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Dual complex of log Fano pairs and its application to Witt vector cohomology

Yusuke Nakamura

Abstract

We prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro-Fujino type and a rational point formula in dimension three.

Dual complex of log Fano pairs and its application to Witt vector cohomology

Abstract

We prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro-Fujino type and a rational point formula in dimension three.

Paper Structure

This paper contains 13 sections, 27 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Results on minimal model program
  5. Dual complexes
  6. Results on the Witt vector cohomologies
  7. Dual complex of weak Fano varieties
  8. Dual complex of dlt pairs with a Mori fiber space structure
  9. Dual complex of weak Fano varieties
  10. Vanishing theorem on log canonical Fano varieties
  11. Non-klt locus of log Fano three-folds
  12. Vanishing theorem of Witt vector cohomology of Ambro-Fujino type
  13. Rational point formula

Key Result

Theorem 1.1

Let $(X, \Delta)$ be a projective pair over an algebraic closed field of characteristic zero. Assume that $-(K_X + \Delta)$ is nef and big. Then for any dlt blow-up $g: (Y, \Delta _Y) \to (X, \Delta)$, the dual complex $\mathcal{D}(\Delta _Y ^{\ge 1})$ is contractible, where we define $\Delta _Y$ by

Theorems & Definitions (63)

  • Theorem 1.1: $=$ Theorem \ref{['thm:nefbig0']}
  • Theorem 1.2: $=$ Theorem \ref{['thm:nefbig']}
  • Theorem 1.3: $=$ Theorem \ref{['thm:WAFV']}
  • Theorem 1.4: $=$ Theorem \ref{['thm:RPF']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: Kol13, Fuj07, DH16
  • proof
  • Definition 2.4
  • Theorem 2.5
  • ...and 53 more