Parabolic automorphisms of hyperkähler manifolds

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