Lecture Notes ni Mathematics Edited by .A Dold dna .B Eckmann 812 IIIIIII ! okihikuY Namikawa ladioroT citacifitcapmoC )n fo Siegel Spaces I galreV-regnirpS Berlin Heidelberg New kroY 1980 Author Yukihiko Namikawa Department of Mathematics, Nagoya University Furocho, Chikusa-Ku Nagoya, 464/Japan AMS Subject Classifications (1980): 14 ,? 1L 20 G 20, 32 J 05, 32 M ,51 32N15 ISBN 3-540-10021-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10021-0 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar and means, storage ni data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is to payable the publisher, the amount tohffe e e to be determined agreement by with the publisher. © by Springer-Vertag Berlin Heidelberg 1980 Printed ni Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3t40-543210 To My Parents ~aL t6oaL TaC ~pay~Sa~ aOTO5. Introduction One of the simplest but the richest object to study in mathematics is a unit disc in the complex plane. Among several generalizations of it the notion of hermitian bounded symmetric-domain would be the most meaningful one, which is a generalization in the field of differential geometry (for full exposition see [13] for example). As a generalization of SL(2, ~) acting on D we have a notion of an arithmetic subgroup r which is a discrete subgroup of the Lie group G = Aut(D) of biholomorphic automorphisms of a hermitian bounded symmetric domain D. The quotient space F\V is naturally endowed with a structure of a normal complex analytic space. Two facts stand in the way of studying the geometric structure of r\~. The first is that r may have fixed points in D which give rise to singularities on r\D. This difficulty can be, however, overcome by taking a suitable subgroup 'r of F of finite index which acts on ~ without fixed points. The second is that r\~ may not be compact. Here arises the problem to compactify r\~ suitably. The first answer to this problem was given first by Satake [27 ] in the case of the Siegel upperhalf plane and finally by Baily-Borel [4] in the most general form. The second answer was quite recently given by Mumford and others [2], suggested by the early work by Siegel [30] and Igusa [15]. The aim of this lecture note is to exhibit these theories of compactification of r\D in the case of the Siegel upperhalf plane. Thanks to this restriction one can see the whole theory elementarily and explicitly in this typical example, which would help the reader to understand the general theory developed in [2] written in complete but abstract form. In this respect this book might be considered as an introduction to or supplement of [2]. On account of such expositive character of this book all proofs where one needs general theory (the latter half of Chap.V, Chap.VI-Vll) are omitted but giving a suitable reference, mostly to [2] or [17]. The content is as follows. In Chapter I we introduce the notion of the Siegel upperhalf plane, the symplectic group and its arithmetic VI subgroup and exhibit their fundamental properties. Chapter !I is devoted to the summary of main results concerning the problem of compactification. In Chapter III we explained the idea of the toroidal compactification in the simplest case of F\D with the unit disc D and F ~ SL(2, ~). Two main tools are used for the construction of the compactification, the theory of bounded symmetric domain due to Kor~nyi-Wolf [20] from differential geometry and the theory of torus embeddings due to Mumford [17] from algebraic geometry. The former is exhibited in Chapter IV and V, and the latter in Chapter VI. More precisely, the notion of boundary component of ~ and the structure of the associated parabolic subgroup in G are given in Chapter IV and the realization of the domain as a Siegel domain of the third kind with a boundary component in the first half of Chapter V. Here we give complete proof for the case of Siegel upperhalf plane. For the general case we refer the reader to [2] or [28] (the latter would be more readable). The theory of torus embedding is given without proof in Chapter VI. The complete proof can be found in [17] or [24] except for an elementary construction of torus embedding given in (6.13). The latter half of Chapter V is devoted to the exposition of the Satake compactification. After these preliminaries, in Chapter VII we construct Mumford's toroidal compactification and show funda- mental properties of it, some of which are not explicitly stated in [2] though the proof is already contained there essentially. In Chapter VIII we give some concrete examples of admissible decompositions which relate to the reduction theory of quadratic forms in our case. In the last Chapter IX a particular decomposition, called the 2nd Voronoi decomposition, is treated and we show that the toroidal com- pactification associated with this decomposition admits an algebro- geometric interpretation of this compactification extending the fact that Sp(g, ~)\~g can be considered as the coarse moduli space of principally polarised abelian varieties of dimension g. As an appendix we sum up the abstract theory by giving explicit description in the case of the Siegel upperhalf plane with notations and the course of exposition being in accordance with those in [2] in order that the reader would see where the abstract procedure is going through. There we indicate also a number of missprints in [2]. This note is based on the author's lecture at the Catholic University at Nijmegen in 1978. He would like to express his sincere gratitude to Prof. Looijenga and his colleagues at Nijmegen for their kind hospitality and encouragement, and to Mrs. Kozaki for her neat typing. Table of Contents §i. The Siegel upperhalf plane and the symplectic group. i §2. Main problem and main results. 7 §3, The case of g = .I ii §4. Boundary components and the structure of parabolic 51 subgroups. §5. Realization as a Siegel domain of the third kind, and 29 Satake compactification. §6. Theory of torus embeddings. 39 §7. Toroidal compactification due to Mumford. 58 A) Construction of toroidal compactification. 58 B) Geometric properties of toroidal compactifications 70 (smoothness, projectivity, extension of holomorphic maps). §8. Examples: reduction theory of positive quadratic forms. 58 §9. An application of the Voronoi compactification to 95 the theory of moduli of polarized abelian varieties. A) 2 nd Voronoi reduction theory. 95 B) Mixed decomposition of ~2 × V. i00 C) Compactification of the moduli space of polarized 201 abelian varieties. D) The extension of Torelli map. ii0 Appendix: Abstract theory of bounded symmetric domains bil (with explicit description in the case of Siegel upperhalf plane). .I The structure of bounded symmetric domains. bil A) Definition and realizations. Bil B) The structure of roots of .G 911 IIIV C) The description of D in ~+ via the Harish-Chandra 621 embedding. II. Boundary components. 128 A) Boundary components. 128 B) The normalizer of a boundary component. 134 C) The structure of N(F). 631 D) The natural projection ~F : D + F. 140 E) Rational boundary components. 143 III. Realization of D as a Siegel domain of the third 144 kind. A) The self-adjoint cone C(F) in U(F). 144 B) Realization of D as a Siegel domain. 145 C) Relation of the normalizers of adjacent boundary 152 components. Bibliography. 153 List of notations. 651 Index. 161 .I§ The Siegel upperhalf plane and the symplectic group. We first define two notions which are the main objects studied in this section. They are meaningful generalizations of the usual upperhalf plane H = {T ~ ~ ; Im T > 0} and the linear fractional transformation group SL(2, )ZZ acting on .H Definition (i.I). The complex domain ~g = {T ~ M(g, ~) ; t T = T, Im T > O} is called the Siegel upperhalf plane of degree .g Definition (1.2). The subgroup of GL(2g, )9]] defined as G = {M ~ M(2g, )9~ ; tM M : } g g A B = {M = (C ) D ; tAC = tCA, tBD = tDB, tAD - tCB = ig} A B M-I } = {M = (C ) D ; = _t C A t is called the (real) symplectio group (of degree g) and denoted by Sp(g, ~). (Some denote it by Sp(2g, ~).) Remark (1.3). In general, for a non-degenerate skew-symmetric bilinear form A of degree 2g, we can define Sp(A, ~) = {M ~ M(2g, ~) ; tMAM = A}. However, these groups, called paramodular groups, are conjugate to each other in GL(2g, )gZ hence isomorphic, for there is an element T in GL(2g, ~) with tTAT = , then g Sp(A, m) --+~ Sp(g, m) M ~ . T-1MT. We shall now exhibit fundamental properties of the Siegel upper- half plane and the symplectic groups. We restrict ourselves only in stating results we need later. Much more beautiful and stimulating exposition on these subjects is [31]. Proposition (1.4). i) G acts on ~g biholomorphically as AB M = (C ) D : T ÷ M.T = (AT + B)(CT + D) -I. ii) G acts on ~ transitively. g Bemark (1.5). Actually all biholomorphic automorphisms of g are expressed in this form. Namely Aut(~g) = Sp(g, m)/±l (cf. (1.6)). Note that {±i} is the center of Sp(g, ~) and the quotient is a simple group. Proof of (1.4). i) First we note the following two equalities: a) t(Ax + B)(CT + D) - t(CT + D)(AT + B) = 0 b) t(i~ + B)(C-~) - t(CT + D)(A~ + B) = 2/IT Im T. We shall prove the second equality. left side = (TtA + tB)(C~ + D) - (Ttc + tD)(A~ + B) = TtAC~ + tBCT + xtAD + tBD _ TtCA~ _ tDA ~ _ TtCB _ tDB = -Y + • (of. (1.2)). Next we show that CT + D is invertible. If not, there is a non-zero vector z c Cg such that (CT + D)z = 0. Then 0 = tzt(AT + B)(C-~-~)[ - tzt(CT + D)(A-~)z 2Z---ltz(Im T)z (by b)), = which is impossible because Im T > 0. As the last step, it is shown that ~' = (AT + B)(CT + D) -1 is contained in ~g. The proof, being elementary, is left to the reader. In order to show that T' is symmetric and Im T' > 0 we use the equalities a) and b) respectively. ii) Write T = x + /--T y with real x, y. There is a matrix u ~ GL(g, ~) with y = utu. Then we have Facts (1.6) (differential-geometric background). l) zso(-g~Ylg) := (M ~ a ; M-(~Ig) -- ~Ylg) A B = {(-S ) A ~ G} ~-~ U(g) -- {A + ¢~B) (the unitary group). Hence, as a real analytic manifold, ~g is a homogeneous space Sp(g, ~)/U(g). Here Sp(g, ~)/±I is a simple Lie group and U(g) is a maximal compact subgroup unique up to conjugate. is realized as a bounded domain by the Cayley transformation g g(cid:127) c : ~ ~ Dg = Z( ~ M(g, ~) ; tz = Z, tZZ < 7g} T ~ z = (~ - -~-Zlg)(~ + g~Ylg) -1. c-l(z) = -J~Y(z + l)(-z + l) -I. The expression of ~g as (i.i) is then called an unbounded realization as a tube domain (cf. 4) below). 3) ~g is a symmetric space. Namely ~g or ~g D S = g g -I T +---- -T Z ~ -Z is an involution 2 = (s i) having -~I (or 0 as an isolated g fixed point. Such s is called a symmetry at -~i . As G acts g transitively, every point of ~ has a symmetry. g 4) ~ is a tube domain, i.e. ~g gyv= + _~y~g ÷ where