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A note on varieties of non-negative Kodaira dimension with polarized self maps

Ankit Rai

TL;DR

We address smooth projective varieties $X$ with non-negative Kodaira dimension, a $k$-rational point, and a polarized self-map $(\phi,L)$, proving that $X$ is a finite free quotient of an abelian variety. The approach combines lifting to characteristic zero, Albanese morphisms, and analysis of the étale fundamental group to deduce finiteness of the Albanese map and the existence of a finite étale cover by an abelian variety, handling both char zero and positive char under suitable hypotheses. In characteristic $p$, the argument reduces to the case where the étale fundamental group is virtually abelian and uses spreads-out and specialization to transfer the result from $\overline{\mathbb{F}}_p$ to the base field. The paper also extends these conclusions to non-algebraically closed base fields and sketches a path toward the int-amplified setting, connecting to Lang–Serre, Beauville, and Nakayama–Zhang-type results in algebraic dynamics.

Abstract

In this note we prove that a smooth projective variety (defined over a field $k$) of non-negative Kodaira dimension that has a $k$-rational point and a polarized self map must be a finite free quotient of an abelian variety.

A note on varieties of non-negative Kodaira dimension with polarized self maps

TL;DR

We address smooth projective varieties with non-negative Kodaira dimension, a -rational point, and a polarized self-map , proving that is a finite free quotient of an abelian variety. The approach combines lifting to characteristic zero, Albanese morphisms, and analysis of the étale fundamental group to deduce finiteness of the Albanese map and the existence of a finite étale cover by an abelian variety, handling both char zero and positive char under suitable hypotheses. In characteristic , the argument reduces to the case where the étale fundamental group is virtually abelian and uses spreads-out and specialization to transfer the result from to the base field. The paper also extends these conclusions to non-algebraically closed base fields and sketches a path toward the int-amplified setting, connecting to Lang–Serre, Beauville, and Nakayama–Zhang-type results in algebraic dynamics.

Abstract

In this note we prove that a smooth projective variety (defined over a field ) of non-negative Kodaira dimension that has a -rational point and a polarized self map must be a finite free quotient of an abelian variety.
Paper Structure (11 sections, 9 theorems, 6 equations)

This paper contains 11 sections, 9 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:S-is-0-dim-char-p']}
  3. Preliminary lemmas
  4. Completing the proof of Theorem \ref{['thm:S-is-0-dim-char-p']}
  5. Proof under the assumption $k = \overline{\mathbb{F}}_p$
  6. Reduction to $k = \overline{\mathbb{F}}_p$
  7. Two lemma on the hypothesis of the Theorem
  8. Results over non-algebraically closed fields
  9. Preliminary lemmas
  10. Char(k)=0
  11. Char(k)$>$0

Key Result

Theorem 1.1

Let $X$ be a smooth projective variety of non-negative Kodaira dimension over an algebraically closed field $k$ of characteristic $p>0$, $L$ an ample line bundle on $X$ and $\phi : X \rightarrow X$ a separable map such that $\phi^*L \cong L^{\otimes d}$ for an integer $d>1$. Suppose that the étale f

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2: Katz-Lang-81
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4