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On the termination of flips for log canonical generalized pairs

Guodu Chen, Nikolaos Tsakanikas

Abstract

We prove the termination of flips for 4-dimensional pseudo-effective NQC log canonical generalized pairs. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.

On the termination of flips for log canonical generalized pairs

Abstract

We prove the termination of flips for 4-dimensional pseudo-effective NQC log canonical generalized pairs. As main ingredients, we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs, and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.

Paper Structure

This paper contains 8 sections, 22 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized pairs
  4. Minimal and log canonical models of generalized pairs
  5. Flips and special termination for generalized pairs
  6. The difficulty
  7. Termination of Flips in Dimension 3
  8. Proofs of the Main Results

Key Result

Theorem 1.1

Let $(X/Z,B+M)$ be a $4$-dimensional NQC lc g-pair. If $K_X+B+M$ is pseudo-effective over $Z$, then any sequence of flips over $Z$ starting from $(X/Z,B+M)$ terminates.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 45 more