Classification of Higher Dimensional

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.

Basics	p. 1
Introduction	p. 3
Classification	p. 3
Preliminaries	p. 17
Notation	p. 17
Divisors	p. 18
Reflexive Sheaves	p. 20
Cyclic Covers	p. 21
R-Divisors in the Relative Setting	p. 22
Families and Base Change	p. 24
Parameter Spaces and Deformations of Families	p. 25
Singularities	p. 27
Canonical Singularities	p. 27
Cones	p. 29
Log Canonical Singularities	p. 30
Normal Crossings	p. 32
Pinch Points	p. 32
Semi-Log Canonical Singularities	p. 34
Pairs	p. 36
Vanishing Theorems	p. 39
Rational and Du Bois Singularities	p. 41
Recent advances in the minimal model program	p. 47
Introduction	p. 49
The Main Result	p. 51
The Cone and Basepoint-Free Theorems	p. 51
Flips and Divisorial Contractions	p. 53
The Minimal Model Program for Surfaces	p. 58
The Main Theorem and Sketch of Proof	p. 59
The Minimal Model Program with Scaling	p. 61
Pl-Flips	p. 62
Corollaries	p. 63
Multiplier Ideal Sheaves	p. 67
Asymptotic Multiplier Ideal Sheaves	p. 70
Extending Pluricanonical Forms	p. 73
Finite Generation of the Restricted Algebra	p. 79
Rationality of the Restricted Algebra	p. 79
Proof of (5.69)	p. 80
Log Terminal Models	p. 83
Special Termination	p. 83
Existence of Log Terminal Models	p. 85
Non-Vanishing	p. 89
Nakayama-Zariski Decomposition	p. 89
Non-Vanishing	p. 93
Finiteness of Log Terminal Models	p. 99
Compact moduli spaces of canonically polarized varieties	p. 103
Moduli Problems	p. 105
Representing Functors	p. 105
Moduli Functors	p. 105
Coarse Moduli Spaces	p. 108
Hilbert Schemes	p. 111
The Grassmannian Functor	p. 111
The Hilbert Functor	p. 114
The Construction of the Moduli Space	p. 117
Boundedness	p. 117
Constructing the Moduli Space	p. 123
Local Closedness	p. 124
Separatedness	p. 126
Properness	p. 130
Families and Moduli Functors	p. 133
An Important Example	p. 133
Q-Gorenstein Families	p. 135
Projective Moduli Schemes	p. 139
Moduli of Pairs and Other Generalizations	p. 140
Singularities of Stable Varieties	p. 141
Singularity Criteria	p. 142
Applications to Moduli Spaces and Vanishing Theorems	p. 144
Deformations of DB Singularities	p. 146
Subvarieties of Moduli Spaces	p. 149
Shafarevich's Conjecture	p. 152
The Parshin-Arakelov Reformulation	p. 152
Shafarevich's Conjecture for Number Fields	p. 153
From Shafarevich to Mordell: Parshin's Trick	p. 154
Hyperbolicity and Boundedness	p. 155
Higher Dimensional Fibers	p. 158
Higher Dimensional Bases	p. 160
Uniform and Effective Bounds	p. 162
Techniques	p. 163
Allowing More General Fibers	p. 166
Iterated Kodaira-Spencer Maps and Strong Non-Isotriviality	p. 168
Solutions and hints to some of the exercises	p. 171
Bibliography	p. 185
Index	p. 203
Table of Contents provided by Ingram. All Rights Reserved.

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Classification of Higher Dimensional Algebraic Varieties

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