Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold dna .B Eckmann
812
IIIIIII !
okihikuY Namikawa
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fo Siegel Spaces
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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
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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
