Toroidal Compactification of Siegel Spaces
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812 IIIIIII
Yukihiko Namikawa
Toroidal Compactificatic)n of Siegel Spaces I
Springer-Verlag Berlin Heidelberg New York 1980
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Author Yukihiko Namikawa Department of Mathematics, Nagoya University Furocho, Chikusa-Ku Nagoya, 464/Japan
AMS Subject Classifications (1980): 14 L1 ?, 20 G 20, 32 J 05, 32 M 15, 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 means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Vertag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3t40-543210
To M y P a r e n t s
~aL t 6 o a L
TaC ~ p a y ~ S a ~
aOTO5.
Introduction
One of the simplest but the richest object to study in
mathematics
is a unit disc
in the complex plane.
A m o n g several g e n e r a l i z a t i o n s
of it the n o t i o n
of h e r m i t i a n b o u n d e d s y m m e t r i c - d o m a i n w o u l d be the most m e a n i n g f u l
one, w h i c h is a g e n e r a l i z a t i o n in the field of d i f f e r e n t i a l g e o m e t r y (for full e x p o s i t i o n see [13] for example). As a g e n e r a l i z a t i o n of
of an a r i t h m e t i c group
G = Aut(D)
subgroup
symmetric domain
SL(2, ~ )
r
of b i h o l o m o r p h i c
D.
acting on
w h i c h is a d i s c r e t e
The q u o t i e n t
automorphisms
space
F\V
with a structure of a normal complex analytic
D
we have a notion
subgroup of the Lie
of a h e r m i t i a n b o u n d e d
is n a t u r a l l y endowed
space.
Two facts stand in the way of s t u d y i n g the geometric
of
r\~.
The first is that
rise to s i n g u l a r i t i e s
on
r
may have fixed points in
r\D.
overcome by taking a suitable
w h i c h acts on
~
without
The second is that
problem to compactify
r\~
This d i f f i c u l t y
subgroup
r'
fixed points. r\~
F
of finite index Here arises
the
The first answer to this
[27 ] in the case of the Siegel
u p p e r h a l f plane and finally by B a i l y - B o r e l others
which give
can be, however,
may not be compact.
suitably.
p r o b l e m was given first by Satake form.
of
D
structure
[4] in the most general
The second answer was quite r e c e n t l y given by Mumford and [2], s u g g e s t e d by the early work by Siegel
[30] and Igusa
[15].
The aim of this lecture note is to exhibit these theories of
c o m p a c t i f i c a t i o n of
r\D
in the case of the Siegel u p p e r h a l f plane.
Thanks to this r e s t r i c t i o n one can see the whole theory e l e m e n t a r i l y and e x p l i c i t l y in thi
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