ICM-90 Satellite Conference Proceedings: Algebraic Geometry and Analytic Geometry

264 Pages·1991·23.36 MB·English

by  V. Balaji

ICM-90 Satellite
Conference Proceedings
A. Fujiki . K. Kato . T. Katsura
Y. Kawamata . Y. Miyaoka (Eds.)
Algebraic Geometry and
Analytic Geometry
Proceedings of a Conference held in Tokyo, Japan
August 13-17, 1990
Springer-Verlag
Tokyo Berlin Heidelberg
New York London Paris
Hong Kong Barcelona
Editors
Akira Fujiki
College of General Education, Kyoto University
Yoshida-nihonmatsu, Sakyo-ku, Kyoto, 606 Japan
Kazuya Kato and Yujiro Kawamata
Department of Mathematics, University of Tokyo
Hongo, Bunkyo-ku, Tokyo, 113 Japan
Toshiyuki Katsura
Department of Mathematics, Ochanomizu University
Otsuka, Bunkyo-ku, Tokyo, 112 Japan
Yoichi Miyaoka
Department of Mathematics, Rikkyo University
Nishi-ikebukuro, Toshima-ku, Tokyo, 171 Japan
Mathematics Subject Classification (1980): 14-06, 14F99, 14J28,
14K99,14L99
ISBN-13:978-4-431-70086-9 e-ISBN-13:978-4-431-68172-4
DOl: 10.1007/978-4-431-68172-4
© Springer-Vert~ Tokyo 1991
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The use of registered names, trademarks, etc. in this publication does not imply, even
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Preface
The International Conference "Algebraic Geometry and Analytic Geometry, Tokyo
1990" was held at Tokyo Metropolitan University and the Tokyo Training Center of
Daihyaku Mutual Life Insurance Co., from August 13 through August 17, 1990, under
the co-sponsorship of the Mathematical Society of Japan. It was one of the satellite
conferences of ICM90, Kyoto, and approximately 300 participants, including more than
100 from overseas, attended the conference.
The academic program was divided into two parts, the morning sessions and the
afternoon sessions. The morning sessions were held at Tokyo Metropolitan University,
and two one-hour plenary lectures were delivered every day. The afternoon sessions at
the Tokyo Training Center, intended for a more specialized audience, consisted of four
separate subsessions: Arithemetic Geometry, Algebraic Geometry, Analytic Geometry I
and Analytic Geometry II. This book contains papers which grew out of the talks at the
conference.
The committee in charge of the organization and program consisted of A. Fujiki, K.
Kato, T. Katsura, Y. Kawamata, Y. Miyaoka, S. Mori, K. Saito, N. Sasakura, T. Suwa and
K. Watanabe. We would like to take this opportunity to thank the many mathematicians
and students who cooperated to make the conference possible, especially Professors T.
Fukui, S. Ishii, Y. Kitaoka, M. Miyanishi, Y. Namikawa, T. Oda, F. Sakai and T. Shioda
for their valuable advice and assistance in organizing this conference.
Financial support was mainly provided by personal contributions from Professors M.
Nagata, T. Shioda and S. Iitaka as well as grants and fellowships from the following in
stitutions:
The Inamori Foundation,
Inoue Foundation for Science,
Japan Association for Mathematical Sciences,
The Nikko Securities Co., Ltd.
Tokyo Metropolitan Univ ..
We would like to thank Tokyo Metropolitan University for making available the lecture
hall for the morning sessions. Last, but not least, we would like to express our hearty
gratitude to Daihyaku Mutual Life Insurance Co. for generously providing facilities and
accommodations.
The editors
Tokyo, April 1991
CONTENTS
V. BALAJI and C. S. SESHADRI, Poincare polynomials of some moduli
varieties • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1
F. A. BOGOMOLOV, On two conjectures in birational algebraic geometry·· 26
HeUme ESNAULT and Eckart VIEHWEG, Ample sheaves on moduli
schemes • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 53
Shihoko ISHII, Simultaneous canonical models of deformations of isolated
singularities • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 81
Janos KOLLAR, Cone theorems and cyclic covers •••••••••••••••••••••• 101
L:8 Dung Trang, Depth and perversity •••••••••••••••••••••••••••••••• 111
Noboru NAKAYAMA, Elliptic fibrations over surfaces I ••••••••••••••••• 126
Viacheslav V. NIKULIN, Weillinear systems on singular I<3 surfaces •••••• 138
Takashi SAITO, The Euler numbers of f-adic sheaves of rank 1 in positive
characteristic •••••••••••••••••••••••••••••••••••••••••••••••••••••• 165
Vadim V. SCHECHTMAN and Alexander N. VARCHENKO, Quantum
groups and homology of local systems ••••••••••••••••••••••••••••••••• 182
Carlos T. SIMPSON, A lower bound for the size of monodromy of systems
of ordinary differential equations •••••••••••••••••••••••••••••••••••••• 198
Yuri G. ZARHIN, Abelian varieties of I<3 type and f-adic representations •• 231
List of talks •••••••••••••••••••••••••••••••••••••••••••••••••••••• 256
List of Participants ••••••••••••••••••••••••••••••••••••••••••••••••• 258
POINCARE POLYNOMIALS OF SOME MODULI
VARIETIES
V.BALAJI1 AND C.S.SESHADRI
§l Introduction:
For smooth projective varieties v/~, the Weil conjectures as
established by Deligne, tell us that the number of rational
points of the corresponding variety V over IF n, for all n,
q
determine the Betti numbers of V (for precise details cf., §5) •
This theme has been taken up by Harder and Narasimhan in [H-N]
and by Desale and Ramanan in [D-R) to compute the Poincare
pOlynomial of the moduli space M(n,d) of semi-stable vector
bundles of rank n and degree d, where nand d are coprime. More
recently, Atiyah and Bott [A-B) following a geometric approach
compute the Poincare polynomial of the moduli space M(n,d) when
(n,d) l,and also show that there is no torsion in the
cohomology in this case. Let N be the smooth compactification of
M(2,O)~ ( the stable bundles with detE ~ Ox constructed in [5).
x
In [B-S), an approach modelled on [A-B) was studied and this gave
only partial success in the computation of the cohomology of N.
In this paper, we base ourselves on the Harder-Narasimhan
approach; more precisely, we use the strata obtained in [B-S) to
compute the number of IF -rational points of N and we deduce the
q
Poincare polynomial of N in a compact form.
1
Supported by N.B.H.M.
2
The layout of the paper is as follows: in §2 we recall some
facts about N and fix some notations. The heart of the work is
in §3. Here we analyse the Siegel formula (cf. ,
[H-N] , [D-R] , [Bi])to obtain an expression for the number of
rational points of M( 2,0) ~. In §4 we compute the rational
x
points of the remaining strata and this provides the ingredients
for obtaining the number of rational points of N. We end by
obtaining the expression for the Poincare polynomial of N making
use of a lemma of Kirwan (cf., [K] pp., 186 ).
It is easy to see that the computations in this paper give
the intersection Poincare polynomial of M( 2,0) O. It is very
x
likely that these methods generalize to give the intersection
Poincare polynomial for all M(n,d), obtained by Kirwan by other
methods (cf., [K-2] ).
Acknowledgements: We thank D.Indumati for helping us reduce the
polynomial expressions and the referee for his meticulous
reading of the paper.
§2 Preliminaries.
We first recall the basic information on the
desingularization model N defined in [5].
X is a smooth projective curve of genus g 2 • Let M
i!:
denote the normal projective variety of equivalence classes of
semi-stable vector bundles of rank 2 and trivial determinant,
under the equivalence relation V - V' iff gr(V) ~ gr(V'). Let
MS be the smooth open subset of M consisting of the stable
bundles.
3
For a vector bundle V on X, the notion of a parabolic
structure has been defined in [M-S] ( for the purposes of the
present paper (cf., [B-l],[B-2]).
Let PV denote the category of semi-stable vector bundles
4
(V,I1) with parabolic structure 11, as in [B-2], where V is a
vector bundle of rank 4 on X with det V ~ Ox' and 11 is the
parabolic structure on V given at a point P e X with suitable
weights (cf.,[S]) • Then we have
Definition : (cf. , [S] , [B-S]) N is the set of isomorphism classes
of (V,I1) e PV4' such that End V is a specialisation of M2 - the
2 x 2 matrix algebra.
We recall for convenience the definition of the Kummer
variety associated to the Jacobian J of X. The Kummer variety
for our purposes can be defined as follows:
K = { the isomorphism classes of vector bundles of the form
{L @ L-1} where L is a line bundle of degree 0 } .
It is known that K has only 22g nodal singularities which
we denote by Ko' and we have the canonical morphism ~ : J~ K,
which is defined by mapping a line bundle L to ~ (L) = L @ L -1 •
We have the following description of K :
K - Ko = {L @ L-1 I L2 >l: Ox' L e J }
° ,
K = {L eLI L2 ~ L e J }
o x
All that is needed for our purposes is concentrated in the
following theorem. (cf., [B-l],[B-2],[B-S])
Theorem 2.1. There is a natural structure of a smooth
projective variety on N and there exists a canonical morphism
IT : N~ M, which is an isomorphism over MS. Further, there is
4
a natural stratification of N by locally closed subvarieties
{NI} i = 1,2,3, which can be described as follows:
a) IT :N - N ---7 MS, is an isomorphism.
1
b) NI - N2 which we denote by Y is a IP g-2 x bundle over
K - K, where K is the Kummer variety defined above which has
o
been identified with the singular locus of M indeed
K = M - MS. (cf., [N-R] ).
c) Let p e Ko' then the fibre IT-I (p) is the disjoint union of
two closed subschemes Rand S where
(i)R is a vector bundle of rank (g - 2) over G(2,g), the
Grassmanian of 2-planes in g-space.
(ii) S is isomorphic to G(3,g).
Thus
II R ( the union is over points of K )
o
and
N II S ( union over points of K ) .
3 0
§3. The Siegel Formula.
Let X be a curve of genus g defined over a finite field ~ .
q
Then the (-function of X takes the form
2g
n
1=1
where WI are algebraic integers (depending on X) with I WI I
q1/2. Let E be a vector bundle of rank n over X defined over ~ ,
q
having fixed determinant, detE = L, of degree d. The Siegel
formula is the following:
5
2
T. N (A1 utE) (q1-1 ) q(n -l)(g-l) Cx (2) ••• Cx (n).
q
where the summation is over isomorphism classes of bundles E of
rank n defined over IF, such that detE = L, and N (AutE) = the
q q
cardinality of the set of IF -valued points of AutE, the group
q
scheme of automorphisms of E (def ined over IF) • (cf . ,
q
[A-B],[H-N],[D-R]).
We shall restrict ourselves to the case of bundles E of
rank 2 with detE = Ox. Then the Siegel formula takes the form:
1 1
L N (AutE) (q-1) (*)
q
the summation being over isomorphism classes of bundles E of
° .
rank 2 defined over IF with detE =
q x
Since the stable bundles admit only scalar automorphisms
these contribute
(q-1)
to the left hand side of the Siegel formula, where the numerator
denotes the number of rational points of M(n,d)~ the open
x
subset of stable bundles of M(n,d)o
x
Following [D-R] we introduce
~(n,d) L N (A1 utE) (1)
q
where the summation is over the semi-stable bundles, and
6
1
(3' (n,d) L N (AutE) (2)
q
summed over the remaining bundles (the non-semi-stable ones ).
=
For the purposes of the present work , we assume that n 2
and d = 0, with detE = Ox' V E.
In (2) we have the contribution of the non-semi-stable
bundles, which have a canonical (Harder-Narasimhan) filtration
def ined over IF :
q
o ---+ L ---+ E ---+ L -1 ---+ 0 (3)
where the degree of the bundles L (defined over IF) ranges from
q
r 1,2,3, ••• , (one must observe that detE!!!! Ox ). Note that
r o is not included here since the corresponding
extensions define semi-stable bundles. To compute N (AutE), we
q
consider (following [A-B) )the following two cases:
(I) the split extensions: For E = L $ L-1 the automorphisms
group is easily seen to be isomorphic as a scheme to the product
IG x IG xlA (allover IF) where IA is the affine space of
m m q
dimension = dim HO(X,L2) = dim Hom(L-1,L)
Hence we have :
N (AutE) (q_l)2 h
q o
».
where h = N (Ho(X,L2
o q
(II) the non-split extensions: Here because of the non-split
nature the automorphism group is isomorphic to the product
IGm x IA (over IFq) where IA is as in (I) above. Hence we have in
this case :
N (AutE) = (q-l) h
q 0
The number of isomorphism classes of bundles E which come from