Introduction to the Minimal Model Problem

Abstract

Introduction Chapter O. Notation and preliminaries § 0-1. Kleiman's criterion for ampleness § 0-2. Definitions of terminal, canonical and (weak) log-terminal singularities § 0-3. Canonical varieties § 0-4. The minimal model conjecture Chapter 1. Vanishing theorems § 1-1. Covering Lemma § 1-2. Vanishing theorem of Kawamata and Viehweg § 1-3. Vanishing theorem of Elkik and Fujita Chapter 2. Non-Vanishing Theorem § 2-1. Non-Vanishing Theorem Chapter 3. Base Point Free Theorem § 3-1. Base Point Free Theorem § 3-2. Contractions of extremal faces § 3-3. Canonical rings of varieties of general type Chapter 4. Cone Theorem § 4-1. Rationality Theorem § 4-2. The proof of the Cone Theorem Chapter 5. Flip Conjecture § 5-1. Types of contractions of extremal rays § 5-2. Flips of toric morphisms

Complex projective 3-fold with non-negative canonical Euler-Poincare characteristic

“…By the 3-dimensional Minimal Model Program (MMP) (see [19,22], for instance), we only have to consider a minimal 3-fold X of general type with Q-factorial terminal singularities. Denote the Cartier index of X by r := r(X) which is the minimal positive integer with rK X a Cartier divisor, where K X is a canonical divisor on X.…”

Lectures on Vanishing Theorems

“…In particular, the applications of vanishing theorems in the birational classification theory and in the minimal model program is left out. The reader is invited to consult the survey's of S. Mori [46] and of Y. Kawamata, K. Matsuda and M. Matsuki [38].…”

A criterion for semiampleness

“…It is a generalization of the Base Point Free Theorem (Kawamata-Shokurov [1]), and it also extends the known fact that the existence of the Cutkosky-Kawamata-Moriwaki decomposition for a log canonical divisor of general type implies the finite generation of the log canonical ring (Kawamata [2]). …”