Parabolic automorphisms of hyperkahler manifolds
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866910454775808000 |
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| author | Amerik, Ekaterina Verbitsky, Misha |
| author_facet | Amerik, Ekaterina Verbitsky, Misha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_01951 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Parabolic automorphisms of hyperkahler manifolds Amerik, Ekaterina Verbitsky, Misha Algebraic Geometry Differential Geometry Dynamical Systems 53C26, 32G20, 37F80 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. |
| title | Parabolic automorphisms of hyperkahler manifolds |
| topic | Algebraic Geometry Differential Geometry Dynamical Systems 53C26, 32G20, 37F80 |
| url | https://arxiv.org/abs/2112.01951 |