Classification of higher dimensional algebraic varieties

✍ Scribed by Hacon C.D., Kovacs S.

• Publisher: Birkhauser
• Year: 2010
• Tongue: English
• Leaves: 221
• Series: Oberwolfach Seminars
• Category: Library

✦ Synopsis

This book focuses on recent advances in the classification of complex projective varieties. It is divided into two parts. The first part gives a detailed account of recent results in the minimal model program. In particular, it contains a complete proof of the theorems on the existence of flips, on the existence of minimal models for varieties of log general type and of the finite generation of the canonical ring. The second part is an introduction to the theory of moduli spaces. It includes topics such as representing and moduli functors, Hilbert schemes, the boundedness, local closedness and separatedness of moduli spaces and the boundedness for varieties of general type. The book is aimed at advanced graduate students and researchers in algebraic geometry.

✦ Table of Contents

Cover......Page 1
Classification of
Higher Dimensional
Algebraic Varieties......Page 4
ISBN 9783034602891......Page 5
Preface......Page 6
Table of Contents
......Page 7
I Basics......Page 13
1.A Classification......Page 15
1.A.1 Curves......Page 16
1.A.2 Kodaira Dimension......Page 19
1.A.3 Surfaces......Page 20
1.A.4 Higher dimensional varieties......Page 22
1.A.6 Varieties with Kodaira dimension 0......Page 25
1.A.8 Moduli spaces of varieties of general type......Page 26
2.A Notation......Page 29
2.B Divisors......Page 30
2.C Reflexive sheaves......Page 32
2.D Cyclic covers......Page 33
2.E R-divisors in the relative se......Page 34
2.F Families and base change......Page 36
2.G Parameter spaces and deformations of families......Page 37
3.A Canonical singularities......Page 39
3.B Cones......Page 41
3.C Log canonical singularities......Page 42
3.E Pinch points......Page 44
3.F Semi-log canonical singularities......Page 46
3.G Pairs......Page 48
3.H Vanishing theorems......Page 51
3.I Rational and Du Bois singularities......Page 53
II Recent advances in the minimal model program......Page 59
CHAPTER 4......Page 61
5.A The Cone and Basepoint-free theorems......Page 63
5.B Flips and divisorial contractions......Page 65
5.C The minimal model program for surfaces......Page 70
5.D The main theorem and sketch of proof......Page 71
5.E The minimal model program with scaling......Page 73
5.F Pl-flips......Page 74
5.G Corollaries......Page 75
CHAPTER 6 Multiplier ideal sheaves......Page 79
6.A Asymptotic multiplier ideal sheaves......Page 82
6.B Extending pluricanonical forms......Page 85
7.A Rationality of the restricted algebra......Page 91
7.B Proof of (5.69)......Page 92
8.A Special termination......Page 95
8.B Existence of log terminal models......Page 97
9.A Nakayama-Zariski decomposition......Page 101
9.B Non-vanishing......Page 105
CHAPTER 10 Finiteness of log terminal models......Page 111
III Compact moduli spaces of canonically
polarized varieties......Page 115
11.B Moduli functors......Page 117
11.C Coarse moduli spaces......Page 120
12.A The Grassmannian functor......Page 123
12.B The Hilbert functor......Page 126
13.A Boundedness......Page 129
13.B Constructing the moduli space......Page 135
13.C Local closedness......Page 136
13.D Separatedness......Page 138
13.E Properness......Page 142
14.A An important example......Page 145
14.B Q-Gorenstein families......Page 147
14.C Projective moduli schemes......Page 151
14.D Moduli of pairs and other generalizations......Page 152
CHAPTER 15 Singularities of stable varieties......Page 153
15.A Singularity criteria......Page 154
15.B Applications to moduli spaces and vanishing theorems......Page 156
15.C Deformations of DB singularities......Page 158
CHAPTER 16 Subvarieties of moduli spaces......Page 161
16.B The Parshin-Arakelov reformulation......Page 164
16.C Shafarevich’s conjecture for number fields......Page 165
16.D From Shafarevich to Mordell: Parshin’s trick......Page 166
16.E.1 Hyperbolicity......Page 167
16.E.2 Weak Boundedness......Page 168
16.E.3 From Weak Boundedness to Boundedness......Page 169
16.F Higher dimensional fibers......Page 170
16.F.3 Weak Boundedness......Page 171
16.G Higher dimensional bases......Page 172
16.H.1 Families of curves......Page 174
16.I.1 Positivity of direct images......Page 175
16.I.2 Vanishing theorems......Page 176
16.I.3 Kernels of Kodaira-Spencer maps......Page 177
16.J Allowing more general fibers......Page 178
16.K Iterated Kodaira-Spencer maps and strong non-isotriviality......Page 180
IV Solutions and hints to some of the exercises......Page 183
Bibliography......Page 197
Index......Page 215