Kummer Rigidity for Hyperkähler Automorphisms

Abstract

We show that a holomorphic automorphism on a projective hyperkähler 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 hyperkähler manifold is shown to be flat. A result in (Greb-Kebekus-Peternell 2016) then implies that our hyperkähler manifold is birational to a torus quotient, giving the Kummer example structure.

arXiv commentTo appear in Journal Of Modern Dynamics, Volume 20, 2024