Table Of Contentde Gruyter Expositions in Mathematics 12
Editors
O. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, Ohio State University, Columbus
R.O. Wells, Jr., Rice University, Houston
de Gruyter Expositions in Mathematics
1 The Analytical and Topological Theory of Semigroups, K. H. Hofmann,
J. D. Lawson, J. S. Pym (Eds.)
2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues
3 The Stefan Problem, A. M. Meirmanov
4 Finite Soluble Groups, K. Doerk, T. O. Hawkes
5 The Riemann Zeta-Function, A.A.Karatsuba, S.M. Voronin
6 Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii,
V. E. Shatalov, B. Yu. Sternin
7 Infinite Dimensional Lie Superalgebras, Yu.A.Bahturin, A.A.Mikhalev,
V. M. Petrogradsky, M. V. Zaicev
8 Nilpotent Groups and their Automorphisms, E. I. Khukhro
9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug
10 The Link Invariants of the Chern-Simons Field Theory, E.Guadagnini
\ 1 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon,
G. Zhao
Moduli Spaces of
Abelian Surfaces:
Compactification,
Degenerations,
and Theta Functions
by
Klaus Hulek
Constantin Kahn
Steven H. Weintraub
W
DE
G
Walter de Gruyter · Berlin · New York 1993
Authors
Klaus Hulek Steven H. Weintraub
Constantin Kahn Department of Mathematics
Institut für Mathematik Louisiana State University
Universität Hannover Baton Rouge, Louisiana 70803-4918
Weifengarten l USA
D-30167 Hannover
1991 Mathematics Subject Classification: 14-02; 11F46, 14F05, 14J15,
14K10, 14K25, 32J05, 57R99
Keywords: Abelian surfaces, toroidal compactification, theta functions,
Horrocks-Mumford bundle
© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Hulek, Klaus.
Moduli spaces of Abelian surfaces : compactification,
degenerations, and theta functions / by Klaus Hulek, Constantin
Kahn, Steven H. Weintraub.
p. cm. — (De Gruyter expositions in mathematics ; 12)
Includes bibliographical references and index.
1. Moduli theory. 2. Abelian varieties. I. Kahn, Constantin,
1960- . II. Weintraub, Steven H. III. Title. IV. Series.
QA564.H85 1993
516.3'53-dc20 93-29681
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Hulek, Klaus:
Moduli spaces of Abelian surfaces : compactification,
degenerations, and theta functions / Klaus Hulek ; Constantin
Kahn ; Steven H. Weintraub. — Berlin ; New York : de Gruyter,
1993
(De Gruyter expositions in mathematics ; 12)
ISBN 3-11-013851-4
NE: Kahn, Constantin P. M.:; Weintraub, Steven H.:; GT
© Copyright 1993 by Walter de Gruyter & Co., D-10785 Berlin.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced in any form or by any means, electronic or mechanical, including photocopy,
recording, or any information storage and retrieval system, without permission in writing from
the publisher.
Printed in Germany.
Disk Conversion: D. L. Lewis, Berlin. Printing: Gerike GmbH, Berlin.
Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.
To Judy and Nancy
Contents
Introduction ix
I Compactified moduli spaces l
1 Moduli spaces 3
2 Torus embeddings and applications 14
2A Torus embeddings 14
2B Shioda and Kummer modular surfaces 20
2C The topology of Shioda and Kummer modular surfaces . . .. 43
3 Toroidal compactification of A( 1, p} 45
3A Boundary components 46
3B The Tits building 58
3C Toroidal compactification 65
3D Partial compactifications of .4(1, p) 83
4 The boundary of A*(\,p) 125
4A Corank 2 boundary components 125
4B Transversality 147
4C The topology of A*(\,p) 151
5 Humbert surfaces and scaffoldings 155
5A The scaffolding 155
5B Geometry of the Humbert surfaces 169
6 The Satake compactification 184
II Degenerations of abelian surfaces 191
1 Mumford's construction 195
1A Outline of the construction 195
IB Relatively complete models 196
1C Construction of G 200
ID Properties of G 202
2 The basic construction for surfaces 204
2A The basic data 204
2B Computations 207
viii Contents
3 Degenerate abelian surfaces (the principally polarized case) . . .. 215
3A Boundary points and Mumford's construction ........ 215
3B Description of singular surfaces .............. 218
3C Global aspects ..................... 230
4 Degenerate abelian surfaces (the case of (!,/?) -polarization) . . .. 235
4A Boundary points .................... 235
4B Degenerate abelian surfaces ................ 237
5 Polarizations on degenerate abelian surfaces ........... 245
5A Theta functions ..................... 245
5B Extending polarizations (p = 1) .............. 249
5C Miscellaneous remarks .................. 257
5D The general case (p > 1) ................. 262
HI The Horrocks-Mumford map ................. 271
1 The Horrocks-Mumford bundle ................ 274
1A Basic properties .................... 274
IB Horrocks-Mumford surfaces ............... 275
1C Geometry in the space of sections ............. 278
2 Construction of the Horrocks-Mumford map ........... 281
2A Heisenberg equivariant embeddings ............ 281
2B Odd theta null values .................. 289
2C Construction of the Horrocks-Mumford map ......... 292
3 Extension of the Horrocks-Mumford map to *4(1, 5) ....... 298
3A Extension to H .................... 298
2
3B Extension to H .................... 300
{
4 Extension of the Horrocks-Mumford map to A* (1, 5) ....... 305
4 A Extension to the central boundary component ........ 305
4B Extension to the peripheral boundary components ....... 309
4C Extension to the corank 2 boundary components ....... 314
Bibliography ......................... 333
Glossary of Notations ...................... 337
Index . . . .. ....................... 343
Introduction
Moduli spaces for abelian varieties of dimension d are obtained as quotients of
the Siegel space 6^ by arithmetic subgroups Γ of the symplectic group Sp(2d, Q).
The subgroup Γ varies, depending on the type of polarization and, when appro-
priate, on the choice of level structure. The resulting quotient space Γ\6/ is a
ί
quasiprojective but not projective variety with at worst finite quotient singularities.
It is natural to ask for suitable compactifications of this space. This problem has
attracted considerable attention and several answers have been given. The first so-
lution was given by Satake ([Sa]), who obtained a projective compactification in
the case of principally polarized abelian varieties. His compactification is minimal
in a certain sense ([Nam4, p. 7]). The boundary of Satake's compactification A/,
i.e., the set Α/Χ^Λ is a subvariety of codimension d. The disadvantage of the Sa-
take compactification is that in case d > 1 it is highly singular along the boundary,
although it is still normal. Satake's compactification was later generalized by Baily
and Borel to compactifications of quotients of symmetric domains by arithmetic
groups. By blowing up along the boundary, Igusa ([II]) constructed a projective,
partial desingularization of Satake's compactification. As a result the boundary of
Igusa's compactification A* has codimension 1. The ideas of Igusa together with
d
work by Hirzebruch on Hubert modular surfaces were the starting point for Mum-
ford's very general theory of toroidal compactifications of quotients of bounded
symmetric domains ([Mu4]), which was described in detail in [AMRT]. Namikawa
showed that the Igusa compactification is a toroidal compactification in Mumford's
sense ([Nam4]). Toroidal compactifications depend on the choice of cone decom-
positions and are, therefore, not unique. However, they have the advantage that
for proper choices of cone decompositions they are almost non-singular, i.e., they
have at worst finite quotient singularities.
In this book we investigate moduli spaces of abelian surfaces. Instead of principal
polarizations we shall more generally consider polarizations of type (l,p). In most
cases we shall assume that p is an odd prime, but many statements remain valid
for p — 1, i.e., the case of principal polarizations. The reason why we consider
(\,p)-polarizations is twofold: On the one hand polarizations other than principal
polarizations have rarely been treated in the literature, and on the other hand, when
one studies embeddings of abelian surfaces into projective spaces, then (l,p)-
polarizations are often the most interesting cases. In particular the case p — 5
leads to abelian surfaces in P4 and hence to the Horrocks-Mumford bundle. The
fascinating geometry of this vector bundle was one of our principal motives for
χ Introduction
studying these moduli spaces. In the (I,/?)-polarized case it is natural to consider
a kind of level structure we call a "level structure of canonical type", and we will
be considering abelian surfaces equipped with such a level structure. In the case of
projective embeddings this is essentially equivalent to a choice of isomorphism of
the group of linear automorphisms of the embedded surface with the Heisenberg
group of level p. Our approach can be extended to arbitrary polarizations of type
(1, n), but in this more general situation the number of cases which have to be
considered and the calculations involved are greatly increased. Finally note that
every polarization on an abelian surface is a multiple of a (1, n)-polarization.
In part I we construct and describe a toroidal compactification A*(\,p) of the
moduli space .4(1,p) of abelian surfaces with (l,p)-polarization and a level struc-
ture of canonical type (for a precise definition see 1.1.7). The toroidal compacti-
fication which we construct generalizes the Igusa compactification, which in turn
corresponds to the case ρ = 1. (Actually, Igusa also studied the case of principally
polarized abelian varieties with full level structure.) Our compactification is con-
structed using the "Legendre decomposition" of the cone of positive semi-definite
real 2x2 matrices. In chapter I.I we describe the moduli problem in question
and determine the corresponding groups. Chapter 1.2 reviews the theory of torus
embeddings. As an application we construct the Shioda and Kummer modular sur-
faces. These surfaces will play a very important role later on when we describe
the boundary surfaces of *4*(l,p). In chapter 1.3 we explain the construction of
the toroidal compactification A*(l,p) of ,4(1,p) in some detail. We first compute
the relevant Tits building, which enumerates the various boundary components and
contains important information about their intersection behavior. After this we give
the actual construction of A*(l,p) and identify the boundary surfaces as images
of Kummer modular surfaces. We also observe that A*(l,p) is projective and al-
most non-singular. We have precisely described the singularities of A*(l,p) in
[HKW]. Chapter 1.4 deals with the corank 2 boundary components, which are con-
figurations of rational curves in the boundary of A*(\,p). This chapter is mostly
of a combinatorial nature, but we also make a few remarks about the topology
of A*(\,p). In chapter 1.5 we describe the geometry of two important Humbert
surfaces and their closures in A*(\,p). These two Humbert surfaces parametrize
products of elliptic curves and bielliptic abelian surfaces respectively, i.e., exactly
those abelian surfaces where the polarization (in case p > 5) is not very ample.
Part II deals with degenerations of abelian surfaces. Ideally the compactification
of a moduli space is itself a moduli space which includes degenerations of the
objects considered originally. In general, however, toroidal compactifications of
moduli spaces of abelian varieties of dimension d > 2 do not represent known
functors. Nevertheless there is a meaningful way to associate to each boundary
point a degenerate abelian surface. The essential tool which we use here is Mum-
ford's construction of degenerating abelian varieties over local rings. We describe
his construction in chapter II. 1, where we follow Mumford's paper [Mu3] very
