Parabolic automorphisms of hyperkahler manifolds

TL;DR

This paper studies parabolic automorphisms of hyperkahler manifolds, proving their ergodic action on fibers of Lagrangian fibrations and extending known results from K3 surfaces to higher dimensions.

Contribution

It establishes ergodic behavior of parabolic automorphisms on hyperkahler manifolds and generalizes previous results from K3 surfaces.

Findings

Parabolic automorphisms act ergodically on Lagrangian fibers.

Invariant Lagrangian fibrations are automatic under the hyperkahler SYZ conjecture.

Groups generated by two such automorphisms act ergodically on the manifold.

Abstract

A parabolic automorphism of a hyperkahler manifold is a holomorphic automorphism acting on $H^2(M)$ by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangian fibration acts on its fibers ergodically. The invariance of a Lagrangian fibration is automatic for manifolds satisfying the hyperkahler SYZ conjecture; this includes all known examples of hyperkahler manifolds. When there are two parabolic automorphisms preserving two distinct Lagrangian fibration, it follows that the group they generate acts on $M$ ergodically. Our results generalize those obtained by S. Cantat for K3 surfaces.