Automorphisms of blowups of threefolds being Fano or having Picard number $1$

TL;DR

This paper investigates automorphisms of blowups of threefolds that are Fano or have Picard number 1, showing that most such automorphisms have equal first and second dynamical degrees and often zero entropy.

Contribution

It demonstrates that for most blowups of these threefolds, automorphisms have constrained dynamical degrees and zero entropy, extending understanding of automorphism behavior in these geometries.

Findings

Most automorphisms have equal first and second dynamical degrees.

Many blowups admit automorphisms of zero entropy.

Constraints on nef classes due to log-concavity and Chern class invariance.

Abstract

Let $X_0$ be a smooth projective threefold which is Fano or which has Picard number $1$. Let $\pi :X\rightarrow X_0$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow X_0$, on which any automorphism is of zero entropy. The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology classes. We will also discuss a possible application of these results to a threefold constructed by Kenji Ueno.