Shokurov’s Reduction Theorem to Pre Limiting Flips

This is an expanded version of a talk given in the Fall 2007 Forschunggseminar at the Universität Duisburg-Essen. It is written for non-specialists, and hence might contain more details as experts would think necessary. Naturally, no claims regarding originality are made. The purpose of this short note is to provide a mostly self-contained exposition of Shokurov's reduction theorem. We will follow closely the presentations found in [8, Chapter 18], [1, Section 4.3] and an early version of [2] 1. In what follows X will denote a normal variety of dimension n. Definition 0.1. (pre limiting flips and elementary pre limiting flips) Let (X, ∆) be a dlt pair, f : X → Z a small extremal contraction. Assume that (a) −(K X + ∆) is f-ample 2 (b) there exists an irreducible component S ⊆ ∆ such that S is f-negative. Then f is called a pre limiting flipping contraction, and the flip of f (if it exists) is called a pre limiting flip. If in addition one assumes that X is Q-factorial and ρ(X/Z) = 1, then f is called an elementary pre limiting flipping contraction, and its flip an elementary pre limiting flip. In this note we will always refer to 'pre limiting' as 'pl'.