Some constraints on positive entropy automorphisms of smooth threefolds

TL;DR

The paper investigates conditions under which smooth threefolds with automorphisms of positive entropy exhibit certain geometric properties, showing that such automorphisms are constrained by the structure of the threefold.

Contribution

It establishes that, after iteration, positive entropy automorphisms on smooth threefolds must satisfy one of three specific conditions, linking dynamics with algebraic geometry.

Findings

Either the canonical class is numerically trivial, the automorphism is imprimitive, or it is not dynamically minimal.

No primitive automorphism of positive entropy exists on varieties obtained by blow-ups of certain threefolds.

Provides examples of terminal threefolds with infinitely many extremal rays, illustrating limitations of the method.

Abstract

Suppose that $X$ is a smooth, projective threefold over $\mathbb C$ and that $\phi : X \to X$ is an automorphism of positive entropy. We show that one of the following must hold, after replacing $\phi$ by an iterate: i) the canonical class of $X$ is numerically trivial; ii) $\phi$ is imprimitive; iii) $\phi$ is not dynamically minimal. As a consequence, we show that if a smooth threefold $M$ does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of $M$ can admit a primitive automorphism of positive entropy. In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a non-uniruled, terminal threefold $X$ with infinitely many $K_X$-negative extremal rays on $NE(X)$.