Classification of Higher Dimensional Algebraic Varieties

by Christopher D. Hacon, Sándor Kovács

Overview

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.

Product Details

ISBN-13:	9783034602891
Publisher:	Birkhäuser Basel
Publication date:	07/16/2010
Series:	Oberwolfach Seminars , #41
Edition description:	2010
Pages:	220
Product dimensions:	6.60(w) x 9.40(h) x 0.60(d)

Table of Contents

I Basics 1

1 Introduction 3

1.A Classification 3

2 Preliminaries 17

2.A Notation 17

2.B Divisors 18

2.C Reflexive Sheaves 20

2.D Cyclic Covers 21

2.E R-Divisors in the Relative Setting 22

2.F Families and Base Change 24

2.G Parameter Spaces and Deformations of Families 25

3 Singularities 27

3.A Canonical Singularities 27

3.B Cones 29

3.C Log Canonical Singularities 30

3.D Normal Crossings 32

3.E Pinch Points 32

3.F Semi-Log Canonical Singularities 34

3.G Pairs 36

3.H Vanishing Theorems 39

3.I Rational and Du Bois Singularities 41

II Recent advances in the minimal model program 47

4 Introduction 49

5 The Main Result 51

5.A The Cone and Basepoint-Free Theorems 51

5.B Flips and Divisorial Contractions 53

5.C The Minimal Model Program for Surfaces 58

5.D The Main Theorem and Sketch of Proof 59

5.E The Minimal Model Program with Scaling 61

5.F Pl-Flips 62

5.G Corollaries 63

6 Multiplier Ideal Sheaves 67

6.A Asymptotic Multiplier Ideal Sheaves 70

6.B Extending Pluricanonical Forms 73

7 Finite Generation of the Restricted Algebra 79

7.A Rationality of the Restricted Algebra 79

7.B Proof of (5.69) 80

8 Log Terminal Models 83

8.A Special Termination 83

8.B Existence of Log Terminal Models 85

9 Non-Vanishing 89

9.A Nakayama-Zariski Decomposition 89

9.B Non-Vanishing 93

10 Finiteness of Log Terminal Models 99

III Compact moduli spaces of canonically polarized varieties 103

11 Moduli Problems 105

11.A Representing Functors 105

11.B Moduli Functors 105

11.C Coarse Moduli Spaces 108

12 Hilbert Schemes 111

12.A The Grassmannian Functor 111

12.B The Hilbert Functor 114

13 The Construction of the Moduli Space 117

13.A Boundedness 117

13.B Constructing the Moduli Space 123

13.C Local Closedness 124

13.D Separatedness 126

13.E Properness 130

14 Families and Moduli Functors 133

14.A An Important Example 133

14.B Q-Gorenstein Families 135

14.C Projective Moduli Schemes 139

14.D Moduli of Pairs and Other Generalizations 140

15 Singularities of Stable Varieties 141

15.A Singularity Criteria 142

15.B Applications to Moduli Spaces and Vanishing Theorems 144

15.C Deformations of DB Singularities 146

16 Subvarieties of Moduli Spaces 149

16.A Shafarevich's Conjecture 152

16.B The Parshin-Arakelov Reformulation 152

16.C Shafarevich's Conjecture for Number Fields 153

16.D From Shafarevich to Mordell: Parshin's Trick 154

16.E Hyperbolicity and Boundedness 155

16.F Higher Dimensional Fibers 158

16.G Higher Dimensional Bases 160

16.H Uniform and Effective Bounds 162

16.I Techniques 163

16.J Allowing More General Fibers 166

16.K Iterated Kodaira-Spencer Maps and Strong Non-Isotriviality 168

IV Solutions and hints to some of the exercises 171

Bibliography 185

Index 203

Editorial Reviews

From the reviews:

“The present text presents the proofs of many results surrounding the minimal model program (MMP) for higher-dimensional varieties. … This text treats the subject in the great generality which is required for getting the most recent results. Hence, it is laden with terminology, all necessary for the modern researcher. … As such, the text will be invaluable for those currently living off of survey articles trying to grasp recent advances in higher-dimensional geometry.” (Michael A. van Opstall, Mathematical Reviews, Issue 2011 f)

“The authors give a detailed account of these new results and the theory of compact moduli spaces of canonically polarised varieties. … the book contains a considerable number of exercises as well as a chapter of hints to solve them. … the authors have made quite an effort to write a text that is both an accessible introduction and a useful reference. … I can only recommend it to researchers and advanced graduate students interested in this highly active field of mathematics.” (Andreas Höring, Zentralblatt MATH, Vol. 1204, 2011)

From the Publisher