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Log Calabi--Yau pairs of complexity zero and arbitrary index

Joshua Enwright, Fernando Figueroa

Abstract

In this article, we give a characterization of log Calabi--Yau pairs of complexity zero and arbitrary index. As an application, we show that a log Calabi--Yau pair of birational complexity zero admits a crepant birational model which is a generalized Bott tower.

Log Calabi--Yau pairs of complexity zero and arbitrary index

Abstract

In this article, we give a characterization of log Calabi--Yau pairs of complexity zero and arbitrary index. As an application, we show that a log Calabi--Yau pair of birational complexity zero admits a crepant birational model which is a generalized Bott tower.

Paper Structure

This paper contains 21 sections, 49 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Geometry of log canonical centers
  3. Behavior with respect to contractions
  4. Preliminaries
  5. Log pairs
  6. Contractions
  7. Mori Dream Spaces
  8. The relative MMP
  9. Toric geometry
  10. Complexity
  11. Degenerate divisors
  12. Canonical bundle formula
  13. Toric boundary arrangements
  14. Associated toric divisors
  15. Toric boundary arrangements
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(X,B)$ be a log canonical pair with $-(K_X+B)$ nef. Then $c(X,B)\geq 0.$ If $c(X,B)<1,$ then there is a toric log Calabi--Yau pair $(X,\Delta)$ with $\lfloor B \rfloor \leq \Delta$ and all but possibly $\lfloor 2c(X,B)\rfloor$ components of $\Delta$ in the support of $B.$

Theorems & Definitions (128)

  • Theorem 1: BMSZ18
  • Corollary 1
  • Theorem 2
  • Definition 1
  • Theorem 3
  • Definition 2
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 118 more