Moduli Spaces of Abelian Surfaces: Compactification, Degenerations and Theta Functions [Reprint 2011 ed.] 3110138514, 9783110138511

Table of contents :
Introduction
I Compactified moduli spaces
1 Moduli spaces
2 Torus embeddings and applications
3 Toroidal compactification of A(1, p)
4 The boundary of A*(1, p)
5 Humbert surfaces and scaffoldings
6 The Satake compactification
II Degenerations of abelian surfaces
1 Mumford’s construction
2 The basic construction for surfaces
3 Degenerate abelian surfaces (the principally polarized case)
4 Degenerate abelian surfaces (the case of (1,p)-polarization)
5 Polarizations on degenerate abelian surfaces
III The Horrocks-Mumford map
1 The Horrocks-Mumford bundle
2 Construction of the Horrocks-Mumford map
3 Extension of the Horrocks-Mumford map to A(1, 5)
4 Extension of the Horrocks-Mumford map to A* (1, 5)
Bibliography
Glossary of Notations
Index

Citation preview

de 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 Constantin Kahn Institut für Mathematik Universität Hannover Weifengarten l D-30167 Hannover

Steven H. Weintraub Department of Mathematics Louisiana State University Baton Rouge, Louisiana 70803-4918 USA

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 2A Torus embeddings 2B Shioda and Kummer modular surfaces 2C The topology of Shioda and Kummer modular surfaces

3

Toroidal compactification of A( 1, p} 3A Boundary components 3B The Tits building 3C Toroidal compactification 3D Partial compactifications of .4(1, p)

. . . .

14 14 20 43 45 46 58 65 83

4 The boundary of A*(\,p) 4A Corank 2 boundary components 4B Transversality 4C The topology of A*(\,p)

125 125 147 151

5

155 155 169

Humbert surfaces and scaffoldings 5A The scaffolding 5B Geometry of the Humbert surfaces

6 The Satake compactification

184

II Degenerations of abelian surfaces

191

1

Mumford's construction 1A Outline of the construction IB Relatively complete models 1C Construction of G ID Properties of G

195 195 196 200 202

2

The basic construction for surfaces 2A The basic data 2B Computations

204 204 207

viii

Contents

3

Degenerate abelian surfaces (the principally polarized case) . . . . 3A Boundary points and Mumford's construction ........ 3B Description of singular surfaces .............. 3C Global aspects .....................

215 215 218 230

4

Degenerate abelian surfaces (the case of (!,/?) -polarization) 4A Boundary points .................... 4B Degenerate abelian surfaces ................

235 235 237

5

Polarizations on degenerate abelian surfaces ........... 5A Theta functions ..................... 5B Extending polarizations (p = 1) .............. 5C Miscellaneous remarks .................. 5D The general case (p > 1) .................

HI The Horrocks-Mumford map

.................

1 The Horrocks-Mumford bundle 1A Basic properties IB Horrocks-Mumford surfaces 1C Geometry in the space of sections 2

274 274 275 278

Construction of the Horrocks-Mumford map ........... 2A Heisenberg equivariant embeddings ............ 2B Odd theta null values .................. 2C Construction of the Horrocks-Mumford map .........

Extension of the Horrocks-Mumford map to A* (1, 5) 4 A Extension to the central boundary component 4B Extension to the peripheral boundary components 4C Extension to the corank 2 boundary components

Bibliography Glossary of Notations Index

. . . . .

......................... ...................... .......................

245 245 249 257 262 271

................ .................... ............... .............

3 Extension of the Horrocks-Mumford map to *4(1, 5) 3A Extension to H2 .................... 3B Extension to H{ .................... 4

. . . .

281 281 289 292

.......

298 298 300

....... ........ ....... .......

305 305 309 314 333 337 343

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 appropriate, on the choice of level structure. The resulting quotient space Γ\6 ί / is a 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 solution 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 Satake 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*d has codimension 1. The ideas of Igusa together with work by Hirzebruch on Hubert modular surfaces were the starting point for Mumford'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 decompositions 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 structure of canonical type (for a precise definition see 1.1.7). The toroidal compactification 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 constructed 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 surfaces. 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 almost 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 configurations 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 Mumford'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

Introduction

xi

closely. We then discuss the basic construction in case d = 2 in some detail. In chapter II.3 we associate a degenerate abelian surface to each boundary point of A*(\, 1), and then generalize this construction to the case of A*(l,p) in chapter II.4. In chapter II.5 we explain how degenerations of theta functions can be used to construct polarizations on the degenerate abelian surfaces. Here our treatment differs from that of Mumford who considers degenerations of twice the given polarization, since multiplying the polarization by 2 ensures a much better behavior with respect to degenerations. Since we want to work with the original polarization, we have to modify Mumford's family over the "deepest points" of A*(\,p) in order to obtain ample line bundles on the degenerate abelian surfaces associated to these points. We briefly discuss global aspects of Mumford's construction which can be used to construct universal families over suitable covers of ,A*(1, p). On the whole, however, the emphasis in part II lies more on the construction of degenerate abelian surfaces, which provide interesting examples. Another approach to the construction of degenerate abelian varieties was developed by Nakamura ([Nakl], [Nak2]) and Namikawa ([Naml], [Nam2], [Nam3]). Their construction is closely related to that of Mumford. In our case they both lead to the same degenerate abelian surfaces except over the deepest points, where one has to blow down Mumford's family in order to obtain that of Nakamura and Namikawa. Mumford's construction is also an essential tool in the construction of moduli spaces of abelian varieties over the integers, as was developed by Fallings and Chai ([FC]). They reverse the classical procedure and use degenerations of abelian varieties in order to construct compactifications of the moduli spaces. On the other hand they are primarily interested in the degeneration of abelian group schemes, i.e., they consider non-complete semi-abelian varieties over the boundary (these have very good functorial properties), whereas our emphasis lies on projective degenerations, i.e., we consider complete singular surfaces over the boundary. Part III is mostly devoted to the Horrocks-Mumford bundle F and its geometry. This bundle was first constructed by Horrocks and Mumford ([HM]) in 1972. It is an indecomposable rank 2 bundle on P4, and every other known indecomposable rank 2 bundle on P4 is derived from this one by easy constructions. The HorrocksMumford bundle gives rise to an amazing wealth of beautiful geometry which has been studied by several authors, see [BHM], [BM], [DS], [HL2], [Hul2], [HulSJ, [K1J, and the references quoted there. For a non-zero section s of F, let Xs denote the zero-set of s. The variety Xs is called a Horrocks-Mumford surface. Horrocks and Mumford already noted that in case Xs is smooth, it is an abelian surface equipped with a (1, 5)-polarization and a level structure of canonical type. This observation yields an isomorphism from a Zariski open set in the projective space ΡΓ of sections of F to a Zariski open set in the moduli space ,4(1, 5). We construct a birational morphism from ,4*(1,5) to ΡΓ which extends the inverse of this isomorphism. This enables us to understand the degenerations of the HorrocksMumford surfaces in the light of degenerations of abelian surfaces. This map also

xii

Introduction

provides an explanation for much of the interesting geometry in the space ΙΡΓ. The essential tool which we use in the construction of this map is degeneration of theta functions. We end this part with a discussion of complete linear systems on degenerate abelian surfaces. Although we restrict ourselves mostly to the case p = 5 where we can immediately compare abstract degenerations of abelian surfaces with projective degenerations of Horrocks-Mumford surfaces, most of our arguments remain valid for general p. Finally, we would like to mention two results which we shall not discuss in this book. The first author and Sankaran have shown that A*(I, p) is of general type for p large ([HSa]). The relation between A*([,p) and degenerations of mixed Hodge structures will be treated in a forthcoming paper by the first author and Spandaw ([HSp]). We are greatly indebted to various institutions who supported us while we were working on this book. We would like to thank the DFG for support under the Schwerpunktprogramm "Komplexe Mannigfaltigkeiten". This enabled the second and third author to each spend a year at Bayreuth University, and the third author to visit Hannover University. It was in Bayreuth that this project was begun. The first author would like to thank Louisiana State University for financial support during a visit in 1989. The first and third author are also grateful to the SFB "Geometrie und Analysis" at G ttingen University for several invitations. (Indeed, it was in G ttingen that they first met, in the fall of 1986.) The first author wants to thank Hannover University for granting a sabbatical. He is also greatly indebted to Trinity College Cambridge and the University of Cambridge for an invitation during this sabbatical. He profited greatly from the pleasant and inspiring atmosphere at Cambridge where he completed a major part of his share of the manuscript. The third author gratefully acknowledges partial support from the National Science Foundation and the Louisiana Educational Quality Support Fund at various times during this project, and an LSU sabbatical and an MSRI visit during which he completed his share. We would also like to thank our colleagues from the Erlangen Algebraic Geometry seminar who have read parts of the manuscript and who have pointed out a number of improvements to us. We are grateful to H.-J. Brasch and J. Spandaw for a number of corrections and suggestions. We would like to thank F. Lehmann who very competently converted our manuscript into a TpX file. We are also grateful to Dr. M. Karbe and de Gruyter publishers for their helpful cooperation. Last but not least we want to thank our families and friends for their patience and support.

Part I Compactified moduli spaces

1 Moduli spaces

In this chapter we describe the moduli spaces of abelian surfaces we are interested in, and the arithmetic groups from which they arise as quotients of the Siegel space of degree 2. For general background we refer the reader to the books of Igusa ([12, paragraphs II.3-II.5]), Mumford ([Mu2]), and Lange-Birkenhake ([LB]). Let Λ be a compact complex 2-dimensional torus, i.e., a quotient A = C2/L for some lattice L of rank 4 in C2. A torus A is called an abelian surface if it is a projective algebraic variety, i.e., if it can be holomorphically embedded in some projective space. The question of which complex tori (of any dimension) are projective varieties was first answered by Riemann. As we now phrase the answer, it is that A is an abelian surface if and only if C2 admits a positive definite Riemann form with respect to the lattice L, cf. [Mu2, p. 35]. A Riemann form on C2 with respect to L is a Hermitian form H > 0 on C2 with the property that its imaginary part is integer-valued on L. In this case H' = Im(//) : L L -»· Z is an alternating bilinear form whose R-linear extension to C2 satisfies H'(ix, z'y) = H'(x,y). Of course, H determines H' = Im(//), but the converse is true as well. One can recover H from H' by

(Here we denote a bilinear form and its R-linear extension by the same symbol.) Note that if a Hermitian form H > 0 and such an alternating form H' are related in this way, H is positive-definite if and only if H' is non-degenerate ([12, p. 65]). The positive definite Riemann form H, or equivalently the bilinear form H', is called a polarization of A. Since H' is a non -degenerate alternating bilinear form on L, it follows from the elementary divisor theorem that with respect to a suitable basis of L it is given by a matrix of the form

Here e\ and ei are uniquely determined positive integers such that e\ divides e^. In other words, with respect to this basis, //' is given by H'(x, y) = JcA'y, for

4

Part I

1 Moduli spaces

vectors x, y 6 L. (Note that when we identify L with Z4 we choose to think of row vectors with integral entries.) We fix the notation Λ and Ε throughout. We call Η (or //') a polarization of type (e\, e2) of A. If (e\,ei) = (1,1) we call Η a principal polarization of A. Definition 1.1. An (e\, e-^-polarized abelian surface is a pair (A, H) with A an abelian surface and Η a polarization of type (e\, 62) of A. If (e\, e^) = (1, 1), an (e\ , £2) -polarized abelian surface is called a principally polarized abelian surface. Remark 1.2. Assume we have chosen a basis of the lattice L. Let Ω e Μ (4 χ 2, C) be the matrix whose rows are the basis vectors of L, expressed in the standard basis of C2. Then Ω is called a period matrix of A. The condition that Η is Hermitian is equivalent to the condition that

and the condition that Η is positive-definite is equivalent to the condition that

These are the famous Riemann bilinear relations. (Here and henceforth we write Η > 0 resp. Η > 0 for the conditions that a matrix Η is positive definite resp. positive semi-definite.) Consider the Siegel space of degree 2, 62 - {τ € M(2x2, C) | r = V, Im(r) > 0}. Fix positive integers e\ and ^2 with e\ \ei. Then we can associate to every element τ € &2 the lattice L spanned by the rows of the normalized period matrix

the complex torus A — C 2 /L, and the Riemann form

With respect to the basis of L given by the rows of Ω, the bilinear form H' = Im(//) has matrix Λ, and so (A, H) is an (e\ , e-ι) -polarized abelian surface. Conversely, every (e\, e^) -polarized abelian surface arises in this way ([12, pp. 73-74]). To see this, note that two complex tori C 2 /L and C2/L' are isomorphic if and only if there is a linear automorphism of C2, i.e., an element of GL(2, C), taking L to L'. Thus, given a basis of L, we may use the action of GL(2, C) (this

1 Moduli spaces

5

action being right-multiplication of vectors by matrices, as we regard C2 as consisting of row vectors) to transform the last two vectors in the basis to (e\, 0) and (0, 62), respectively, to obtain a normalized period matrix Ω as above. It is then easy to check that Ω satisfies the Riemann bilinear relations, and hence A = C2/L is an abelian surface, if and only if τ € 62This correspondence between polarized abelian surfaces and points in 62 is well defined up to the choice of a basis of L, that is up to linear automorphisms of the lattice L which preserve the form Λ. These automorphisms form the symplectic group with respect to A, i.e.,

Sp(A, Z) = (g € GL(4, Z) | gA'g = Λ}. Note that the ordinary sympletic group Sp(4, Z) is defined with respect to the standard sympletic form

/

l

'-- ' \ -i

^

J

in place of A, hence Sp(4, Z) = Sp(7, Z). In analogy to the usual characterization of symplectic matrices with respect to J we note that with respect to A the following holds: Remark 1.3. A matrix (£ £) € GL(4, Z) where A, B, C, Z) are 2x2 matrices is an element of Sp(A, Z) if and only if A p 'r? — Uf>ft_L, '4 U — J\. ^

f~\t^,

( /"'/τ Γ) — Γ)ρ tf \_s jl_/ i_X —- t J l,, \_, _

anH dl HJ.

4 f '/") 1-J

£\L-i

1 PF1 V" — P \^s —— *-» τ

^^iJl—t

The action of the group Sp(A, Z) on the lattice L induces an action on 62 which is perhaps best understood as follows. The lattice L is the image of Z4 under the map ν ι-> νΩ, ν € Ζ4, Ω a period matrix of L, and g e Sp(A, Z) gives an automorphism of L by v M·· vg Μ» (ν#)Ω = ν(#Ω). Thus Sp(A, Z) acts on Λ/(4χ2, C) by matrix multiplication on the left, and we have seen that GL(2, C) acts by matrix multiplication on the right; clearly these actions commute. Consider the "right projective space Ρ of GL(2, C)" which is obtained from the set of all complex 4x2 matrices of rank 2 by dividing out the equivalence relation

Note that Ρ is isomorphic to the Grassmannian Gr(2, C4). We denote the resulting equivalence classes, i.e., the points of Λ by [$' ]. The group Sp(A, Z) acts on Ρ

6

Part I

1 Moduli spaces

by ordinary matrix multiplication from the left, i.e., writing an element of Sp(A, Z) as a matrix of 2x2-blocks,

A B\ [M,] _ [AM, +BM2 C

DJ ' [M2J ~~ [CM, -f DM2

If we embed ©2 into P by τ Η» [|], then the action of Sp(A, Z) restricts to an action on the image of 62- In particular,

A B\ C D

(Ατ + ΒΕ)(€τ + DE)~} Ε Ε

E

As an action of Sp(A, Z) on &2 this can be written as

Ά j~

I

.

(,

I

-"

\t L L· l

UM-IJ\^

L·

f

-^i-/^

C

*

^l.T·^

Note that in the principally polarized case, where E — \^, this becomes

(1.5) Here and throughout, 1* denotes the &x£ identity matrix. Where no misunderstanding is possible we will simply write 1 instead of 1*. For future reference we note that (1 .4) is equivalent to the following: If τ' = #(τ) for τ, t! e &2 and g = (£ £) e Sp(A, Z), there is a commutative diagram

E-l(Ci+DE)

We will be interested in polarized abelian surfaces carrying an additional structure. Denote by Lv the dual lattice of L with respect to the form Im(//), where H is a polarization of type (e\, 62), i.e., Lv = [y e I ®z K I Im(//)(x, y) e Z for all Λ; 6 L}. Then an easy computation shows that L V /L is isomorphic to (Z^ xZ e2 ) 2 . This latter space has a Q/Z-valued skew-symmetric form AQ which with respect to the canonical generators is given by the matrix 0 -£-'

E0

1 Moduli spaces

7

Definition 1.7. Let A = C2/L be an abelian surface and H a polarization of type (e\ , e-i) on A. A level structure of canonical type on (A, //) is an isomorphism a :L / > (

which takes Im(//), regarded as a form on Lv C L z E, to the form AQ. These level structures are called level structures of canonical type since their definition depends only on the type of the polarization, which is already given. Since we are not considering other kinds of level structures in this book we will refer to level structures of canonical type simply as level structures. Just as before there is a correspondence between polarized abelian surfaces with level structures of canonical type, i.e., triples (A, //, a), and points of the Siegel space &2 defined via period matrices, but now the indeterminacy in the choice of a basis of L is expressed by the subgroup of Sp(A, Z) consisting of automorphisms which induce the identity on L V /L. Each of these two correspondences (without and with level structure) between polarized abelian surfaces and points in &2 is bijective modulo the action of the appropriate group on 62, so that the respective quotients of &2 will be moduli spaces for these objects. To be precise, let us identify L = Z4,

hence

Lv = — Ζ Θ — Ζ Θ — Ζ Θ — Ζ e\

62

e\

62

(1.8)

with respect to A, and L V /L is isomorphic to (Z f , xZ e ,) 2 . Furthermore, the automorphisms in Sp(A, Z) which induce the identity on L V /L are just those g e Sp(A, Z) with vg = v mod L for all v e L v . Then, in the notation we are about to introduce in definition 1.9, our above discussion is summarized by theorem 1.10. Definition 1.9. (i) For p an odd prime and A as above let

rei,e, - {g 6 f ° £,, | vg = v mod L for all v β 2 = Γ β ,, β2 = Sp 3, then —1 ^f e|j£ , 2 , and this group acts effectively. Remark 1.13. We have denned a polarization as a positive definite Riemann form, and from this interpretation have arrived at our moduli spaces. This approach leads most directly to our results in part I of this book. However, for our work in parts II and III we will need an alternate definition of a polarization, namely as the class of an ample line bundle. We shall describe here how these notions correspond. We follow [Mu2, chap. I], but see also [12, chap. II]. Any line bundle £ on A = C 2 /L may be constructed as follows: Let u be a factor of automorphy on L, i.e., a function u: LxC 2 —> C* with the property that H(/I + / 2 , z ) = w ( / i , z + /2)«(/2,z)

for all / i , / 2 e L, z € C2.

Then L acts on C 2 xC by

l(z, w) = (l + z, u(l, z)w)

for / € L, z e C2, w e C,

covering its action on C2 by translation, and so the quotient (C 2 xC)/L is a line bundle £ on A = C 2 /L. Given a positive definite Riemann form H, let δ: L —» R be a function satisfying (—ζ, τ) which is a reflection. Then D°(Hi) is smooth locally around a generic fiber by Chevalley's theorem (see [Sp, chapter 4.2]).

2B Shioda and Kummer modular surfaces

23

Finally, suppose that the stabilizer of a point τ € &\ in Η contains elements other than ±1. The above analysis essentially remains unchanged except for two differences: First, the fiber over such a point becomes a multiple fiber (so that π ceases to be a fiber bundle in the topological sense). Second, the elements of finite order in H L may give rise to isolated quotient singularities where they have fixed points in the fiber over τ. D Notation 2.17. In the case —l g H the surface Z)°(///,) is called an (open) elliptic modular surface and is denoted by S°(Hi). If — 1 e H, then Dc(Hi) is called an (open) Kummer modular surface and is denoted by K°(Hi). The surfaces D°(k) for k > 1 are known as the (open) Shioda modular surfaces of level k. For k > 2, they are elliptic modular surfaces denoted by 5°(/c); for k = 1 or k = 2 they are of Kummer type and are denoted by K°(k). Finally, for k > 2 we denote the surface D°(HL) with Η = ±Γι(&) and L = {(m, n) 6 Q 2 1 m, n e kl] by K°(k) since it is of Kummer type. Remark 2.18. From proposition 2.16 we see that the surfaces Kc(k), k > 2, and S°(k), k > 2, are non-singular. The surface K°(\) is known to be singular ([Be]). Remark 2.19. If H c H and L c L such that L is invariant under H and L is invariant under H then there is a natural commutative diagram of fiber spaces in which the horizontal arrows represent ramified covers: D°(HL)

l X°(H) If either — 1 e H and — 1 € //, o r — l g H and —l g H, then this cover has degree [H : H] on the base and degree [L : L] on the fibers. If —1 g H but — 1 e H, then the degree on the base is \[H : H] and the degree on the fibers is 2[L : L]. In particular, for k > 2 the Shioda modular surface S°(k) is a ramified double cover of the Kummer modular surface K°(k) defined in 2.17 above.

Our next aim is to compactify the spaces D°(Ht) to spaces D(HL) in such a way that the fiber maps π: D°(HL) -» X°(H) extend to maps π: D(HL} -* X(H) where X(H) is the compactification of the modular curve X°(H). We will see, however, that the fibers of π over the cusps of X(H) will in general be singular fibers, i.e., π will fail to be a fiber bundle map there. The case where D°(HL) is an elliptic modular surface was extensively studied by Kodaira. The compactification of the surfaces 5°(k), k > 2, was studied in

24

Part I

2 Torus embeddings and applications

detail by Shioda ([Shio]). The other case, i.e., of Kummer modular surfaces, has not attracted an equal interest, although see [Bu]. The way we construct the compactification of £>° (///.) is by means of suitable torus embeddings. This construction will serve as a paradigm for many of the steps necessary to build toroidal compactifications of moduli spaces of abelian surfaces which is the content of the following chapter. We start by compactifying the modular curve X°(H) = H\ 0.

(b) If-l g H, then either „

(b,)

;

Ι

n € Z \ , for some bo € Q, bo > 0, or

ι \ η

Κ

~ _° J

Λ

n € Z L /or some fc0 e Q, ^o > 0.

As far as the compactification of X°(H) is concerned the cases (a) and (bj) are equivalent because —1 acts trivially on Q\. In these cases oo is called a regular cusp, while in the case (b2) it is called an irregular cusp ([Shim, p. 29]). We assume henceforth that H gives only rise to regular cusps. Then P(oo) = < ε

i 0 1

η e Ζ, ε ε S

where 5 = { ± l } i f — l e / ί , and S = {1} otherwise. The group P(oo) leaves a neighborhood

yv(oo) = {re6!| Im(r) > r0} of oo invariant. (More precisely, /V(oc) is the intersection of a neighborhood of oo in ΡΊ with ΘΙ.) For r0 large we have that h(N(oo)) Π N(oo) = 0 for any h e H, h $ P(oo). It then follows that P(oo)\N(oc) embeds as an open subset into H\&i= X°(H).

2B Shioda and Kummer modular surfaces

25

On the other hand, the exponential map gives rise to an embedding e: P(oo)\N(oo) ^ C*,

τ Η* «?27Γίτ/*°

with image D' = D Π C* where D = {ζ \ \z\ < e~2mn/h°}. Then by using these two embeddings we may form the identification space Λ /λ

This space contains one point in addition to X ° ( H ) , namely the image of 0 e D which is then called the cusp "oo". The resulting space is smooth at oo since D is smooth at 0. Now in general there will be more than one //-equivalence class of points in QU {oo}, each class yielding a different cusp by a procedure analogous to the one above. However, it is not necessary to describe all these procedures separately, as any rational boundary point of 61 can be mapped to oo by an element of SL(2, Z). Note that the rational number bo need not be the same for all cusps, although it will be if H is normal in SL(2, Z). Finally, although this case goes back to the nineteenth century, and it is a very simple example, we remark that this construction of X(H) is indeed a toroidal compactification of X°(H) in the sense of [AMRTJ. We are now going to examine the toroidal compactification £>(///,) of D°(///,) which is less trivial. This example was worked out in [AMRT, §1.4] (see also [LW1, 2.5.9]) but here we will be considerably more explicit. We begin by constructing a partial compactification of D°(HL) by "adding a fiber" over the cusp oo of X(H). The group HL acts on C x © i as noted in (2.14). Recall that we have restricted ourselves to the case where H gives rise to regular cusps only. In order to simplify the discussion let us now assume that the //-invariant lattice L is of the form

L = {(m, n) € Q2 l m e moZ, n e «oZ},

mo, «ο ε Q> WQ, no > 0.

We will call these lattices, which have fundamental domains that are rectangles with sides parallel to the coordinate axes, rectangular lattices. Note that this additional assumption causes no loss of generality since there always exists a g in SL(2, Q) fixing oo such that L = Lg"1 is rectangular. Then H-L = HL, where H = gHg~}, and the compactification problem for D°(HL) is equivalent to that of D°(HL). (We will see in lemma 2.28 below that the compactification is well defined, i.e., independent of the choice of g.) The action (2.14) of HL on Cx6i extends to an action of HL on Cx i C i. The stabilizer P = P(Cx{oo}) of the fiber over oo € 61 is then

Ρ= L

PH(oo)

26

Part I

2 Torus embeddings and applications

where /*# (oo) is the stabilizer of oo in H, PH(oo) =

ε b Ο ε

b€

as in lemma 2.20. Either S = {±1} or 5 = {1} depending on whether H contains —1 or not. We now choose a sufficiently small neighborhood N of Cχ {oo} in Cx ΘΙ which is invariant under P, e.g., N = {(ζ, τ) € Cx©i | I m r > f 0 },

f 0 » 0.

We choose to large enough such that P\N embeds into Hi\(Cx&\). This is the case if g(N) ΓΊ Ν φ 0 for g € HL implies g € P. Just as before we will add to P\N a compact "fiber at infinity" and then form the identification space with Hi\(Cx&\). This gives us a partial compactification of D°(Hi) over the cusp oo. Doing this for all cusps we obtain a compact space D(HL) which contains D°(Hi) as an open and dense subset. Since P//(oo) can be considered as a subgroup of P it is tempting to simply add a fiber Cx{0} to the space / > #(oo)\(Cx6i) = Cx£>' where D' is a punctured disk, and then try to obtain the partial compactification over oo as a quotient of this space. But since P//(oo) is not a normal subgroup of P this simplistic approach must fail. However, we shall see that a modest modification of this idea leads to success. Consider the following subgroup of P, P' =

n e /ioZ, b e bQZ } ,

(2.21)

which is indeed normal in P. The quotient P" = P/P' can be identified with P" =

m e m0Z, ε e 5 > .

(2.22)

The group P' then acts on the neighborhood Ν of Cx{oo} and the corresponding "partial" quotient map can conveniently be expressed using exponential maps, e:N-> (C*)2,

e(z, τ) =

Hence, P'\N = e(N) = C*xD' where D' is the punctured disc of radius e~27I'0/b° centered around 0. The induced action of P" on e(N) is easily seen to extend to an action of P" on the torus Τ = (C*)2 containing e(N). Let r0 = mob0/n0. Note that the fact that P is a group implies that TO is an integer.

2B Shioda and Kummer modular surfaces

27

then the images in P" of either h and t or of h alone generate P". The induced action of h on T is defined by the requirement that the diagram

N —h-+ N

τ —

τ

commutes, and similarly for i. Let (w, v) = e(z, τ). Then e(h(z, τ)) = e(z + m0r, τ)

= (uvr°, ν).

This together with an analogous computation for ι gives the following actions on the torus 7: h:T-^T,

(M, v) (-» (wv r °, v)

t: Γ -» Γ, (w, ν) Μ- (tT1, ν)

We will now define a torus embedding Γ C T^ using a fan Σ which satisfies the condition of proposition 2.8 with respect to the group P" acting on T. This will enable us to form P"\T^ and then also to form the identification space

in the analytic category where ΧΣ denotes the interior of the closure of e(N) in ΓΣ. In our case this is just X^ — e(N) U (7^ \ Γ). This identification space will serve as the partial compactification of D°(Hi) over the cusp oo of X(H). We continue to denote by u and v the coordinates on T = (C*)2. Moreover, u and v can be considered as characters on the torus Γ, and as such they generate the lattice M of characters. Denote these generators of M by U and V, and let U* and V* be the corresponding dual basis of the lattice M* of 1-parameter subgroups. This choice of generators results in an explicit identification of the lattices M and M* with Z2 in such a way that U = (1, 0) and V = (0, 1), and also U* = (1, 0) and V* = (0, 1).

Part I

28

2 Torus embeddings and applications

According to the remarks preceding proposition 2.8 we observe that the actions of the generators h and ι of P" on Τ are characterized by the matrices of exponents A(h) = ( J l ° ) and A(i) = (~ 0 ' ?)· Then' the induced actions on Μ = Z2 and Μ* = Z2 are given by

h: M -»· M, Λ:

M*,

m i->'A( )~ J m = m*h+A(/z)m*= ( ^

m

-r0 ^

(2.24) l

~*

and similarly for i,

i:M-+ M,

m h+ *Α(ιΓι m (2.25)

i : M* -* M*,

m* h^ A(t) m* -

nf

Let Σ be the infinite fan in M^ = M2 which consists of the 2-dimensional cones crk = R> 0 (k, 1) + E>0 (k+\, 1), for all Λ 6 Z, together with their 1-dimensional faces £* = R>o (&, 1), k e Z, and also the vertex {(0, 0)} at the origin, see figure 2.1. In other words, £* is the closed ray extending from the origin going through (k, 1), and ak is the closed region bounded by t-k and

Fig. 2.1 A polyhedral decomposition of the upper half-plane

The torus embedding ΓΣ defined by Σ is the union of its affine subvarieties Tak, k € Z, corresponding to the cones of maximal dimension. From remark 2.6(iii) we 2 2 know that Τσ* σ. = C and TV f* = CxC* for every k e Z, and Γ (0} = T = (C*) . while if \k - l\ > 1, then : Π Τσ, = T m = Τ. Moreover, Γσί Π Γσ^ — Tt ' ?*+!' Finally we note that TV, Π Γξ; = Γ{0| = Tifk^l. In order to proceed further we need to introduce coordinates on the various affine torus embeddings. If we let U*k = (k + 1, 1) and V*k = (k, 1), then we have ok = R>QU*k + R>oV£ and %k = R>oU*k. Next we observe that the dual basis to the basis [Uk, V*k] of M^ is the basis [Uk, V k } of MR given by Uk = (1, —Λ) and

2B Shioda and Kummer modular surfaces

29

Vk = (— 1, k+ 1), and hence σ^ = R>o U k + IR>o Vk as a cone in MR. Since {/A and Vjt give rise to generators of the regular ring C[a^ Π Μ], there are corresponding coordinate functions w* and v* on T„k = SpecQa^ Π Λ/] = C2. Moreover, the natural inclusion Τ = Γ{0} °^· Γσ(. is with respect to the respective coordinates given by 7,0} ^ Tak, («, V) H» (W*, V*) = (MV-*, iT 1 /* 1 )

which follows from the unique expression of U k and Vk in terms of i/ and V in the additive group M, namely i/* - U - kV and Vfc = -U + (k + 1)V. Similarly, ^+1 = E> 0 i/* + R V * = R> 0 i/t + M> 0 V* + K> 0 (-V*), and hence 7^+1 = SpecC[^+] Π Μ] = CxC* is again naturally coordinatized by uk and v*. Using these coordinates the inclusion TV, T0k,

(uk, vk) M- (uk, vk)

is trivial while the second inclusion

is determined by the unique expressions Uk+\ = —Vk and V^+i — U k + 2V* of ί/fc+i and Vk+\ in terms of U k and V* in the additive group M. Now note that all sets T^k are contained in the sets T„k. It therefore suffices to consider the identifications imposed among the Γσ/8 by the gluing process which leads to Γχ. When gluing together T„t and Τσΐί+ι along 7^_, we have to identify the open subset of Ta>. = C2 determined by ν* φ Ο and the open subset determined by Uk+\ φ 0 in TOM = C2 along the bijective map («*, v^) H^· (v^" 1 , ukv\) between these subsets. Denote by C°k — Cx{0} and D°k+l = {0}xC the two axes which make up the complement of Τ — (C*)2 in T„k. Then the gluing of T„k and Tfft^ along r tt+1 identifies D°k+l ^ {(0,0)} c T„k with C°k+l \ {(0,0)} C Tn+l. The net effect is that to the affine line C°k+l in T„k+i there is added a point at infinity corresponding to (0,0) e D°k+l in T0k, resulting in a projective line C*+1 = PI in Γχ. As there is one such line for every k € Z and as successive lines intersect transversely we find that the complement of T = (C*)2 in ΓΣ is an infinite string C = Ui:ez C* of rational curves, i.e.,

It is now easy to see that the curves C*, k e Z, all have self-intersection numbers C , = -2 in 7Σ. Namely, Γσ|_, = C2 can be identified with the restriction of the normal bundle Nck/r^ of Ck in ΓΣ to C°k. The holomoφhic section {Μ^_Ι = 1} in ci tnen extends to a meromorphic section in Λ/"Ο*/ΓΣ which has a pole of 2

30

Part I

2 Torus embeddings and applications

order 2. Hence, C\ = degA/cyrr = —2. (This argument is very similar to that of Hirzebruch, see [Hi, p. 17]. For a different argument that makes better use of the conceptual framework of torus embeddings see [Od, proposition 1.19 and p. 80].) We have noted in (2.24) and (2.25) how P" acts on M* (and hence on M^), and we see from there that this group action satisfies the condition of proposition 2.8 for the fan Σ. Hence the action of P" on T extends in a unique way to an algebraic action of P" on Γχ. We now wish to determine the quotient space P"\Tz. As we already know that P"\e(N) = P\N embeds into D°(HL), it remains to describe the action of P" locally around and in particular on the infinite string C = ΓΣ \ T of rational curves. We first consider the action of the element h e P". From (2.24) we see h(ak) = &k+rQ for every k e Z. By the proof of proposition 2.8 the map h: ΤΣ —>· T^ can be obtained by gluing together the various restrictions hk = h\Ta : Tak —>· T0k+r . We compute these maps in the respective coordinate systems that have been introduced above. A point χ e Τσ/ί = C2 with coordinates (uk, Vk) = (s, t) corresponds to an algebra homomorphism φ^ι C[ak Π Μ] —> C such that (flk(Uk) = s and (pk(Vk) — t. According to the proof of proposition 2.8 we need to take the composition of h ~ ] : C[a^+r C[a^ Π Μ] with φχ in order to find φ^Χ) which determines h(x) € Tak+r . By (2.24) the generators Uk+r0 and Vk+r0 of C[a^+ro Π Μ] are mapped to Uk and V*, respectively. Hence, ψΐι(χ) '· C[cr^_|_r Π Μ] —> C is determined by (f>h(x)(Uk+ro) — s an Τσ_^}, (uk, vk} = (s, Ο Η» («_*-i, v_jt_i) = (r, 5). This has the effect that for every k e Z the curve, Cyt C ΓΣ is mapped bijectively onto C_£. In particular CQ is mapped onto itself and this mapping corresponds to z H» z~' on PI where Ο, σο e PI correspond to the points of intersection of CQ with C\ and C_i, respectively. Hence, in C C PO \^Σ the curve Co is mapped onto itself with two fixed points, and the curves C\,... , C[( ro _i)/2] and C _ i , . . . , C_[( r o _i)/2j are pairwise identified. If the number TO of sides of C is odd then there are no other curves in C but we find that ί has another fixed point where C (/ - 0 _D/2 and C-(ro-\) = C( ro+ D/2 intersect. If /"o is even then the curve C rfl /2 is also mapped onto itself by ί and a simple computation shows that this situation is analogous to the map induced by ι on CQ. Thus in either case the action of t on C corresponds to a reflection in the vertical axis in the above figure of C, and the resulting quotient space C = (t)\C = P"\C is a string consisting of [r0/2] + 1 smooth rational curves intersecting transversely, i.e.,

(2.27)

Of course the case r0 = 1 deserves special attention again. Here ί acts on

32

Part I

2 Torus embeddings and applications

Co = PI in ΓΣ as before by z H> z ~ { , and all that is different from the other cases is that the points 0 and oo have been identified in C beforehand under the action of PQ = (A). Thus C turns out to be a single smooth rational curve, which is consistent with our above analysis of P"\C. We finally need to work out the self-intersection numbers of the irreducible components Cj of C. We first note that locally around its fixed points on C the map L is equivalent to a reflection. Hence by Chevalley's theorem P"\T^ is a smooth space locally around C. Then it is also clear that the curves C i , . . . ,C[ ro / 2 j-i not at one of the ends of C will have the same self-intersection numbers as C i , . . . ,C [ro/2] -i on Ρ'0\ΤΣ, that is Cj = -2 for 1 < i < [r0/2] - 1. For the remaining curves a simple topological argument shows that C\ — ^C\ — —1 and either Cf ro/2 , = |cfr()/21 = -1 if r0 is even or Cf ro/2) = i(C ko/2) + C lro/2 , +1 ) 2 = -1 if r0 is odd. Hence in any case the curves at the ends have self-intersection numbers — 1. Finally, the case where r o = 1 constitutes an exception since here CQ = 0 because it is a fiber. This, however, is only a special case of C2 = 0 which is always true. This concludes our description of the partial compactification of D°(Hi) over the cusp oo of the modular curve X(H), as we can now form the identification space Κ Σ (οο)=/>"\Χ Σ U/>w D°(HL) where ΧΣ = ( e ( N ) ) ° is the open subset of ΓΣ obtained by taking the interior of the closure of e(N) in ΓΙ. A full compactification £>(//£,) of D°(HL) is then obtained by performing this procedure for all of the finitely many cusps of X(H). If XQ is an arbitrary cusp in X(H), then there exists a transformation go in SL(2, Z) which maps JCQ to the cusp oo. It then suffices to consider the partial compactification of D°(H'L,} over the cusp oo of X (//'), where H' = goHgo1 and L' = Lg^. Although L is a rectangular lattice, L' need not enjoy this property anymore. However, using a suitable rational transformation go e SL(2, Q) we may assume that L' is rectangular, and then proceed as before. It remains to be shown that the partial compactification of D°(Hi) over XQ obtained in this way does not depend on the choice of go, which is not a priori clear. Suppose that g'0 and g'o are two elements of SL(2, Q) such that g'Q(xQ) = 00 = g'o(xo) and such that L' = Lg'0~* and L" = Lg'u~l are rectangular lattices invariant under H' = g'QHg'^ resp. H" = g'^Hg^1. Then (//", L") = (gH'g~l, L'g~l) where g — gQg'Q ', and g(oo) = oo. The following slightly more general lemma implies that the two partial compactifications of D° (///.) over XQ obtained using g'0 and go' are naturally isomorphic. Lemma 2.28. Let (H, L) and (H, L) be two pairs consisting of an arithmetic subgroup of SL(2, Q) and a rectangular lattice invariant under the group, and suppose that (H,L) = (gHg~l,kLg~l) for some g € GL(2, Q), det(g) > 0,

2B Shioda and Kummer modular surfaces

33

g(oo) = oo, and k e Q, k Φ 0. Then g = ( ^ β & ) , αδ > 0, and the map ψ: C χ Θ, -* C χ 6,,

^r(z, τ) = ( k S ~ } z , g(r)),

induces an analytic isomorphism if. D°(HL) -> D'i/^) which extends in a natural and unique way to an analytic isomorphism ψ: ΧΣ(οο) -> F£(oo) where Y^(oo) resp. Fg(oo) denote the partial compactifications of the surfaces D°(HL) resp. D°(HL) over the cusp oo of X(H) resp. of X(H). Proof. As g fixes oo it is of the form g = ( £ ^ ) . Let

Then //£ = MH^M

l

, and ι/^(ζ, τ) = Λ/ ? , where we regard points (z, r) in

C x © i as points in I?2 with homogeneous coordinates [ζ : τ : 1]. Let Φ: //^ —> W^ be given by Φ (Α) = ΜΛΜ"1. Then it is straightforward to check that there is a commutative diagram (φ.«/ο HLx(Cx 6]) —U HLx(Cx 6,) I

I

4-

-I-

Cx6,

^^

(2.29)

Cx6i

which yields an analytic isomorphism ψ: D°(HL) —>· D°(HL) between the quotients. Now we must show that ψ extends to the partial compactifications over oo. We let Ρ resp. Ρ denote the stabilizer of the fiber over oo in HL resp. in HL. Then Ρ is given by

P= and similarly for from /l M O VO

m € m Γ Σ . Since ψ·°: Τ -+ Τ identifies the fans Σ and Σ in the sense of proposition 2.9, it follows from this proposition that ·φ°{ can be extended to an isomorphism \jjr\: ΓΣ —> Τ%. Furthermore, as ty\ is given by multiplication with an element of T, it is a consequence of remark 2.6(i) that ^ extends to an isomorphism ^ : ^£ ~^ ^£· Then ψ = fa ο ψ ι: ΤΣ -> T£ is an analytic isomorphism which extends ψ°. It remains to check that the actions of P" on ΤΣ and of P" on Γ£ agree under the identification by \jr. For this it is sufficient to check agreement on the tori T

2B Shioda and Kummer modular surfaces

35

and T. That is, we need to verify commutativity of the diagram Ρ" χ Τ

{Φ Ψ

' \ Ρ" χ Τ

where Φ is induced by Φ, and ψ\τ = ψ° by construction. It is straightforward to check this for generators of P" and P" since Φ(/ζ) = h and Φ (t) = l. In particular, by (2.23) (M, ν)) = Λ(Μ, μν) = (Μ(μν/°, μν)

and

, ν)) = ^°(wv r °, ν) = (wv r °, μν), and these are equal as ?Q = r o and μ'° = 1. Agreement for the other generator is trivial. This completes the proof since uniqueness of the extension of ψ to ψ follows from the fact that D°(HL) is dense in ΥΣ(οο). D Λ

In general the integers r o which determine the singular fibers will be different for different cusps. If, however, H is a normal subgroup of SL(2, Z), which in particular is true for the principal congruence subgroups Π(&), then the group SL(2, Z) permutes the cusps transitively, and hence all singular fibers in D(HL) will be isomorphic. The cases which are the most important for us are those of the Shioda and Kummer modular surfaces derived from the principal congruence subgroups Y\ (k) in SL(2, Z). Before we summarize our results about their compactifications we make an additional definition by which we acknowledge the close relationship between the two configurations of rational curves drawn in (2.26) and (2.27). Definition 2.30. A finite string of [/Ό/2] + 1 smooth rational curves arising as the quotient of a ro-gon as in (2.27) is called Ά folded r$-gon. The following theorem gives a summary of the results about the toroidal compactifications of D°(Hi) obtained in this chapter for the cases in which we will need them later. Theorem 2.31. (i) For k > 2 the Shioda modular surface S°(k) admits a compactification to a smooth surface S(k) fibered over the modular curve X(k) such that its singular fibers over each of the cusps ofX(k) are k-gons of smooth rational (—2)-curves.

36

Part I

2 Torus embeddings and applications

(ii) For k > 1 the Kummer modular surface K°(k) can be compactified to a surface K(k) fibered over X(k) whose singular fibers over the cusps of X(k) are folded k-gons, i.e., strings of [k/2] + 1 smooth rational curves intersecting transversely. If k = 1 then there is only one such fiber which is a single smooth rational curve with self-intersection number 0; if k > \ then the two curves at the ends of the folded polygon are (-l)-curves while all of its other curves are (—2)-curves. Remark 2.32. (i) The surfaces S(k) and K(k) of theorem 2.31 are called (compact) Shioda resp. Kummer modular surfaces of level k. (ii) In the case of the surface K(\) the term singular fiber for its fiber over the only cusp of X(l) is actually a misnomer as this fiber is in fact a smooth rational curve. The following geometric observation will be useful later for comparison with the results of chapter 5. For now it may help to illustrate some aspects of the compactification £>(///,). For brevity we shall only state the results for the Shioda and Kummer modular surfaces S(k) and K(k). The generalization to arbitrary surfaces D(HL) is straightforward. Consider for any pair of integers i\ , ii € {0, 1 } the subset

of Cx6i. We denote the image of