Table Of Content134
BIRATIONAL
GEOMETRY OF
• ALGEBRAIC VARIETIES
JANOS KOLLAR & SHIGEFUMI MORI
CAMBRIDGE TRACTS IN MATHEMATICS
General Editors
B. BOLLOBAS, F. KffiWAN, P. SARNAK, C.T.C. WALL
134 Birational Geometry
of Algebraic Varieties
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 United Kingdom
lRP,
CAMBRIDGE UNIVERSITY PRESS
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Road,
@ English edition Cambridge University Press 1998
Translated from Souyuuri Kikagaku published by Iwanami Shoten,
Publishers, Tokyo, 1998
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 1998
Printed in the United Kingdom at the University Press, Cambridge
Typeset in Computer Modern 10/13pt
A catalogue record for this book is available from the British Library
Library of Congress Cataloguing in Publication data
Kollar, Janos.
Birational geometry of algebraic varieties Janos Kollar and
/
Shigefumi Mori, with collaboration of C.H. Clemens and A. Corti.
p. cm. -(Cambridge tracts in mathematics; 134)
Includes bibliographical references and index.
ISBN 0 521 63277 3
1. Surfaces, Algebraic. 2. Algebraic varieties. I. Mori,
Shigefumi. II. Title. III. Series.
QA571.K65 1998
516.3'5-dc21 98-24732 CIP
ISBN 0 521 63277 3 hardback
Contents
Preface vii
page
Introduction 1
1 Rational Curves and the Canonical Class
1
1.1 Finding Rational Curves when Kx is Negative 8
1.2 Finding Rational Curves when Kx is not Nef 16
1.3 The Cone of Curves of Smooth Varieties 18
1.4 Minimal Models of Surfaces 26
1.5 Ampleness Criteria 29
2 Introduction to the Minimal Model Program
36
2.1 Introduction to Mori's Program 37
2.2 Extensions of the Minimal Model Program 45
2.3 Singularities in the Minimal Model Program 50
2.4 The Kodaira Vanishing Theorem 62
2.5 Generalizations of the Kodaira Vanishing Theorem 67
3 Cone Theorems 4
7
3.1 Introduction to the Proof of the Cone Theorem 75
3.2 Basepoint-free Theorem 78
3.3 The Cone Theorem 81
3.4 The Rationality Theorem 86
3.5 The Non-vanishing Theorem 91
3.6 Relative versions 93
3.7 Running the MMP 96
3.8 Minimal and Canonical Models 106
4 Surface Singularities of the Minimal Model Programlll
4.1 Log Canonical Surface Singularities 112
v
Contents
vi
4.2 Du Val Singularities 122
4.3 Simultaneous Resolution for Du Val Singularities 128
4.4 Elliptic Surface Singularities 136
4.5 Deformations of Hypersurface Singularities 144
5 Singularities of the Minimal Model Program 152
5.1 Rational Singularities 153
5.2 Log Terminal Singularities 158
5.3 Canonical and Terminal Threefold Singularities 164
5.4 Inversion of Adjunction 172
5.5 Duality Theory 179
6 Three-dimensional Flops 187
6.1 Flips and Flops 188
6.2 Terminal Flops 192
6.3 Terminalization and Q-factorialization 195
6.4 Canonical Flops 201
Semi-stable Minimal Models 207
1
7.1 Semi-stable MMP 208
7.2 Semi-stable Reduction Theorem 214
7.3 Special Semi-stable Flips 220
7.4 Semi-stable Flips 224
7.5 Applications to Families of Surfaces 229
7.6 A Survey of Further Results 236
Bibliography 241
Index 249
Preface
One of the major discoveries of the last two decades in algebraic geome
try is the realization that the theory of minimal models of surfaces can be
generalized to higher dimensional varieties. This generalization is called
the minimal model program or Mori's program. While originally the
program was conceived with the sole aim of constructing higher dimen
sional analogues of minimal models of surfaces, by now it has developed
into a powerful tool with applications to diverse questions in algebraic
geometry and beyond.
So far the program is complete only in dimension 3, but large parts
are known to work in all dimensions.
The aim of this book is to introduce the reader to the circle of ideas de
veloped around the minimal model program, relying only on knowledge
of basic algebraic geometry.
In order to achieve this goal, considerable effort was devoted to make
the book as self-contained as possible. We managed to simplify many of
the proofs, but in some cases a compromise seemed a better alternative.
There are quite a few cases where a theorem which is local in nature
is much easier to prove for projective varieties. For these, we state the
general theorem and then prove the projective version, giving references
for the general cases. Most of the applications of the minimal model
program ultimately concern projective varieties, and for these the proofs
in this book are complete
Acknowledgments
The present form of this book owes a lot to the contributions of our two
collaborators.
H. Clemens was our coauthor in [CKM88]. Sections 1.1-3, 2.1, 2.2, 2.4
and 3.1-5 are revised versions of sections of [CKM88]. We owe special
vii
viii Preface
thanks to H. Clemens and to Asterisque for allowing us to use this
material.
A. Corti showed us unpublished note 'Semi-stable 3-fold flips'
his
which utilized the reduction ideas of [Sho92] (as explained in [K+92,
Chap. 18]) to semi-stable flips. This allowed him to eliminate from the
proof the use of the classification of surface quotient singularities. For 3-
folds this is a relatively small difference, but each time we eliminate the
use of a special low-dimensional result, we hopefully a step closer to
are
flips in higher dimensions. Corti's note formed the basis of our treatment
of semi-stable flips in section 7.4.
In the past five years, we have given lecture series about the mate
rial presented in this book at the Regional Geometry Institute at Park
City Utah, at RIMS Kyoto University and at the University of Utah.
We would like to thank our audience, colleagues and students who con
tributed many observations and improvements to the present form of this
book. We received especially helpful comments from D. Abramovich, A.
Bertram, J. Cheah, Fujino, Y. Kawamata, H. Kley, S. Kovacs, T.
0.
Kuwata, Y. Lee, R. Mayer, Y. Miyaoka, R. Morelli, N. Nakayama, Th.
Peternell, E. Szab6 and N. Tziolas.
We also thank the financial support of the NSF (grant number DMS-
9622394), the University of Utah and the Japanese Ministry of Educa
tion, Science and Culture under a Grant-in-Aid for International Sci
entific Research (Joint Research Program 08044078) and Scientific
Re
search (B) Program 09440010.
Introduction
From the beginnings of algebraic geometry it has been understood that
birationally equivalent varieties have many properties in common. Thus
it is natural to attempt to find in each birational equivalence class a
variety which is simplest in some sense, and then study these varieties
in detail.
Each irreducible curve is birational to a unique smooth projective
curve, thus the investigation of smooth projective curves is equivalent
to the study of all curves up to birational equivalence.
For surfaces the situation is more complicated. Each irreducible sur
face is birational to infinitely many smooth projective surfaces. The the
ory of minimal models of surfaces, developed by the Italian algebraic
geometers at the beginning of the twentieth century, aims to choose a
unique smooth projective surface from each birational equivalence class.
The recipe is quite simple. If a smooth projective surface contains a
smooth rational curve with self-intersection -1, then it can be con
tracted to a point and we obtain another smooth projective surface.
Repeating this procedure as many times as possible, we usually obtain
a unique 'minimal model'. In a few cases we obtain a model that is not
unique, but these cases can be described very explicitly.
A search for a higher dimensional analogue of this method started
quite late. One reason is that some examples indicated that a similar
approach fails in higher dimensions.
The works of Reid and Mori in the early 1980s raised the possibility
that a higher dimensional theory of minimal models may be possible if
we allow not just smooth varieties but also varieties with certain mild
singularities. This approach is called the Minimal Model Program or
Mori's Program. After many contributions by Benveniste, Kawamata,
1
2 Introduction
Kollar, Reid, Shokurov, Tsunoda, Viehweg and others, the program was
completed in dimension three by Mori in 1988.
Since then this program has grown into a method which can be applied
successfully to many problems in algebraic geometry.
The aim of this book is to provide an introduction to the techniques
and ideas of the minimal model program.
Chapter 1 gives an introduction to the whole program through a ge
ometric approach. Most of these results are not used later, but they
provide a useful conceptual foundation.
Chapter 2 is still introductory, discussing some aspects of singularities
and the relevant generalizations of the Kodaira Vanishing Theorem.
The first major part of the program, the Cone Theorem, is proved in
Chapter 3. These results work in all dimensions.
The rest of the book is essentially devoted to the study of 3-dimension
al flips and flops. Flips and flops are new types of birational transforma
tions which first appear, in dimension 3. Most major differences between
the theory of surfaces and 3-folds can be traced back to flips and flops.
Chapter 4 is devoted to the classification of certain surface singular
ities. These results are needed in further work on the 3-dimensional
theory.
The singularities appearing in the course of the minimal model pro
gram are investigated in Chapter 5. The results are again rather complete
in all dimensions.
Flops are studied in Chapter 6. Flops are easier to understand than
flips, and, at least in dimension 3, their description is rather satisfactory.
Chapter 7 is devoted to 3-dimensional flips. The general theory is still
too complicated and long to be included in a textbook, thus we restrict
ourselves to the study of a special class, the so-called semi-stable flips. We
have succeeded in simplifying the proofs in this case considerably. Semi
stable flips appear naturally in many contexts, and they are sufficient
for several of the applications.
A more detailed description of the contents of each chapter is given
at its beginning.
Sections 4.5 and 5.5 are each a side direction, rather than being part
of the main line of arguments. In each case we felt that the available
references do not adequately cover some results we need, and that our
presentation may be of interest to the reader.
