Classification of Higher Dimensional Algebraic Varieties

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.- Preliminaries.- Singularities.- Recent advances in the minimal model program.- The main result.- Multiplier ideal sheaves.- Finite generation of the restricted algebra.- Log terminal models.- Non-vanishing.- Finiteness of log terminal models.- Compact moduli spaces of canonically polarized varieties.- Moduli problems.- Hilbert schemes.- The construction of the moduli space.- Families and moduli functors.- Singularities of stable varieties.- Subvarieties of moduli spaces.