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Polarized endomorphisms of Fano varieties with complements

Joaquín Moraga, José Ignacio Yáñez, Wern Yeong

Abstract

Let $X$ be a Fano type variety and $(X,Δ)$ be a log Calabi-Yau pair with $Δ$ a Weil divisor. If $(X,Δ)$ admits a polarized endomorphism, then we show that $(X,Δ)$ is a finite quotient of a toric pair. Along the way, we prove that a klt Calabi-Yau pair $(X,Δ)$ with standard coefficients that admits a polarized endomorphism is the quotient of an abelian variety.

Polarized endomorphisms of Fano varieties with complements

Abstract

Let be a Fano type variety and be a log Calabi-Yau pair with a Weil divisor. If admits a polarized endomorphism, then we show that is a finite quotient of a toric pair. Along the way, we prove that a klt Calabi-Yau pair with standard coefficients that admits a polarized endomorphism is the quotient of an abelian variety.
Paper Structure (7 sections, 15 theorems, 48 equations)

This paper contains 7 sections, 15 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof
  3. Preliminaries
  4. Lifting polarized endomorphisms to finite covers
  5. Varieties with torus actions
  6. Proof of the Main Theorems
  7. Examples and Questions

Key Result

Theorem 1

Let $X$ be a Fano type variety and let $(X,\Delta)$ be a log Calabi--Yau pair with $\Delta$ reduced. If the pair $(X,\Delta)$ admits a polarized endomorphism, then $(X,\Delta)$ is a finite quotient of a toric log Calabi--Yau pair.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 29 more