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Endomorphisms of Fano 3-folds and log Bott vanishing

Burt Totaro

TL;DR

This work develops a logarithmic generalization of Bott vanishing for endomorphisms with totally invariant divisors, proving that log Bott vanishing holds for normal projective varieties carrying an int-amplified endomorphism of degree invertible in the base field. By establishing a robust trace/compatibility framework for reflexive differentials with log poles, the authors extend Bott vanishing techniques to a broader class of varieties, including Fano 3-folds, and apply these results to characterize which varieties are images of toric varieties. They reprove and extend known results in characteristic zero to positive characteristic, showing that smooth Fano 3-folds with such endomorphisms must be toric, and that images of toric varieties remain toric under degree-invertible morphisms. The approach also yields a parallel classification for del Pezzo surfaces, and provides a systematic method to rule out non-toric cases via log Bott vanishing, thereby advancing the toric-detection program in algebraic dynamics and birational geometry.

Abstract

Kawakami and the author showed that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. That was a new way to analyze which varieties have nontrivial endomorphisms. In this paper, we extend that result to a logarithmic version of Bott vanishing for an endomorphism with a totally invariant divisor. We apply this to Fano 3-folds. Meng-Zhang-Zhong showed that the only smooth complex Fano 3-folds that admit an int-amplified endomorphism are the toric ones. Also, Achinger-Witaszek-Zdanowicz showed that the only smooth complex Fano 3-folds that are images of toric varieties are the toric ones. Using log Bott vanishing, we reprove both results and extend them to characteristic p, for morphisms of degree prime to p.

Endomorphisms of Fano 3-folds and log Bott vanishing

TL;DR

This work develops a logarithmic generalization of Bott vanishing for endomorphisms with totally invariant divisors, proving that log Bott vanishing holds for normal projective varieties carrying an int-amplified endomorphism of degree invertible in the base field. By establishing a robust trace/compatibility framework for reflexive differentials with log poles, the authors extend Bott vanishing techniques to a broader class of varieties, including Fano 3-folds, and apply these results to characterize which varieties are images of toric varieties. They reprove and extend known results in characteristic zero to positive characteristic, showing that smooth Fano 3-folds with such endomorphisms must be toric, and that images of toric varieties remain toric under degree-invertible morphisms. The approach also yields a parallel classification for del Pezzo surfaces, and provides a systematic method to rule out non-toric cases via log Bott vanishing, thereby advancing the toric-detection program in algebraic dynamics and birational geometry.

Abstract

Kawakami and the author showed that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. That was a new way to analyze which varieties have nontrivial endomorphisms. In this paper, we extend that result to a logarithmic version of Bott vanishing for an endomorphism with a totally invariant divisor. We apply this to Fano 3-folds. Meng-Zhang-Zhong showed that the only smooth complex Fano 3-folds that admit an int-amplified endomorphism are the toric ones. Also, Achinger-Witaszek-Zdanowicz showed that the only smooth complex Fano 3-folds that are images of toric varieties are the toric ones. Using log Bott vanishing, we reprove both results and extend them to characteristic p, for morphisms of degree prime to p.
Paper Structure (7 sections, 12 theorems, 18 equations, 1 table)

This paper contains 7 sections, 12 theorems, 18 equations, 1 table.

Table of Contents

  1. Notation
  2. The trace of a differential form with log poles
  3. Endomorphisms and log Bott vanishing
  4. Morphisms
  5. Images of toric varieties: log Bott vanishing
  6. Endomorphisms of del Pezzo surfaces and Fano 3-folds
  7. Images of toric varieties: del Pezzo surfaces and Fano 3-folds

Key Result

Lemma 2.1

Let $f\colon Y\to X$ be a finite surjective morphism of normal varieties over a perfect field $k$. Let $A$ be a Weil divisor on $X$. Let $E_X$ and $D_X$ be reduced divisors on $X$ with $0\leq E_X\leq D_X$. Let $D_Y$ be the sum of the components of $f^{-1}(D_X)$ along which the ramification degree $e with composition equal to multiplication by $\deg(f)$.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Theorem 3.1
  • Remark 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • ...and 14 more