Hermitian-symplectic and Kahler structures on degenerate twistor deformations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916696385650688 |
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| author | Soldatenkov, Andrey Verbitsky, Misha |
| author_facet | Soldatenkov, Andrey Verbitsky, Misha |
| contents | Let $(M, Ω)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $π: M \to B$, and $η$ a closed $(1,1)$-form on $B$. Then $Ω+ π^* η$ is a holomorphically symplectic form on a complex manifold which is called the degenerate twistor deformation of $M$. We prove that degenerate twistor deformations of compact holomorphically symplectic Kähler manifolds are also Kähler. First, we prove that degenerate twistor deformations are Hermitian symplectic, that is, tamed by a symplectic form; this is shown using positive currents and an argument based on the Hahn--Banach theorem, originally due to Sullivan. Then we apply a version of Huybrechts's theorem showing that two non-separated points in the Teichmüller space of holomorphically symplectic manifolds correspond to bimeromorphic manifolds if they are Hermitian symplectic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07867 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hermitian-symplectic and Kahler structures on degenerate twistor deformations Soldatenkov, Andrey Verbitsky, Misha Algebraic Geometry Complex Variables Differential Geometry 53C26, 14J42 Let $(M, Ω)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $π: M \to B$, and $η$ a closed $(1,1)$-form on $B$. Then $Ω+ π^* η$ is a holomorphically symplectic form on a complex manifold which is called the degenerate twistor deformation of $M$. We prove that degenerate twistor deformations of compact holomorphically symplectic Kähler manifolds are also Kähler. First, we prove that degenerate twistor deformations are Hermitian symplectic, that is, tamed by a symplectic form; this is shown using positive currents and an argument based on the Hahn--Banach theorem, originally due to Sullivan. Then we apply a version of Huybrechts's theorem showing that two non-separated points in the Teichmüller space of holomorphically symplectic manifolds correspond to bimeromorphic manifolds if they are Hermitian symplectic. |
| title | Hermitian-symplectic and Kahler structures on degenerate twistor deformations |
| topic | Algebraic Geometry Complex Variables Differential Geometry 53C26, 14J42 |
| url | https://arxiv.org/abs/2407.07867 |