Kummer Rigidity for Hyperk\"ahler Automorphisms

TL;DR

This paper proves that certain automorphisms with positive entropy on projective hyperk"ahler manifolds are necessarily Kummer examples, extending known results and showing these manifolds are birational to torus quotients.

Contribution

It extends Kummer rigidity results to hyperk"ahler automorphisms with positive entropy, using a novel approach involving Jensen's inequality and stability analysis.

Findings

Automorphisms with positive entropy are Kummer examples.

Hyperk"ahler manifolds with such automorphisms are birational to torus quotients.

The method involves uniform contraction and expansion rates of stable and unstable distributions.

Abstract

We show that a holomorphic automorphism on a projective hyperk\"ahler manifold that has positive topological entropy and has volume measure as the measure of maximal entropy, is necessarily a Kummer example, partially extending the analogous results in (Cantat-Dupont 2020)(Filip-Tosatti 2018) for complex surfaces. A trick with Jensen's inequality is used to show that stable and unstable distributions exhibit uniform rate of contraction and expansion, and with them our hyperk\"ahler manifold is shown to be flat. A result in (Greb-Kebekus-Peternell 2016) then implies that our hyperk\"ahler manifold is birational to a torus quotient, giving the Kummer example structure.