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arXiv · 2112.01951

Parabolic automorphisms of hyperkahler manifolds

Ekaterina Amerik·Misha Verbitsky
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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.

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Ekaterina Amerik, Misha Verbitsky. 2021-12-03. Parabolic automorphisms of hyperkahler manifolds. https://doi.org/10.1016/j.matpur.2023.09.006

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