Parabolic automorphisms of hyperkahler manifolds

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.

34 pages, version 4.0, accepted by JMPA, many changes

References in corpus (2)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• Base manifolds for fibrations of projective irreducible symplectic manifolds