Curves with decomposable normal vector bundles and automorphism groups

TL;DR

This paper studies conditions under which automorphisms of certain threefolds have zero entropy, focusing on properties preserved under blowups and providing explicit examples with controlled dynamical behavior.

Contribution

It introduces Property A and shows its stability under specific blowups, leading to new results on automorphism entropy and dynamical degrees for complex threefolds.

Findings

Automorphisms of threefolds with Property A have zero entropy.

Blowups at points or curves with certain properties preserve Property A.

Explicit examples demonstrate automorphisms with equal first and second dynamical degrees.

Abstract

If a smooth projective threefold $X$ satisfies a certain Property A (see below for definition), then any automorphism of $X$ has zero entropy. Let $Y$ be a smooth projective threefold satisfying Property A. Let $\pi :X\rightarrow Y$ be a blowup at either a point or at a smooth curve $C\subset Y$ with the following two properties: i) $c_1(Y).C$ is an odd number, and ii) the normal vector bundle $N_{C/Y} $ is decomposable. Then we show that $X$ also satisfies Property A. As a further application of Property A we prove the following result. Let $X_1$ be the blowup of $X_0=\mathbb{P}^3$ at a finite number of points, and let $X=X_2$ be the blowup of $X_1$ at a finite number of pairwise disjoint smooth curves (here the images of these curves in $X_0$ may intersect). Then any automorphism of $X$ has the same first and second dynamical degrees. Under some further conditions, then any automorphism of $X$ has zero entropy. The result is also valid for threefolds $X_0$ satisfying a certain condition on the second Chern class. Some explicit examples are given.