Introduction to the log minimal model program for log canonical pairs

TL;DR

This paper introduces the foundational aspects of the log minimal model program for log canonical pairs, extending key theorems and proving cone and contraction theorems for quasi-log varieties.

Contribution

It generalizes Kollár's vanishing and torsion-free theorems and establishes the cone and contraction theorems for quasi-log varieties and log canonical pairs.

Findings

Generalization of Kollár's theorems for embedded simple normal crossing pairs

Proof of cone and contraction theorems for quasi-log varieties

Foundation of the log minimal model program for log canonical pairs

Abstract

We describe the foundation of the log minimal model program for log canonical pairs according to Ambro's idea. We generalize Koll\'ar's vanishing and torsion-free theorems for embedded simple normal crossing pairs. Then we prove the cone and contraction theorems for quasi-log varieties, especially, for log canonical pairs.