Introduction

Abstract

Let X be a smooth complex projective variety. The minimal model program aims to show that if K _X is pseudo-effective (respectively if K _X is not pseudo-effective), then there exists a finite sequence

of well-understood birational maps known as flips and divisorial contractions such that $\bar X$ is a minimal model, i.e., $K_{\bar X}$ is nef (respectively $\bar X$ has the structure of a Mori fiber space, i.e., there is a morphism $f:\bar X \to Z$ such that $- K_{\bar X}$ is relatively ample over Z).