紹介
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.
目次
I Basics.- 1 Introduction.- 1.A. Classification.- 2 Preliminaries.- 2.A. Notation.- 2.B. Divisors.- 2.C. Reflexive sheaves.- 2.D. Cyclic covers.- 2.E. R-divisors in the relative setting.- 2.F. Vanishing theorems.- 2.G. Families and base change.- 2.H. Parameter spaces and deformations of families.- 3 Singularities.- 3.A. Canonical singularities.- 3.B. Cones.- 3.C. Log canonical singularities.- 3.D. Normal crossings.- 3.E. Pinch points.- 3.F. Semi-log canonical singularities.- 3.G. Pairs.- 3.H. Rational and du Bois singularities.- II Recent advances in the MMP.- 4 Introduction.- 5 The main result.- 5.A. The cone and base point free theorems.- 5.B. Flips and divisorial contractions.- 5.C. The minimal model program for surfaces.- 5.D. The main theorem and sketch of proof.- 5.E. The minimal model program with scaling.- 5.F. PL-flips.- 5.G. Corollaries.- 6 Multiplier ideal sheaves.- 6.A. Asymptotic multiplier ideal sheaves.- 6.B. Extending pluricanonical forms.- 7 Finite generation of the restricted algebra.- 7.A. Rationality of the restricted algebra.- 7.B. Proof of (5.69).- 8 Log terminal models.- 8.A. Special termination.- 8.B. Existence of log terminal models.- 9 Non-vanishing.- 9.A. Nakayama-Zariski decomposition.- 9.B. Non-vanishing.- 10 Finiteness of log terminal models.- III Compact moduli spaces.- 11 Moduli problems.- 11.A. Representing functors.- 11.B. Moduli functors.- 11.C. Coarse moduli spaces.- 12 Hilbert schemes.- 12.A. The Grassmannian functor.- 12.B. The Hilbert functor.- 13 The construction of the moduli space.- 13.A. Boundedness.- 13.B. Constructing the moduli space.- 13.C. Local closedness.- 13.D. Separatedness.- 14 Families and moduli functors.- 14.A. An important example.- 14.B. Q-Gorenstein families.- 14.C. Projective moduli schemes.- 14.D. Moduli of pairs and other generalizations.- 15 Singularities of stable varieties.- 15.A. Singularity criteria.- 15.B. Applications to moduli spaces and vanishing theorems.- 15.C. Deformations of DB singularities.- 16 Subvarieties of moduli spaces.- 16.A. Shafarevich's conjecture.- 16.B. The Parshin-Arakelov reformulation.- 16.C. Shafarevich's conjecture for number fields.- 16.D. From Shafarevich to Mordell: Parshin's trick.- 16.E. Hyperbolicity and boundedness.- 16.F. Higher dimensional fibers.- 16.G. Higher dimensional bases.- 16.H. Uniform and effective bounds.- 16.I. Techniques.- 16.J. Allowing more general fibers.- 16.K. Iterated Kodaira-Spencer maps and strong non-isotriviality.- IV Solutions and hints to some of the exercises.
