# Fundamental Theorems for the Log Minimal Model Program

### Osamu Fujino

Osaka University, Japan

## Abstract

In this paper, we prove the cone theorem and the contraction theorem for pairs $(X,B)$, where $X$ is a normal variety and $B$ is an effective $R$-divisor on $X$ such that $K_{X}+B$ is $R$-Cartier.

## Cite this article

Osamu Fujino, Fundamental Theorems for the Log Minimal Model Program. Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, pp. 727–789

DOI 10.2977/PRIMS/50