Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The log minimal model program for Kähler $3$-folds

Omprokash Das, Christopher Hacon

Abstract

In this article we show that the Log Minimal Model Program for $\mathbb{Q}$-factorial dlt pairs $(X, B)$ on a compact Kähler $3$-fold holds. More specifically, we show that after finitely many divisorial contractions and flips we obtain either a (log) minimal model or a Mori fiber space. We also prove a base point free theorem Kähler $3$-folds.

The log minimal model program for Kähler $3$-folds

Abstract

In this article we show that the Log Minimal Model Program for -factorial dlt pairs on a compact Kähler -fold holds. More specifically, we show that after finitely many divisorial contractions and flips we obtain either a (log) minimal model or a Mori fiber space. We also prove a base point free theorem Kähler -folds.

Paper Structure

This paper contains 12 sections, 55 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projectivity criteria
  4. Resolution of singularities and Kawamata-Viehweg vanishing theorem
  5. MMP for Kähler $3$-folds
  6. Technical lemmas
  7. Minimal model program for dlt pseudo-effective pairs
  8. The minimal model program for uniruled pairs
  9. Cone Theorem
  10. Existence of divisorial contractions and flips
  11. Towards the Existence of Mori fiber spaces
  12. Main Theorems

Key Result

Theorem 1.1

Let $(X,B)$ be a dlt pair where $X$ is a $\mathbb{Q}$-factorial compact Kähler $3$-fold. If $K_X+B$ is pseudo-effective, then there exists a finite sequence of flips and divisorial contractions such that $K_{X_n}+\phi _* B$ is nef.

Theorems & Definitions (139)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.5
  • Conjecture 1.6: Base point freeness
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 129 more