Table Of ContentAbelian Varieties
Abelian Varieties
Proceedings of the International Conference
held in Egloffstein, Germany, October 3-8, 1993
Edited by
Wolf Barth
Klaus Hulek
Herbert Lange
w
DE
CL
Walter de Gruyter · Berlin · New York 1995
Editors
Wolf Barth Klaus Hulek
Herbert Lange Institut für Mathematik
Mathematisches Institut Universität Hannover
Universität Erlangen-Ni Weifengarten 1
Bismarckstr. la D-30167 Hannover
D-91054 Erlangen Germany
Germany
1991 Mathematics Subject Classification: 14-06; 14Kxx, 14Dxx, 14Cxx, 14Gxx,
14Hxx, llFxx, llGxx
Keywords: Abelian varieties, moduli of abelian varieties, linear systems,
modular forms, Prym varieties, Hamiltonian systems
© 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
Abelian varieties: proceedings of the international conference,
held in Egloffstein (Germany), October 3-8. 1993 / edited
by Wolf Barth, Klaus Hulek, Herbert Lange,
p. cm.
ISBN 3-11-014411-5 (acid-free)
1. Abelian varieties - Congresses. I.Barth, Wolf.
II. Hulek, Klaus. III. Lange, Η. (Herbert), 1943-
QA564.A214 1995
516.3'5—dc20 94-47103
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Abelian varieties: proceedings of the international conference in
Egloffstein (Germany), October 3-8, 1993 / ed. by Wolf
Barth ... - Berlin ; New York : de Gruyter, 1995
ISBN 3-11-014411-5
NE: Barth, Wolf [Hrsg.]
© Copyright 1995 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. Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin.
Cover design: Thomas Bonnie, Hamburg.
Preface
Abelian Varieties are in many ways a unique class of compact complex manifolds:
- Abelian Varieties of dimension one (elliptic curves) were of major historical
importance in the development of the theory of complex-analytic functions.
This became a mathematical theory in its own right only after the inversion of
the elliptic integral.
- The modern development of Algebraic Geometry began when Riemann intro-
duced topological and transcendental analytic techniques. The abelian integrals
used by him and his followers to understand algebraic curves have nowadays
been translated into geometrical language as the canonical map of a curve to
its Jacobian.
- Periods and moduli of abelian varieties have always been of interest in the de-
velopment of number theory. In the creation of "arithmetical geometry" during
the last two decades abelian varieties were one of the essential tools.
These three links between abelian varieties and other fields are, historically
speaking, the most important. There are others, some still developing at the mo-
ment, such as the use of abelian varieties in algebraically completely integrable
Hamiltonian systems.
Thus it is not surprising that the mathematical treatment of abelian varieties
has concentrated on many diverging aspects, which often differ widely in their
methods, aims and even language.
The Egloffstein conference was organized with the intention of bringing together
mathematicians working on different aspects of the field.
As a result the talks given at the conference, as well as the contributions col-
lected in this volume cover various aspects of present-day developments. It is the
hope of the organizers that the publication will be useful for all those wishing to
study ideas or problems in this large and expanding area of research.
We are indebted to the Deutsche Forschungsgemeinschaft for supporting the
conference in the framework of its "Forschungsschwerpunkt Komplexe Mannig-
faltigkeiten" and to the de Gruyter Verlag for publishing these proceedings. We
are grateful to S. Schmiedl and R. Wetke for help with the preparation of TßX-files.
Table of contents
W. Barth
Quadratic equations for level-3 abelian surfaces 1
Th. Bauer and T. Szemberg
Abelian threefolds in (P )3 19
2
H.-J. Brasch
Branch points in moduli spaces of certain abelian surfaces 25
F. Catanese
Compact complex manifolds bimeromorphic to tori 55
V. Gritsenko and K. Hulek
V. Gritsenko
Irrationality of the moduli spaces of polarized abelian surfaces 63
Appendix to the paper "Irrationality of the moduli spaces of polarized
abelian surfaces" 83
L. Haine and E. Horozov
Tau-functions and modules over the Virasoro algebra 85
C. F. Hermann
Some modular varieties related to P4 105
C. F. Hermann
New relations between the Fourier coefficients of modular forms of
Nebentypus with applications to quaternary quadratic forms 131
K. Hulek and G. K. Sankaran
The fundamental group of some Siegel modular threefolds 141
V. Kanev
Spectral curves and Prym-Tjurin varieties I 151
J. Kramer
On a generalization of the Neron-Tate height on abelian varieties 199
viii
D. W. Masser and G. Wüstholz
Refinements of the Tate conjecture for abelian varieties 211
F. Oort
Complete subvarieties of moduli spaces 225
G. P. Pirola
Abel-Jacobi invariant and curves on generic abelian varieties 237
B. Runge
The Schottky ideal 251
Sir Peter Swinnerton-Dyer
Rational points on certain intersections of two quadrics 273
Y. Tai
On the Kodaira Dimension of moduli spaces of abelian varieties with
nFo. n-Tporvienncaip al polarizations 293
Abelian surfaces with a polarization of type (1,4) 303
S. H. Weintraub
Symmetries of a moduli space of abelian surfaces 323
List of contributors 343
Quadratic equations
for level-3 abelian surfaces
W. Barth
0. Introduction
Let A be an abelian surface with principal polarization Οα(Θ)· It is well-known
that the line bundle Ö (3Θ) defines an embedding of A into Ρβ· A theorem
A
of G. Kempf [K] states that the ideal of A is generated by quadrics and cubics
[BL, chap. 7, thm. (4.1) b]. If A — E\ χ £2 is a product of elliptic curves
with product polarizations, then the embedding E\ χ i?2 > Pe is the product
embedding Εχ χ E2 '—t P2 x P2, with Ei C P2 a cubic curve, followed by the Segre
embedding P2 x P2 c—• Ρβ· So each quadric vanishing on the image of A in Pg
obviously vanishes on the Segre fourfold P2 x P2 C Ρβ· Hence in this case cubic
equations are definitely needed to define A.
The quadrics vanishing on A are well-known classically [C, AR], It is usually
assumed (cf. e.g. [vG]) that they define the surface if A is general. Coble even
claims that A is the complete intersection of these quadrics [C, p. 357]. His ar-
gument seems to be that the parameters in these equations in some sense, indeed
determine the surface. I cannot recognize there, however, a proof for the complete
intersection property. Probably Coble did not use the notion 'Complete Intersec-
tion' in the way we do it nowadays.
In fact, the quadrics certainly do not generate the homogeneous ideal of the
abelian surface A C P4: This follows comparing
Λ°(Γα(3)) > /i°(Öp (3)) - h°(ö (99)) = 165 - 81 - 84
8 A
with the rank of the multiplication map
H0(O (l)) ® H°(I (2)) -> H°(I (3)) .
Pa A A
dim = 9 dim = 9 dim > 84
Now the aim of this note is to prove the
Theorem. If A C Pe is as before, not a product of elliptic curves with product
polarizations, then the quadrics vanishing on A generate the ideal-sheaf ΙΑ/Ψ ·
6
I find this fact interesting, because general theory provides quadratic equations
