Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds

TL;DR

This paper studies the structure of non-isomorphic surjective endomorphisms on smooth projective threefolds, proving their behavior aligns with minimal model programs and confirming the Kawaguchi-Silverman conjecture in this context.

Contribution

It establishes that such endomorphisms become equivariant after iteration and verifies the Kawaguchi-Silverman conjecture for threefolds, advancing understanding of their dynamical properties.

Findings

Minimal model program becomes $f$-equivariant after iteration.

Kawaguchi-Silverman conjecture proven for all non-isomorphic surjective endomorphisms of threefolds.

Reduces Zariski dense orbit conjecture to terminal threefolds with $f$-equivariant Fano contractions.

Abstract

Let $f:X\to X$ be a non-isomorphic (i.e., $\text{deg } f>1$) surjective endomorphism of a smooth projective threefold $X$. We prove that any birational minimal model program becomes $f$-equivariant after iteration, provided that $f$ is $\delta$-primitive. Here $\delta$-primitive means that there is no $f$-equivariant (after iteration) dominant rational map $\pi:X\dashrightarrow Y$ to a positive lower-dimensional projective variety $Y$ such that the first dynamical degree remains unchanged. This way, we further determine the building blocks of $f$. As the first application, we prove the Kawaguchi-Silverman conjecture for every non-isomorphic surjective endomorphism of a smooth projective threefold. As the second application, we reduce the Zariski dense orbit conjecture for $f$ to a terminal threefold with only $f$-equivariant Fano contractions.