') --+ HomA(M, NjN' + x e+s+1 N) --+ Ext~(M, NjN')

II j 1" (2)

HomA(M, NjN') --+ HomA(M, NjN' + xe+sN) - Ext~(M, NjN')

with exact lines. Since the last column is zero by hypothesis we obtain a R-linear map 0(: M --+ N j N' such that the following diagram is commutative:

(3)

Note that in the diagram

(4)

the small square is cartesian and so there exists 'I' which makes (4) commutative. Now apply Lemma (2.1.1).

Proof of Lemma (2.1.1). If YEN' n x e Nand zEN satisfy y = xez then 0= xey = x 2ez and so zN', i.e. y = xez = O.

(2.2) LEMMA. Let (R, m) be a local CM-ring which has bound properties on MCM modules, a c R an ideal and y E Is(R). Suppose that Reg R is open. Then there exists a function v: N --+ N, v ~ 11'.1 such that for every SE N, all MCM R-modules M, N and every R-linear map cp: M --+ Nj(a, xv(s»)N there exists a

R-linear map '1': M --+ NjaN such that the following diagram commutes:

M qJ ,Nj(a, xv(s»)N

I 1 'PI

1 N jaN -----+ N j (a, xS)N

Proof Let r, e be the positive integers given by (U)(i), (U)(ii), respectively. Define v(s) = r(l +max{e,s}). Let M,N,cp be given. Since y'Exti(M,P)=O


for everyf.g. R-module P and (aN: ye)N = (aN: ye+ l)N we get 'I' by Lemma (2.1). (2.3) Let A be a local CM-ring and a c A an ideal. As in [Po2] , (3.6) the couple

(A, a) is a CM-approximation ifthere exists a function v: N -+ N, v ~ 11'11 such that for every sEN, every two MCM A-modules M, N and every A-linear map cp: M -+ N/av(s)Nthere exists an A-linear map '1': M -+ N such that A/as ®A cp ~ A/as ®A'I', in other words: the following diagram is commutative

M ~N/av(S)N

~:l 1 N----N/aN

(2.4) LEMMA. Let (R, m) be a local CM-ring which has bound properties on MCM modules. Suppose that Reg R is open. Then for every ideal a c Is(R) the couple (R, a) is a CM-approximation.

Proof Let y 1, ... ,Yt be a system of generators of a. Apply induction on t. If t = 1 then use Lemma (2.2) for a = O. Suppose t > 1 and let v' be the function given by induction hypothesis for b := (y l' ... , Yt - 1)' Let v; be the function given by Lemma (2.2) for bV'(s) and Yt, SE N. Then the function v given by v(s) = v'(s) + v;(s) works. Indeed, let M,N be two MCM R-modules, sEN and cp: M-+ N/aV(S)N a R-linear map. Then there exists a R-linear map oc: M -+ N/bV'(S)N such that (R/ y:R) ®R oc ~ (R/ y:R)®R cp (note that aV(s) c (b"'(S), yr'(s»)). Moreover,

there exists a R-linear map '1': M -+ N such that (R/bS)®R'" = (R/bS)®R OC. As

bS c as we get R/as®R '" ~ R/as®R cp. (2.5) Let b be an ideal in a local CM-ring A. Then b is a CM-reduction ideal

if the following statments hold:

(i) A MCM A-module M is indecomposable iff M/bM is indecomposable over A/b.

(ii) Two indecomposable MCM A-modules M, N are isomorphic iff M/bM and N/bM are isomorphic over A/b.

(2.6) LEMMA. Let R be a Henselian local CM-ring and a c R an ideal such that (R, a) is a CM-approximation. Then ar is a CM-reduction ideal for a certain positive integer r.

The proof follows from [Po2] (4.5), (4.6).

(2.7) PROPOSITION. Let (R, m) be a Henselian local CM-ring which has bound properties on MCM modules. Suppose that Reg R is open. Then for every ideal a c Is(R) there exists a positive integer r such that ar is a CM-reduction ideal.

The proof follows from Lemmas (2.4), (2.6).

(2.8) COROLLARY ([P02] (4.8)): Let (R, m) be a reduced quasi-excellent Henselian local CM-ring, k:= Rjm and p:= char k. Suppose that

(i) [k: kP] < 00 if p>O (ii) Reg(RjpR) = {qjpR I qE Reg R, q ::::) pR} if p ;6 char R.

Then Is(R)' is a CM-reduction ideal for a certain positive integer r.

The result follows from Propositions (1.7), (2.7).

3. Stability properties of Auslander-Reiten quivers under base change

(3.1) Let (R, m) be a local CM-ring, k:= Rjm, p:= char k and M, N two indecomposable MCM R-modules. The R-linear map f is irreducible if it is not an isomorphism, and given any factorization f = gh in CM(R), g has a section or k has a retraction. The AR-quiver r(R) of R is a directed graph which has as vertices the isomorphism classes of indecomposable MCM R-modules, and there is an arrow from the isomorphism class of M to that of N provided there is an irreducible linear map from M to N. Let r o(R) be the set of vertices of r(R). The multiplicity defines a map eR: ro(R) --+ N, M f-+eR(M). If R is a domain and K is its fraction field then let rankR : ro(R) --+ N be the map given by M f-+dimK K Qh M. Clearly eR = e(R) rankR by [Ma2] (14.8).

(3.2) PROPOSITION. Suppose that (i) R is an excellent henselian local ring,

(ii) R has bound properties on MCM modules.

Let A be the completion of R with respect to Is(R). Then the base changefunctor A ®rinduces a bijection ro(R) --+ ro(A).

Proof Clearly A is a CM-ring because the canonical map R --+ A is regular by (i). First we prove that a f.g. R-module M is an indecomposable MCM module iff A ®R M is an indecomposable MCM A-module. Using the following elementary Lemma, it is enough to show that if M is an indecomposable MCM R-module then A ®R M is indecomposable over A.

(3.2.1) LEMMA. Let B be a local CM-ring, M a f.g. B-module and C a fiat local B-algebra. Suppose that C is a CM-ring. Then

(i) M is a MCM B-module iff C ®B M is a MCM C-module, (ii) if C ®BM is indecomposable then M is so.

By Proposition (2.7) Is(R)' is a CM-reduction ideal for a certain rE N and so Rj I.(RY ®R M is still indecomposable. As Rj Is(RY ~ Aj I.(RYA it follows that A ®R Mj Is(RYA ®R M is indecomposable and so A ®R M is indecomposable by Nakayama's Lemma. If M, N are two indecomposable MCM R-modules such

that A ®RM ~ A ®RN then M/I.(R)'M ~ A ®RM/I.(R),A ®RM ~ A ®RN/ I.(R), A ®RN ~ N/I.(R)' Nand so M ~ N, becauseI.(R)'is a CM-reduction ideal. Thus A ®rinduces an injective map 0(: ro(R) --+ ro(A). But 0( is also surjective by [El] Theorem 3 because a MCM A-module is locally free on Spec A \ V(I.(A)) and I.(A) = JI.(R)A (see (1.2) and (1.5)), the map R --+ A being regular by (i).

Using (1.7) we get

(3.3) COROLLARY. Suppose that

(i) (R, m) is an excellent reduced H enselian local ring and k: kPJ < 00 if p > 0,

(ii) Reg(R/pR) = {q/pRlqERegR,q::::> pR} ifp i= char R.

Let A be the completion of R with respect to I.(R). Then the base change functor A ®rinduces a bijection ro(R) --+ ro(A). In particular # ro(R) = # ro(A).

The above Corollary can be improved if we use Artin approximation theory

(see [Ar] or [Pol]). (3.4) Let n: B --+ C be a morphism of rings. We call n algebraically pure (see

[Po 1] §3) if every finite system of polynomial equations over B has a solution in B ifit has one in e. Also n is called strong algebraically pure (see [Pol] (9.1)) if for every finite system of polynomials f from B[X], X = (Xl, ... , X.) and for every finite set of finite systems of polynomials (gih "i"r in B[X, Y], Y = (Yl , ... , l't) the following conditions are equivalent:

(1) f has a solution x in B such that for every i, l:S;;i:S;;r the system gi(X, Y) has no solution in B,

(2) f has a solution x in C such that for every i, 1:S;; i:S;; r the system gi(X, Y) has no solutions in e.

Suppose that B is Noetherian and let b c B be an ideal, C the completion of B with respect to band n the completion map. Then (B, b) has the property of Artin approximation (shortly (B, b) is an AP-couple) if for every finite system of polynomials f in B[X], X = (Xl, ... ,X.), every positive integer e and every solution x of fin C, there exists a solution Xe of fin B such that Xe == x mod bee. It is easy to see that (B, b) is an AP-couple iff n is algebraically pure (see [Po 1] § 1). When B is local and b its maximal ideal then (B, b) is an AP-couple too iff n is strong algebraically pure (see [BNP] (5.1) where these morphisms are called T"-existentially complete). If (B, b) is a Henselian couple and n is regular then (B, b) is an AP-couple by [Pol] (1.3).

(3.5) LEMMA. Let A --+ B be an algebraically pure morphism of Noetherian rings and M, N two f.g. A -modules. Then B ® A M ~ B ® A N over B iff M ~ N over A.

Proof. Let M ~ An/(u), N ~ Am/(v), Ui = ~j=lUijej, i = l, ... ,n', Vr =

~='= 1 vr.e~, r = 1, ... ,m', where Uij' vr• E A and (ej) resp. (ej) are canonical bases in An resp. Am. Let ((J: An --+ Ambe a linear A-map given by ej 1-+~~lXj.e~, where


Xjs E A. Then ((J induces a map f: M -+ N iff there exist (Zir) in A such that ((J(Ui) = L~~1ZirV" i.e.

n m'

I UijXjs = I ZirVrs> 1 ~ i ~ n', 1 ~ s ~ m. (1) j=1 r=1

Clearly f is an isomorphism iff there exists 1jJ: Am -+ An given by e~ f-+

LJ=1 Ysjej, YsjEA such that 'P(v) c (u), (((J'P -1)(e') c (v) and ('P((J -l)(e) c (u), i.e. there exist t ri , W." WJi E A such that

m n'

I VrsYsj = I triUij, 1 ~r~m', 1 ~j~n s=1 i=1

m n'

I Xjs Ysj' - l>jj' = I WjiUij', 1 ~j, j' ~ n (2) s=1 i= 1

n m'

I YsjXjs' - l>ss' = I W~rVrs" 1 ~ s, s' ~ m j=1 r=1

where l>ss' denotes Kronecker's symbol. Thus M ~ N iff the following system of polynomial equations:

n m'

I UijXjs = I ZirV,., 1 ~ i ~ n', 1 ~ s ~ m j= 1 r= 1

m n'

I Vrs ¥.j = I T,.iUij, 1~r~m', 1~j~n s=1 i=1

m n'

I X js ¥.j' - l> jj' = I ltjiUij' , 1 ~j, j' ~ n s= 1 i= 1

n m'

I ¥.jX js ' - l>ss' = I W~rVrs" 1 ~ s, s' ~ m j=1 r=1

has a solution in A. Similarly B ® A M ~ B ® A N as B-modules iff (*) has a solution in B. But (*) has a solution in A iff it has one in B because A -+ B is algebraically pure.

(3.6) LEMMA. Let h: A -+ B be a morphism of Noetherian rings and M a f.g. A-module. Suppose that either

(i) h is strong algebraically pure, or (ii) h is algebraically pure and B is the completion of A with respect to an ideal

a c A contained in the Jacobson radical of A.


Then B ® A M is an indecomposable B-module iff M is an indecomposable A-module.

Proof Conserving the notations from the proof of (3.5) for M = N we note that

(1) f is idempotent iff (cp2 - cp)(e) c (u), i.e. there exist djiEA such that

n n'

I XjsXsr - Xjr = I djiUin 1 ~j, r ~ n s=l i=l

(2) f # 0, 1 iff the following two systems of polynomials

n

I QjiUis = Xjs i= 1

n'

1 ~j, s ~ n

I QjiUis = Xjs - bjs 1 ~j, s ~ n i= 1

have no solutions in A with X = x. Thus M is decomposable iff the following system

n n'

I UijXjs = I ZirU,., 1 ~ i ~ n', 1 ~ s ~ n j=l r=l

n n'

I XjsXsr - X jr = I fljiUin 1 ~j, r ~ n (F) s=l i=l

has a sol uti OIl (x, z, d) for which the systems G1(x, Q), G2 (x, Q') have no solutions in A. A similar statement is true for B 0 A M and they are equivalent if h is strong algebraically pure.

Now suppose that (ii) holds. Then h is faithfully flat and we obtain: M is indecomposable if B ® A M is so (cf. (3.2.1)). If B 0 A M is decomposable then it has an idempotent endomorphism # 0, 1 which gives a solution (i, z, (1) of F in B. Since (A, a) is an AP-couple, h being algebraically pure (see (3.4)) there exists a solution (x, z, d) of F in A such that (x, z, d) = (i, z, (1) mod aB. Let f be the idempotent endomorphism of M given by x. Then AI a ® A f ~ B I a 0 B f because Ala ~ BlaB. By Nakayama's Lemma (BlaB) ®BI # 0, 1 and so f # 0, 1, i.e. M is decomposable.

(3.7) REMARK. When A is a local ring, a its maximal ideal and B the completion of A with respect to a then h is strong algebraically pure if h is algebraically pure. Thus in the above Lemma (ii) may be a particular case of (i).

(3.8) PROPOSITION. Let (A, a) be an AP-couple and B the completion of A with


respect to a. Suppose that A, B are local CM-rings. Then the base change functor B ® A -induces an injection ro(A) --+ ro(B).

The result follows from Lemmas (3.5), (3.6).

(3.9) LEMMA. Conserving the hypothesis of the above Proposition, let M, N be two indecomposable MCM A-modules and f: M --+ N an irreducible A-map.

Suppose that the base change functor B ® A -induces a bijection ro(A) --+ ro(B). Then B ® A f is an irreducible B-map.

Proof By faithfully flatness B ® A f is not bijective. Let B ® A f = gli be a factorization in the category of MCM B-modules, Ii: B ® AM --+ P, g: P --+ B ® AN. By hypothesis P = B ® A P for a MCM A-module P. Suppose that g has no section and Ii has no retraction. Then Bias B ® A g has no section and Bias B ® A Ii has no retraction for a certain SEN by the following:

(3.9.1) LEMMA. Let (B, n) be a Noetherian local ring, b c B an ideal and u: M --+ NaB-linear map. Then there exists a positive integer sEN such that u has a retraction (resp. a section) iff it has one modulo bS.

As (A, a) is an AP-couple we can find a factorization f = gh, h: M --+ P, g:P--+N such that (BlaSB)®Ah~(BlaSB)®Ah, (BlaSB)®Ag~(BlaSB)®Ag (the idea follows the proofs of (3.5), (3.6)). Since Alas ~ BlasB it follows that (Ala") ® h has no retraction and (Ala") ® g has no section. Thus h has no retraction and g has no section. Contradiction (f is irreducible)!

Proof of (3.9.1). As in the proof of (3.5) we see that u has a retraction (resp. a section) iff a certain linear system L of equations over B has a solution in B. Let B be the completion of B with respect to n. Then by a strong approximation Theorem (cf. e.g. [Pol] (1.5) there exists a positive integer SE N such that L has solutions in B iff it has solutions in Bins B.

If u has a retraction (resp. a section) modulo bS then L has a solution in BibS. Thus L has a solution in Bins B and so a solution in B. Then by faithfully flatness L has a solution in B, i.e. u has a retraction (resp. a section).

(3.10) THEOREM. Suppose that (R, m) is an excellent Henselian local ring and A is the completion of R with respect to Is(R). Then the base change functor A ®rinduces an inclusion r(R) c r(A) which is surjective on vertices. In par

ticular # ro(R) = # ro(A). Proof By hypothesis (R,Is(R)) is an AP-couple (cf. [Pol] (1.3)) and thus

A ®R -induces an inclusion ro(R) c ro(A) (cf. (3.8)) which is in fact an equality by [EI] Theorem 3 (cf. the proof of ((4.2)). Now it is enough to apply Lemma (3.9).

(3.11) COROLLARY. Conserving the hypothesis of Theorem (3.10) suppose that

(i) R is a Gorenstein isolated singularity and p = char R (i.e. R is of equal characteristic)

(ii) k is algebraically closed.


Then R is oj finite CM-type iff its completion A is a simple hypersurJace singularity.

Proof Note that R is of finite CM-type iff A is a simple hypersurface singularity by [Kn], [BGS] Theorem A and [GK] (1.4) since # ro(R) = # ro(A).

(3.12) COROLLARY. Conserving the hypothesis oj Theorem (2.10), suppose that k = C and the completion B oj R with respect to m is a hypersurJace. Then R has countable infinite CM-type iff B is a singularity oj type Aoo,Doo'

The result follows from [BGS] Theorem B and our Theorem (3.10).

(3.13) REMARK. Concerning Theorem (3.10), it would be also nice to know when r(R) = r(A). Unfortunately it seems that Artin approximation theory does not help because the definition of irreducible maps involves in fact an infinite set of equations corresponding to all factorizations.

(3.14) PROPOSITION. Let A be a fiat local R-algebra such that mA is the maximal ideal oj A. Suppose that

(i) R is an excellent Henselian local ring, (ii) A is a CM-ring,

(iii) the residue field extension oj R --+ A is strong algebraically pure (e.g. if k is algebraically closed).

Then the base change Junctor A ®rinduces an injective map ro(R) --+ ro(A). In particular # ro(R) :::::; # ro(A).

Proof (R, m) is an AP-couple by [Pol] (1.3) and so the map RA is strong algebraically pure by (iii) (cf. [BNP] (5.6)). Now apply Lemmas (3.5), (3.6).

(3.15) REMARK. If R is not Henselian or (iii) does not hold then our Proposition does not hold in general:

(3.16) EXAMPLE: (i) Let R = R[X, YJ(x,Y)/(X 2 + y2), A := C[X, Y](x,Y)1 (X2 + y2).

Then M:= (X, Y)R is an indecomposable MCM R-module but A ®R M ~ (X + iY)A EEl (X - iY)A is not. Moreover, # ro(R) = 2 and # ro(A) = 3 by [BEH] (3.1).

(ii) Let R:= C[X, Y](x,y)/(y2 - X 2 - X 3 ) and A its henselization. Clearly A contains a unit u such that u2 = 1 + X. Then M:= (X, Y)R is an indecomposable MCM R-module but A ®R M ~ (Y - uX)A EEl (Y + uX)A is not. Also note that # r o(A) = # r 0(',4) = 3, A being the completion of A (see (3.10)).

4. The Brauer-Thrall conjectures on isolated singularities

Let (R, m) be a Henselian local CM-ring, k:= Rim, p:= char k. We suppose that R is an isolated singularity, i.e. Is(R) = m.

(4.1) PROPOSITION. Let rO be a connected component of r(R). Suppose that (i) R has bound properties on MCM modules,

(ii) rO is of bounded multiplicity type, i.e. all indecomposable MCM R-modules M whose isomorphic classes are vertices in rO have multiplicity e(M) :;;; s for a certain constant integer s = s(ro). Then r(R) = rO and r(R) is a finite graph.

Proof By Proposition (2.7) there is a positive integer r such that mr is a Dieterich ideal, i.e., a CM-reduction ideal which is m-primary. Now it is enough to follow [P02] (5.4) (in fact the ideas come from [Di] Proposition 2 and [Yo] Theorem (1.1)).


(i) R has bound properties on MCM modules, (ii) R has infinite CM-type.

Then there exist MCM R-modules of arbitrarily high multiplicity (or rank if R is a domain).

(4.3) COROLLARY. ([P02] (1.2)) Suppose that

(i) R is an excellent ring and [k: k P] < 00 if p > 0, (ii) Reg(R/pR) = {q/pRlqeRegR,q::::l pR} ifpR #- O.

Then the first Brauer-Thrall conjecture is valid for R, i.e., if R has infinite CM -type then there exist MCM R-modulas of arbitrarily high multiplicity (or rank if R is a domain).

(4.4) PROPOSITION. Suppose that

(i) (R, m) is a two dimensional excellent Gorenstein ring (ii) R has bound properties on MCM modules,

(iii) the divisor class group CI(R) of R is irifinite.

Then for all n e N, n ~ 1, there are irifinitely many isomorphism classes of indecomposable MCM R-modules of rank n over R. In particular, the second Brauer-Thrall conjecture holds for R, i.e., if R is of irifinite CM-type then for arbitrarily high positive integers n, there exist infinitely many vertices in r o(R) with multiplicity n (or rank n if R is a domain).

Proof. Let K be the fraction field of R an!i ~ a Weil divisor on Spec R. Then J.<1 := {x e K I div x ~ ~} is a reflexive R-module of rank one and the correspondence ~ f-+ J defines an injective map u: CI(R) -+ r ° (R).

Fix IXe 1m u. Let r", be the connected component of IX in r(R), I the set of vertices of r", and M an indecomposable MCM R-module whose isomorphism class !VI belongs to I. If M '* R then there exists an almost split sequence

O-+P-+E-+M-+O

because R is an isolated singularity (cf. [Au3]). As R is a two dimensional Gorenstein ring we obtain P ~ M by [Au1] III Proposition (1.8) (cf. also [Yo]A14 or [Re]).

Let N be an indecomosable MCM R-module. If there is an irreducible map N -+ M (resp. P ~ M -+ N) then it factorizes by E -+ M (resp. M -+ E) and thus N is a direct summand in E. The converse is also true (cf. e.g. [Re] (3.2)) and in particular,

(1) there exists an irreducible map N -+ M iff there exists an irreducible map M -+ N (so we can consider instead of r a its undirected graph Ir al obtained by removing all the loops and forgetting the direction of the arrows).

(2) 2 rankRM = rankRE ?: L&rankRN, where N runs through the vertices of Ir al which are incident to M.

For M = R the fundamental sequence (an exact sequence of this form corresponding to a nonzero element of Ext~(k, R) ~ k, cf. [Au2] Section 6 or [AR] Section 1 or [Re]) shows that. (2) is valid also in this case.

We proceed with a combinatorial remark (cf. also [HPR]). Let r = (r 0, r d be an undirected graph, r 0 the set of vertices, r 1 <;; r 0 x r 0 the set of edges «i, i) rt r 1 for all i E r 0 and (i,j) E r 1 iff (j, i) E r 1)' r' = (ro, r'1) is a subgraph of rifro c r 0 and r'1 = {(i,j)E r 1li,jE ro}. A function r: ro -+ N is subadditive if

2r(i)?: I r(j) for all i E r o. (i,j)ef,

(4.5) LEMMA. Assume that

(i) r is connected. (ii) r is subadditive,

(iii) r is not bounded and has minimal value 1.

Then r is A 00: • • • ----- and r(i) = i for all i. 123

Proof. Note that (*) implies:

(a) 2r(j)?:dj ,=#{iEro l(i,j)Er 1 }, jEro, (b) (Monotony) Let. • • be a subgraph of r. Then r(i)?: r(j) implies

I } S

r(j) ?: r(s) (Indeed by (*) we have 2r(j) ?: Llt.ilef, r(t) ?: r(i) + r(s), thus 2r(j) ?: r(j) + r(s)). The assertion of the Lemma immediately follows from (**) For all n ?: 1 there is a sub graph

.. -_.-........ ----- .. - ........ 1 2 3 n-1 n

of r such that (c) for 1 ,,;; j ,,;; n - 1, dj = 1 if j = 1 and dj = 2 if j > 1, (d) r(j) = j for 1 ,,;; j ,,;; n.

(Then r contains a subgraph A ox" which must be r itself by (c) since r is connected.)

We prove (**) by induction on n. If n = 1, choose an element 1 E r 0 such that r(l) = 1. Now assume (**) is true for n ~ 1. Then we find a subgraph

__ ... __ --.E--------ei, 1 n -1

{d - 1

where t = d:

t

for n> 1, for n = 1.

2n ~ n - 1 + L r(is), s= 1

i,

and one of the numbers r(is) is > n (otherwise r is bounded on r 0 by (b)). This implies n + 1 ~ L~ = 1 r(is) > n, i.e. t = 1, dn = 1 for n = 1, respectively, dn = 2 for n> 1. If we denote il by n + 1, we obtain r(n + 1) = n + 1, which completes the proof.

Back to our Proposition, we want to apply (4.5) to !r~1 for r:= rank. By (2) ris subadditive. If r is bounded we obtain r~ = r(R) finite (cf. (5.1)) and thus CI(R) finite, contradiction! Thus r is unbounded. By (4.5) Ir ~I = Aoo and rankR(i) = i for every i.

Let rl E 1m u, a' -=1= a. Then a' 1: I because r ~ contains only one vertex of rank one. Thus r~ n r~, = ~ and so for each nE N we find # rank-l({n})~ #CI(R).


(i) (R, m) is a two dimensional excellent Gorenstein ring, (ii) [k: kP] < 00 if p > 0,

(iii) Reg(R/pR) = {q/pR I qE Reg R, q :::J pR} if p -=1= char R, (iv) CI(R) is infinite.

Then for all n E N, n ~ 1 there are infinitely many isomorphism classes of indecomposable MCM R-modules of rank n. In particular the second Brauer- Thrall conjecture holds.

The result follows from (1.7) and (4.5).


(4.6.1) REMARK. Dieterich shows in [Di] Theorem 20, that the second Brauer-Thrall conjecture is valid in arbitrary dimension for complete isolated hypersurface singularities over algebraically closed fields of characteristic # 2. Using our (3.10) we are able to extend it for a large class of excellent henselian local rings. However in the two dimensional case our Corollary (4.6) gives a sharper version.

(4.7) COROLLARY. Let n be a positive integer. Suppose that

(i) (R, m) is a two dimensional excellent Gorenstein ring (ii) R is a R-algebra and k algebraically closed.

Then the following conditions are equivalent: (1) R is not a rational double point, (2) There are infinitely many isomorphism classes of indecomposable MCM R-modules of rank n.

Proof. By a well-known result (cf. e.g. [Ba]) R is not a rational double point iff R is not rational (R is Gorenstein!) and thus iff CI(R) is infinite (cf. [Li] (17.4), (16.2)). Thus (1)=(2) follows from the above Corollary. On the other hand, (2) implies: R is of infinite CM-type and thus R is not a rational double point by [A V] (1.11) (see also [EK]) and our (3.3) or (3.10)).

Note added in proof

In (3.9.1), the argument can be simplified by a recent result of V. Nica (Stud. Cere. Mat., Vol. 41, Nr. 3 (1989), 179-184), who gives a direct proof for the "linear strong approximation".

References

[An] M. Andre, Localisation de la lissite formelle, Manuscripta Math. 13 (1974) 297-307. [Ar] M. Artin, Algebraic approximation of structures over complete local rings, Publ. Math.

IHES 36 (1969) 23-58.

[A V] M. Artin and J. L. Verdier, Reflexive modules over rational double points, Math. Ann. 270 (1985) 79-82.

[Au1] M. Auslander, Functors and morphisms determined by objects, Proc. Conf. on Representation Theory, Philadelphia 1976, Marcel Dekker, 1978, 1-244.

[Au2] M. Auslander, Rational singularities and almost split sequences, Trans. Arner. Math. Soc. 293 (1986) 511-531.

[Au3] M. Auslander, Isolated singularities and existence of almost split sequences, Proc. IeRA IV, Lect. Notes in Math. 1178 (1986) 194-241.

[AR] M. Auslander and 1. Reiten, Almost split sequences for rational double points, Trans. Arner. Math. Soc. 302 (1987) 87-97.

[Ba] L. Badescu, Applications of the Grothendieck duality theory to the study of normal isolated singularities, Rev. Rourn. Math. Pures Appl. 24 (1979), 673-689.

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[BR1]

[BR2]

[BEH]

[BGS]

[Di]

[El]

[EK]

[GK]

[EGA]

[He]

[HPR]

[Kn]

[Li]

[Mal] [Ma2] [NP]

[Pol]

[P02]

[Re]

[Sc]

[Yo]

Dorin Popescu and Marko Roczen

S. Basarab, V. Nica and D. Popescu, Approximation properties and existential completeness for ring morphisms, Manuscripta Math. 33 (1981) 227-282. A. Brezuleanu and N. Radu, Sur la localisation de la lissite formelle, C.R. Acad. Sci. Paris 276 (1973) 439-441. A. Brezuleanu and N. Radu, Excellent rings and good separation of the module of differentials, Rev. Roum. Math. Pures Appl. 23 (1978) 1455-1470. R.-O. Buchweitz, D. Eisenbud and J. Herzog: Cohen-Macaulay modules on quadrics, in Proceedings, Lambrecht 1985, Lect. Notes in Math. 1273 (1987) 58-116. R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer: Cohen-Macaulay modules on hypersurface singularities II, Invent. math. 88 (1987) 165-183. E. Dieterich: Reduction of isolated singularities: Comment. Math. Helvetici 62 (1987) 654-676. R. Elkik: Solutions d'equations a coefficients dans un anneau henselien, Ann. Sci. Ecole Norm. Sup. 4' serie 6 (1973) 533-604. H. Esnault and H. Knorrer: Reflexive modules over rational double points, Math. Ann. 272 (1985) 545-548. G.M. Greuel and H. Kroning: Simple singularities in positive characteristic, Preprint, Bonn, Max-Planck Institut jUr Math. 19 (1988). A. Grothendieck and J. Diendonne, Elements de geometrie algebrique, Publ. Math. IHES (1964). J. Herzog: Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay Moduln, Math. Ann. 233 (1978) 21-34. D. Happel, U. Preiser and C.M. Ringel, Vinberg's characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules, Proceedings of the third international conference on representations of algebras, Carleton University, Ottawa 1979, Springer Lecture Notes in Mathematics 832, 280-294. H. Knorrer, Cohen Macaulay modules on hypersurface singularities I, Invent. Math. 88 (1987) 153-165. J. Lipman, Rational singularities with applications to algebraic surfaces and unique factorization, Publ. Math. IHES 36 (1969) 195-297. H. Matsumura, Commutative Algebra, Benjamin, New York, 1970. H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986. V. Nica and D. Popescu, A structure theorem on formally smooth morphisms in positive characteristic, J. of Algebra 100 (1986) 436-455. D. Popescu, General Neron desingularization and approximation, Nagoya Math. J. 104 (1986) 85-119.

D. Popescu, Indecomposable Cohen Macaulay modules and their multiplicities, to appear in Trans. AMS. I. Reiten, Finite dimensional algebras and singularities, in Proceedings, Lambrecht 1985, Springer Lecture Notes in Mathematics, 1273 (1987) 35-57. F.-O. Schreyer, Finite and countable CM-representation type, in Proceedings, Lambrecht 1985, Springer Lecture Notes in Mathematics, 1273 (1987) 9-34. Y. Y oshino, Brauer-Thrall type theorem for maximal Cohen-Macaulay modules, J. Math. Soc. Japan 39 (1987) 719-739.


Fano bundles of rank 2 on surfaces

MICHAL SZUREK & JAROSLA W A. WISNIEWSKI Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556; (on leave from Warsaw University, PkiN IX p., 00-901 Warszawa, Poland)

Received 4 November 1988; accepted 9 October 1989

O. We say that a vector bundle tC on X is Fano if\P(tC)is a Fano manifold. In [12], we classified rank-2 Fano vector bundles over the complex projective space \p3 and over a smooth quadric (Q3 c \P4. This paper is thus complementary to [12].

Ruled Fano 3-folds were classified by Demin already in 1980. However, his paper [2] contains gaps and omissions, one of them corrected in the addendum. Here we correct the other. Moreov~r, we present a vector-bundle approach to the problem, i.e., we try to get our results by pure vector bundle methods. Even if the proofs that utilize Mori-Mukai-Iskovskich list of all Fano threefolds are sometimes shorter and even if we did not succeed, in some cases, to avoid using the classification list mentioned above (namely, to exclude the cases Ci = 0, c2 = 6 over \p 2 and ci = (-1, -1), c2 = 5 or 6 over \pi x \Pi), we believe that our method is worthy of dealing with, also because it works for ruled Fano 4-folds, as well, see [12]. Besides, we are able to draw some conclusions on the geometry of stable vector bundles with Ci = 0, C2 = 2 or C2 = 3, one of them being the 1-ampleness of such bundles.

THEOREM. There are 21 types of ruled Fano 3-folds V = \P(tC) with a 2-bundle

tC on a surface. Up to a twist, they are the following:

tC _K3 # in M-M-I list, [9] over \p2:

1 (9(1) E9 (9( -1) 62 36, Table 2 2 (9 E9 (9(1) 56 35 3 (9E9(9 54 34 4 T1J'2( -2) 48 32 5 o -+ (9 -+ tC -+ Jx -+ 0, XE \p2 46 31 6 tC stable with Ci = 0, C2 = 2, thus:

0-+ (9( _1)2 -+ (94 -+ tC(1) -+ 0 38 27 7 tC stable, tC(1) spanned, Ci = 0, C2 = 3,

thus: 0 -+ (9( - 2) -+ (93 -+ tC(1)-+ 0 30 24

296 Michal Szurek and laroslaw A. Wisniewski

over [p>l X [p>l:

8 (9(-1,-1)EB(9 9 (9 EEl (9

10 (9(0,1) EB (9 11 (9(0, -1) EEl (9( -1,0) 12 tff stable, tff(1, 1) spanned,

CI = (-1, -1),c2 = 2, thus: 0--+ (9( -1, -1) --+ (93 --+ tff(1, 1) --+ °

52 48 48 44

36

31, Table 3 27 28 25

17

over the Hirzebruch surface F I , with a blow-down map {3: FI --+ [p>2:

13 {3*((9 EEl (9( -1)) 50 30, Table 3 10 (9 EB (9 48 28 14 {3*(T"2(-2)) 42 24

over a Del Pezzo surface being a blow-up of k ~ 8 points on [p>2:

15-21 (9 EB (9 6(9 - k) 10/4.

1. Preliminaries

Throughout the paper tff is a rank-2 vector bundle on a smooth complex projective surface S and ~8 is the antitautologicalline bundle on V = [P>(tff). By p we denote the projection morphism p: [P>(tff) --+ S and by F - the fibre of p. We have the following exact sequence on V = [P>(tff):

(1.1)

with the relative tangent bundle Tv1s fitting in

(1.2)

We call (1.1) the relative Euler sequence. We then obtain

(1.3) COROLLARY. cl(V) = p*(cl(S) - cl(tff)) + 2~8.

The theorem of Leray and Hirsch yields that in the cohomology ring of V the following holds

(1.4)

We infer that S must be a Del Pezzo surface:

F ano bundles of rank 2 on surfaces 297

(1.5) PROPOSITION. If V = 1P(t9') is a ruled Fano threefold, then S is a Del Pezzo surface.

Proof See [2]; and [12] for a more general version.

(1.6) PROPOSITION. In the above notation, we have:

Proof The first statement is obvious. To prove that b3 vanishes, let us observe first that h3,O(V) = h3(l!J y ) = hO(Ky ) = Obythe Kodaira Vanishing Theorem. To show that b3 = 0, it is then sufficient to prove that bl ,2 = b2,l = 0, i.e., that hl(Q2) = h2(Ql ) = O. Dualizing (1.2), we get

Because h2(P*(Qs) = h2(Qs) = hl(Ks) = hl(l!Js) = 0 by rationality, we have to show only that h2(Tvls) = O. Recall that T v1s fits in a short exact sequence

Since p*(t9') restricted to fibres of p: 1P(t9') -+ 1P2 is a trivial 2-bundle and C is l!J( -1) on fibres, we have Rip*(p*(tf) ® C) = 0 for i > 0 and Leray's spectral sequence gives h2(p*(tf) ® ~V) = O. Finally, vanishing of Hl(l!JV ) follows directly from the rationality of V.

2. Fano bundles over 1P2

This case is the most interesting one. We may assume tf is normalized, i.e., Cl(tf) = 0 or -1.

(2.1) PROPOSITION. If tf is a normalized Fano bundle on 1P2, then tf(2) is ample. Proof Let H = p*(l!J(l)). By (1.3), Cl(V) = - H + 2~8(2) if Cl = 0 and Cl(V) =

2~8(2) if Cl = - 1; but His nef and we are done.

(2.2) PROPOSITION. Let t9' be a Fano bundle on 1F2 with Cl(t9') = O. Then C2 :::;; 3 and in the cohomology ring of 1P(t9') we have H3 = H~~ = 0, H2~8 = 1, ~~ =

- C2(t9'). Proof. Vanishing of H3 and the equality H2~8 = 1 is obvious. The relations

between the generators of 1P(t9') are then easy consequences of the LerayHirsch formulae. To prove that C2 :::;; 3, we calculate: 0 > K~ = - d(V) =

(3H + 2~8)3 = 8C2 - 54, hence C2 :::;; 6. However, as follows from the Mori-

298 Michal Szurek and Jaroslaw A. Wisniewski

Mukai-Iskovskich classification, [9J, there are no Fano threefolds with K~ = 6, 14,22 and b2 (V) = 0, cf. (1.6); see also Remark in Section 3.

(2.3) PROPOSITION. Let S be a normalized Fano bundle on !P 2 . Then (a) if S is not semis table, then either S = (9(1) EB (9( - 1) or S = (9 EB (9( -1); (b) if Sis semistable, but not stable, then either S = (9 EB (9 or S fits in 0 -+ (91"2 -+

S -+ oF x -+ 0, where oF x is the sheaf of ideals of a' point x E !p 2 ;

(c) if cl(S) = -1, then either S = (9 EB (9( -1) or S = T1J>2( -2), Proof. We start from (c). Let us take a line L and let S(2)IL be (9L(ad EB

(9L(aZ). We then have al + a2 = cl(S(2)) = 3 and al, a2 > 0 by (2.1). Therefore al = 2, a2 = 1 up to permutation. By the Van de Ven theorem on uniform bundles, [13J, S is as we claim. This proves (c). To show that (a) holds we may then assume that Cl(S) = o. If S is not semistable, there is a non-trivial section s E HO(S( -1)). Assuming s does not vanish anywhere we find a trivial subbundle (9 c S( -1), hence the quotient of S( -1) by this trivial bundle is (9(2) and we get (a). Assume now that s vanishes at a point x. Let us take a line that contains a finite number of zeros of s. The bundle S( - l)IL then splits as (9(k) E9 (9( - 2 - k)

with k ~ 1, so that S(2)IL = (9(k + 3) EB (9(1 - k)in contradiction with (2.1). This concludes the proof of (a). To show (b), assume S is not stable. Since for Cl (S) = -1 a semistable rank-2 bundle is stable, we infer that Cl(S) = O. Pick a non-trivial section sEHO(S).lfthe set {s = O} is empty, we get an embedding (9 c S whose cokernel is also a trivial bundle and then S = (9 EB (9. If s vanished at two points (not necessarily distinct), the line through these points would be ajumping one of type (- 2,2), contradicting (2.1). Finally, if s vanishes at a single point x, then C2(S) = 1 and (b) follows. It is known that any non-trivial extension as in (b) is a bundle, (see [10J, ch.l, §1.5).

This proves 1 through 5 of our theorem. To conclude the case of !p 2 we must, in view of (2.2) and (2.3), study bundles with Cl = 0, C2 = 2 or 3 in more detail.

Case Cl = 0, C2 = 2. Let S be a stable bundle with Cl(S) = 0, C2(S) = 2. The twisted bundle S(l) is then generated by global sections, though not ample, since there are lines L such that S(l)IL = (9 EB (9(2), [1]. Then e8(1) is globally generated, hence nef. Recall that the cone of numerically effective divisors on V is generated by two (classes of) divisors. Because H and em) are not numerically equivalent, their sum must be then in the interior of the cone, i.e., - Kv =

H + 2e8(l) is ample. This gives 6 of our theorem.

REMARK. To see that any stable 2-bundle on !p 2 with Cl = 0, C2 = 2 fits in the exact sequence as in (6), observe first that h2(S(1)) = h2(S) = hO(S) = 0 by stability of S and of its dual. Then hl(S) = - X(S) = 0 by Riemann-Roch, so h 1 (S) = 0 by the Castelnuovo criterion. By Horrock's criterion of decomposability the kernel of the evaluation (94 -+ S(l) -+ 0 splits. Computing the Chern classes of the kernel then gives the sequence as in 6 of the Theorem.

Fano bundles of rank 2 on surfaces 299

APPLICATION 1. For 8 as above, the resulting Fano threefold arises from blowing up a twisted cubic in ~3, [9]. The generic section s of 8(1) vanishes at three points whose associated lines in (~2r form a triangle inscribed in the non-singular conic of jumping lines of 8, [1]. Let Z = zero(s) and 0 -+ m -4 8(1) -+ J z(2) -+ 0 be the corresponding exact sequence. It gives rise to an embedding

83 := ~(J z(2» c ~(8(1» = ~(8)

(over ~2) of the Del Pezzo surface 83 (the blow-up of three points in ~2). Since e8(1) is spanned, 83 is the inverse image of a plane in ~3 = ~(r(8(1))). In other words:

(2.4) The Fano threefold ~(8) with 8 as above, admits two projections p:

~(8) -+ ~2, q: ~(8) -+ ~3 such that q-l is the blow-up of a twisted cubic and p is a ~l-bundle. On a generic plane P c ~3 = ~(r(8(1))), the rational map pq-l: P -+ ~2 is a quadratic map

that blows up the points where P meets the twisted cubic and contracts the three exceptional curves that arise. The rational map pq - 1 is then a family of such "elementary" quadratic maps.

APPLICATION 2. Stable rank-2 vector bundles on ~2 with C1 = 2, c2 = 3 are I-ample, see [11] for the definition of l-ampleness.

Case C1 = 0, C2 = 3.

(2.5) Let 8 be a stable rank-2 vector bundle on ~2 with C1 = 0, C2 = 3. Then hO(8(1» = 3.

Proof As a rank-2 bundle with c1 = 0,8 is autodual, hence hO(8) = h2(8) = h2(8(1»=0. From the Riemann-Roch formula we obtain hl(8) = -X(8) = 1. As a stable bundle, 8 has the generic splitting type m EB m (the theorem of Grauert and Miilich) and then an easy lemma of LePotier, [3], prop. 2.17, gives h1(8(k» = 0 for k ~ 1. Hence hO(8(k» = X(8(k» if only k ~ 1; in particular hO(8(1» equals 3.

(2.6) PROPOSITION. Let 8 be a stable, rank-2 vector bundle on ~2 with c1 = 0, C2 = 3. Then V:= ~(8) is a Fano threefold if and only if 8(1) is spanned.

REMARK. For a general bundle 8 E Jt(0,3), 8(1) is spanned and from Barth's description of Jt(0.3) it follows that there are stable rank-2 vector bundles 8 with C1 = 0, C2 = 3 and 8(1) not spanned (namely, type "a" in Section 7 of [1]).

To prove (2.6), assume first that 8(1) is spanned. Then by the same arguments

300 Michal Szurek and J aroslaw A. Wisniewski

as we used in the case of bundles with C1 = 0, C2 = 2 (namely, that the sum H + ~8(1) lies in the interior of the cone of numerically effective divisors and hence is ample) we conclude that - Kv is ample. Assume then that 8(1) is not spanned, so that the base point locus BslH + ~I is not empty.

(2.7) Claim. I ~ + HI has no base components. Proof Assume then that I H + ~ I has a fixed component Bo and consider

a divisor D in the system I H + ~ I. Let D = Bo + U, with U in the variable part of I H + ~ I. Because the fibres of p: jp>(8) -+ jp>2 are curves, p(Bo) and p(U) are at least one-dimensional. If p(Bo) and p(U) were curves, they would give rise to sections of 8(1) ® fp(Bo) and of 8(1) ® fp(u), contradicting stability. Hence P(Bo) and P(U) are the whole jp>2, so that BoF ~ 1, UF ~ 1, where F is the fibre of p. Then 1 = 0 + 1 = HF + ~F ~ BoF + UF ~ 2, a contradiction.

Since (H + ~)3 = 0, Bsl ~ + HI is not zero-dimensional thus in view of (2.7) it contains one-dimensional components. Let B the sum of them counted with multiplicities so that we can write Dl 0 D2 = B + C where 1-cycle C does not contain one-dimensional components of the base point locus Bsl ~ + HI and D 1 ,D2 are general divisors from I~ + HI.

(2.8) Claim. The cycle C contains at least one fibre F of the projection p: jp>( 8) -+ jp>2.

Proof Let x E p(B). We show that hO(8(1) ® fx) = 2. Indeed, if hO(8(1) ® fx)

were 1, then at x there would exist two independent sections of 8(1) (cf. (2.5)), hence it would be generated by global sections at x which is not the case. Let us now take a line L through x such that 8(1) I L = £'9(1) EB £'9( -1). The inclusion HO(8(1)) -+ HO(8(1) I L) induces the embedding HO(8(1) ® fx) -+ HO(8(1) ® fx I L) and from hO(8(1) ® fx I L) = 2 we infer that hO(8(1) ® fx) ~ 2 and therefore equals 2. To conclude the proof of (2.8). let us take two sections of 8(1) that vanish at x. The corresponding divisors of I~ + HI then vanish along the fibres over x.

(2.9). In the above notation, (H + ~) 0 C ~ 1. Proof In general, if a curve C is not contained in the base point set of a linear

system A, then A 0 C ~ O. This shows that in our situation (H + ~) 0 C ~ o. As C contains the fibre, then (H + ~) 0 C ~ 1.

(2.10). HB ~ 2. Indeed, we know already that (H + ~)2 = B + c. Since (H + ~)2 H = 2 and H is nef, we have HoC ~ 0, so that HB ~ 2.

(2.11). (H + ~)B ~ -1. Indeed, let us take two divisors as in (2.8). Since (H + ~)3 = 0, then by inequality (2.9) we get (2.11).

To conclude the proof of (2.6), let us notice that, by (2.11) and (2.10), c1(V)B = 2(H + ~)B + HB ~ 0, i.e., c1(V) cannot be ample. 0


To study the structure of iP'($) more closely, we consider the evaluation morphism (93 --+ $(1). Its kernel is a line bundle with c 1 = - 2, i.e., we have an exact sequence

(2.12)

Now, $(1) admits a section s with four ordinary zeros at points Xl' X 2, X3, X4; see e.g. [5], proposition l.4b. No three of these points are collinear, since otherwise $(1) would have a jumping line of type (3, -1) or (4, -2) which contradicts (2.1), Hence, in the terminology of [1], $ is a Hulsbergen bundle. By standard arguments (see e.g. [1], §5.2), every such $ is obtained as an extension 0--+ (9 --+ $(1) --+ .1(2) --+ 0 with J c (9 the ideal sheaf of Z = {Xl' X 2 , X 3 ' X4}.

Recall that Y:= iP'(Jz(2)) is the blow-up of iP'2 at Xl' X 2 , X 3 , X 4 and is then a Del Pezzo surface. The above extension gives rise to a iP'2 -embedding of Y into iP'($(1)). Let C p C2 ,C3 and C4 be the blow-ups of X l ,X2 ,X3 ,X4 and H' be the pull-back of the divisor of a line in iP'2. We have: (a) - Kr = 3H' - ~Ci' [7], Proposition 25.1(i); (b) e.J"(l) IY = 2H' - ~Ci and (e.J"(l) ly)2 = ei(l) = (H + e.J")3 = o. (c) because - Ky is ample, by Mori's Cone Theorem the cone of curves is

spanned by the extremal ones, [8], Theorem 1.2. Since Y is neither iP'2, nor a iP'l-bundle over a curve, all extremal rational curves are exceptional, [8], Theorem 2.1., hence they are of the form Ci or H' - Ci - C j , as follows directly from [7], Proposition 26.2. From the above discussion, the following geometrical interpretation of (2.12) emerges:

(2.13). Let s be a generic section of $(1) such that zero(s) = {X l ,X2 ,X3 ,X4 }. Let L c iP'2 = iP'(r($(1))) be a line corresponding to the section s viewed at as an element of HO($(l)). Then Y --+ L is a conic bundle whose fibres are (strict transforms of) conics through X l ,X2 ,X3 ,X4 . In particular, three fibres are reducible - they correspond to pairs of lines through Xi' Xj and X k , X" (i,j, k, 1) being a permutation of the indices (1,2,3,4). It follows that the map p - 1

composed with Y --+ L contracts all conics through our points.

COROLLARY. $(1) is I-ample.

(2.14) REMARK. Globally generated bundles are dense in the moduli space .A 1'2(0,3). Indeed, by standard cohomological algebra (cf. e.g. [10], ch. II, Lemma 4.1.3) the bundle $(1) is uniquely determined by choosing an embedding (9( - 2) C (93. Giving such an embedding is, in turn, equivalent to picking a three-dimensional linear system in I (9(2) I at each point X E iP'2. Such systems are in an 1-1 correspondence with an open set of 2-planes in iP'5, i.e., with points of an open set in Grass (3, 6). Since dim Grass (3, 6) = 9 = dim .A [1>2(0, 3) and the latter

302 Michal Szurek and laroslaw A. Wisniewski

space is irreducible, globally generated bundle form a dense subspace. From the (proof of) Lemma (5.4) in [1] we also infer that a general bundle in.# 1'2(0,3) has a non-singular cubic as its curve of jumping lines.

3. Fano bundles over pl x pl

Let Dl and D2 be the divisors corresponding to the two rulings of n i : pl x pl -+ pl and let D = Dl + D2 • It is easy to derive the Riemann-Rochformulafor rank-2 bundles on pl x pl:

To study whether a bundle tff on pl x pl is Fano, we may assume tff to be normalized, i.e., c1(tff) = (a1,a2 ) with -1:::; ai :::; 0, i = 1,2. Let us denote H = p*(D), Hi = p*(D;), i = 1,2. As in Section 1, we obtain a formula for C 1 (V), where V= P(tff): if c1(tff) = (0,0), then c1(V) = 2H + 2~; if c1(tff) = -Hi' then C1 (V) = 2H + Hi + 2~ and if C1 (tff) = (-1, -1), then c1 (V) = 3H + 2~. Because H is numerically effective, we easily derive an analogue of (2.l):

(3.1). If tff is normalized 2-bundle on pl x pl such that P(tff) is a Fano manifold, then tff(2, 2):= tff ® l'D(2,2) is ample. If c1 (tff) = ° then already tff(l.1) is ample.

Because a bundle iF = 61 l'D(a;) on a line is ample iff all a;'s are positive, we easily obtain the following corollaries:

(3.3). If C1 (tff) = 0, then tffl Di = l'D 61 l'D for i = 1,2;

(3.4). If C1 (tff) = (-1,0), then tffl D2 = l'D 61 l'D( -1) and tff I Dl = l'D EB l'D with the obvious symmetry when c1(tff) = (0,1);

(3.5). If c1(tff)=(-1,-1), then tfflDl =l'D61l'D(-1) and tfflD2 =l'DEB l'D( -1).

Let us notice that in cases (3.3) and (3.4) the push-forward ni*(tff) is a rank-2 vector bundle on pl (for i chosen such that tff I Di = l'D 61 l'D). Moreover, the natural morphism ntni*(tff) -+ tff is an isomorphism and hence we have

(3.6) If c1 (tff) = (0,0), then tff = l'D 61 l'D;

(3.7) If c1(tff)=(-1,0) or C1(tff) =(0,-1), then either tff=l'D(-1,0)61l'D or tff = l'D(0, -1) 61 l'D, respectively.

It remains then to study the cases c1 (tff) = ( -1, -1). If this is the case, the (1.3) reads as c1(V)= 3H + 2(.f and in the cohomology ring of pl x pl the following relations hold

and thus 0 < (C 1(V))3 = (3H + 2~)3 = 56 - 8c2 , i.e., C2 ~ 6. Combining this with (1.7) and Table 3 in [9], we get c2 ~ 4. Then we show that cases c2 = 3 or 4 cannot occur:

(3.8). If cff is a Fano 2-bundle on p1 x p1 with C1 (cff) = (-1, -1), then c2(cff) ~ 2. Proof. First we show that HO(cff( -1, -1)) = O. Indeed, assume that s is

a non-zero section of cff( -1, -1) and (9 ~ cff( -1, -1) is the corresponding inclusion. On a line L we have then the exact sequence

where rank-l quotient sheaf Q has degree 0 in contrary with the ampleness of cff(2,2).

In general, for a 2-bundle :F we have :F = :F v ® det:F, hence in our situation H2(cff(l, 1)) = HO cff« -1, -1)) = 0 by Serre duality and then the Riemann-Roch formula gives

X(cff(l, 1)) = 5 - c2.

We see that if C2 ~ 4, then HO(cff(l; 1)) -# O. Let s be a non-zero section. Claim. If c2 were 3 or 4, then it would exist a curve CE I {9(1, 1)1 such that the

multiplicity of the zero set Z of s on C was at least three. Indeed, let us consider the Segre embedding of p1 x p1 into p3 determined by

the linear system I {9(1, 1)1. Let us pick a plane in p 3 meeting Z at three points, counted with multiplicities. The intersection of the plane and the cubic surface p1 x p1 in p3 is the curve C, therefore C is a conic. If C were reducible, e.g. C = C1 U C2 , the zero set of s would meet one of C;'s with multiplicity at least two. It would give rise to an embedding (91Jl>2(2) ~ cff(l, 1), in a contrary with cff I Di = {9 EB (9( -1). Hence C must be a smooth conic. Assume then that cff(l, 1) I C = (9(a 1 ) EB (9(a2), a1 ~ a2. As C1 (cff(l, 1)) = (1,1), we have a1 + a2 = 2, but siC gives an embedding (9(3) I C ~ cff(l, 1) I c, so that a1 ~ 3 and then a1 - a2 ~ 4, which contradicts Lemma 1.5 in [12] and hence proves (3.8).

REMARK. A similar method may be used to exclude (without using the M.-M.-1. classification) the case c1 = 0, c2 = 4,5 on p2.

(3.9} PROPOSITION A Fano bundle cff on p1 x p1 with C1 (cff) = (-1, -1) fits into the exact sequence

(3.10)

and for C2 ~ 1 the sequence splits. Proof. We know that cff I Di = r9 EB (9( -1), i = 1,2, hence the push-forward

304 Michal Szurek and Jaroslaw A. Wisniewski

ni*(c&") is a line bundle on [pll, say 0(k). The natural morphism n1ni*(c&") ~ c&" is an evaluation on every fibre and because the sections of 0 EB 0( -1) are constant (in particular, they do not vanish), we have an exact sequence

with a line bundle Q as a cokernel. Calculating the Chern classes we obtain (3.10). Finally, the fact that for C2 ~ 1 this sequence splits follows immediately from the vanishing of first cohomology groups of appropriate bundles on [pll x [pll. This proves (3.9).

COROLLARY. For c&" as above, c2 (c&") ~ o. Proof (3.10) gives the exact sequence

o ~ 0(2,.2 - c2 ) ~ C&"(2, 2) ~ 0(1, c2 + 1) ~ 0

and C2 < 0 would contradict the ampleness of C&"(2, 2). This proves (8) and (11) of the Theorem.

(3.11) PROPOSITION. If c&" is a Fano bundle and Cl (c&") = (-1, -1), c2 (c&") = 2, then C&"(1, 1) is globally generated and fits in an exact sequence

o ~ 0( -1, -1) ~ 0$3 ~ C&"(1, 1) ~ O.

Proof By (3.9) we have hi(c&"(O, 1)) = hi(C&"(1,0)) = 0, all i, and hi(C&"(1, 1)) = 3 if i = 0 and 0 otherwise. Restricting c&" to the ruling D 1 gives

o ~ c&"(0, 1) ~ C&"(1, 1) ~ C&"(1, 1)ID1 ~ O.

The induced evaluation morphism HO(C&"(1, 1)) ~ HO(C&"(1, 1)IDJ is then an isomorphism. But C&"(1, 1)IDi is globally generated, so is C&"(1, 1). Since hO(C&"(1, 1)) = 3, computing the Chern classes of the kernel of the evaluation 0 3 ~ C&"(1, 1) gives (3.11). Conversely, if the inclusion 0 ~ 0(1,1)3 corresponds to a non-vanishing section of 0(1,1)3, the quotient is a 2-bundle. To complete our discussion of the case c2 = 2, we must show that [pl(C&") is Fano. Because C&"(1, 1) is globally generated, it is nef and H + 2~.J'(1.1) is nef, as well. Therefore, to prove that it is ample, it is sufficient (by the theorem of Moishezon and Nakai) to check that H + 2~.J'(l.1) has positive intersections with curves in [pl(C&"). However, if H· C = 0, then C is contained in a fibre and then ~"(l.l)· C > O.

4. Fano bundles over non-minimal Del Pezzo surfaces

Let us recall that any non-minimal Del Pezzo surface Sk is a blow-up of k points x i(1 ~ i ~ k ~ 8) on the plane, no three on one line and no six of them on a conic.


The canonical divisor of Sk has the self-intersection number equal to 9 - k. S 1 is the same as the Hirzebruch surface F l' Let {3: Sk --+ [p2 be the blow-down morphism, Ci be the exceptional divisors of {3 and H be the inverse image of the divisor of a line of [p2. Let C be a Fano bundle on Sk' As in the preceeding sections, we may assume C to be normalized, i.e.,

Since K· Ci = 1, we may apply the same methods as in Section 3 (using Lemma 1.5 from [12]) to conclude easily that Cle, = (!) EB (!) and consequently C = {3*(C') with a 2-bundle C' on [p2. Moreover, if C1 (C')· H = 0, then C is trivial. Indeed, let L be the strict transform of a line L c [p3 that passes through one of the points Xi'

Then K sk • L ~ 2 and in virtue of Lemma 1.5 in [12] we have CIL = (!) EB (!),

therefore CIL = (!) EEl (!) and Van de Ven's theorem shows that C' is trivial, so is C. Let us notice that for k ~ 2 we can always choose a line L that passes through

two of the points Xi' so that -KSk • L = -1 and, as above, c1(C)· L = 0, implying c1 (C')· H = 0. In other words, we have proved that for k ~ 2, the only ruled Fano 3-fold over a Del Pezzo surface Sk is [pI X Sk' Finally, on the Hirzebruch surface Fl we have (a) if c1(C') = 0, then, as above, C = (!) EB (!),

(b) if c1(C') = -1, then, as in (2.3), we infer that C' = (!) EB (!)( -1) or C' = T,,2( - 2).

References

1. Barth, W., Moduli of Vector Bundles on the Projective Plane. Inv. Math. 42, (1977), 63-91. 2. Demin, I.V., Three-dimensional Fano manifolds representable as line fiberings (Russian). Izv.

Acad. Nauk SSSR, 44, no. 4 (1980). English translation in Math. USSR Izv. 17. Addendum to this paper in Izv. Acad. Nauk SSSR. 46, no. 3. English translation in Math. USSR Izv. 20.

3. Elencwajg, F. and Forster, 0., Bounding Cohomology Groups of Vector Bundles on P", Math. Ann. 246 (1980) 251-270.

4. Hartshorne, R., Ample Subvarieties of Algebraic Varieties. Lecture Notes 156 (1970). 5. Hartshorne, R., Stable Vector Bundles of Rank 2 on p 3 • Math. Ann. 238 (1978) 229-280. 6. Kawamata, Y., The cone of curves of algebraic varieties. Ann. Math. 119, 603-633 (1984). 7. Manin, Yu.l Cubic fOfnls, Algebra, Geometry, Arithmetic. North Holland 1974. 8. Mori, Sh., Threefolds Whose Canonical Bundle is not Numerically Effective. Ann. Math. 116,

133-176 (1982). 9. Mori, Sh. and Mukai, Sh.: Classification of Fano 3-folds with B2 ;:. 2. Manuscripta Math. 36,

147-162 (1981). 10. Okonek, Ch., Schneider, M. and Spindler, H.: Vector Bundles on Complex Projective Spaces,

Birkhauser, 1981. 11. Schiffman, B, and Sommese, A. J., Vanishing theorems on complex manifolds. Birkhiiuser 1985. 12. Szurek, M., Wisniewski, J .A., Fano Bundles on p 3 and Q 3. Pacific J ourn. Math. 140, no. 2. (1989). 13. Van de Ven, A., On unifofnl vector bundles. Math. Ann. 195 (1972) 245-248.

algebraic geometry: proceedings of the conference at berlin 9â€“15 march 1988

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