Higher Dimensional Birational Geometry

ADVANCED STUDIESIN PURE MATHEMATICS
Chief Editor:

35

Eiichi Bannai (Kyushu University)

Higher Dimensional BirationalGeometry

Edited byShigefumi Mori (Kyoto University) andYoichi Miyaoka (University of Tokyo)

Mathematical Society of Japan

This book was typeset by AMS-TEX and AMS-LATEX, the TEX macro systemsof the American Mathematical Society, together with the style files aspm.styand aspmfm.sty for AMS-TEX written by Dr. Chiaki Tsukamoto andaspmproc.sty for AMS-LATEX written by Dr. Akihiro Munemasa.
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All rights reserved. No part of this publication may be reproduced, stored ina retrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording or otherwise, without the prior permissionof the copyright owner.
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ISBN 4-931469-19-1
2000 Mathematics Subject Classification.
Primary 14-06; Secondary 14Dxx, 14Exx, 14Jxx, 14Kxx.

Advanced Studies in Pure Mathematics 35
Chief Editor
Eiichi Bannai(Kyushu University)

Editorial Board of the Series
Hitoshi Arai

Eiichi Bannai

Yoshikazu Giga

(Univ. of Tokyo)(Kyushu Univ.)(Hokkaido Univ.)

Ki-ichiro Hashimoto

Kazuya KatoToshitake Kohno(Waseda Univ.)(Kyoto Univ.)(Univ. of Tokyo)

Yoichi Miyaoka

Seiki Nishikawa

Junjiro Noguchi(Univ. of Tokyo)(Tohoku Univ.)(Univ. of Tokyo)

Toshiyuki Tanisaki(Osaka City Univ.)
PRINTED IN JAPANby Tokyo Shoseki Printing Co., Ltd.

Preface
The present volume consists of four research articles contributed byparticipants of the international conference Higher Dimensional Algebraic Varieties, 1997, at the Research Institute of Mathematical Sciences(RIMS), Kyoto University during June 2–6 and 9–13, 1997.
The conference, held as part of the special year activities of RIMS
during the academic year 1996/97 and cosponcored by the MathematicalSociety of Japan, represented the high level of the birational geometryat the time. The invited speakers and their talks were:
Y. Miyaoka (U. Tokyo), On a numerical characterization of Pn .
N. Nakayama (RIMS), Projective manifolds whose universal covering
spaces are Cn .
A. Corti (U. Cambridge), Birationally rigid Q-Fano hypersurfaces I(Excluding maximal singularities).
M. Reid (Warwick U.), Birationally rigid Q-Fano hypersurfaces II(Constructing birational involutions and untwisting).
J. Kollár (Princeton U.), Polynomials with integral coeﬃcients.
F. Bogomolov (Courant Inst.), Weak Hironaka theorem.
Y. Kawamata (U. of Tokyo), Unobstructed deformations and index 1
covers.
S. Keel (1) (U. Texas, Austin), Eventual freeness in char p, withapplications to Mg and to 3-fold MMP.
L. Ein (U. Illinois at Chicago), Singularities of theta divisors andbirational geometry of irregular varieties.
S. Helmke (RIMS), On global generation of adjoint linear systems.
K. Matsuki (Purdue U.), Surface log terminal singularities in positive
characteristic.
S. Mukai (RIMS), Degeneration of Pn × Pn with application to delPezzo ﬁbration.
V. Alexeev (U. Georgia), Log canonical singularities and complete
moduli.
G. Sankaran (U. Bath), Abelian surfaces in toric 4-folds.
S. Keel (2) (U. Texas, Austin), Rational curves on open surfaces I.
J. McKernan (U. C. Santa Barbara), Rational curves on open surfaces
II.
S. Mori (RIMS), On Reid’s conjecture on general elephants.
M. Hanamura (Kyushu U.), Mixed motivic sheaves
In the second week, more informal talks were given:

A. Corti, Introduction to the Sarkisov program.
J. Kollar, Real algebraic threefolds.
S. Keel and J. McKernan, Rational curves on open surfaces.
V. V. Shokurov (Johns Hopkins U.), Log ﬂips.
We believe that it would have been a great volume if we could havecollected the lectures delivered at the conference. However, some ofthem had already been submitted elsewhere and some were a little too
technical to be included in the ASPM series, which is supposed to be ofexpository nature. By these twofold reasons, the editors decided that
this volume shouild not be the ordinary proceedings of the conference
but a selection of independent full expositions on diﬀerent topics of fundamental importance in algebraic geometry: moduli spaces of abeliansurfaces, rational curves on algebraic vareities, 3-dimensional ﬂips andthe theory of elliptic ﬁbrations.
The authors made their best to update the articles to ﬁll the timelag between the conference and the publication, and the editors hopethat this volume will provide accessible and systematic treatments ofseveral aspects of recent exciting developments in algebraic geometry.
The editors express their gratitude to the contributors who have
patiently awaited the belated publication, and apologise to the Editorial
Board of the ASPM for the delay of the preparation and the unusual
style of this volume.

Shigefumi Mori
Yoichi Miyaoka

All papers in this volume have been refereed and are in final form.
No version of any of them will be submitted for publication elsewhere.

CONTENTS
Koji Cho, Yoichi Miyaoka and N. I. Shepherd-Barron — Characterizations of Projective Space and Applications to Complex
Symplectic Manifolds

1

Klaus Hulek and G. K. Sankaran — The Geometry of Siegel
Modular Varieties
89
Shigefumi Mori — On Semistable Extremal Neighborhoods

157

Noboru Nakayama — Local Structure of an Elliptic Fibration

185