Sections of Lagrangian fibrations on holomorphic symplectic manifolds
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914095473623040 |
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| author | Bogomolov, Fedor Kamenova, Ljudmila Verbitsky, Misha |
| author_facet | Bogomolov, Fedor Kamenova, Ljudmila Verbitsky, Misha |
| contents | Let $M$ be a holomorphically symplectic manifold, equipped with a Lagrangian fibration $π:\; M \to X$. A degenerate twistor deformation (sometimes also called ``a Tate-Shafarevich twist'') is a family of holomorphically symplectic structures on $M$ parametrized by $H^{1,1}(X)$. All members of this family are equipped with a holomorphic Lagrangian projection to $X$, and their fibers are isomorphic to the fibers of $π$. Assume that $M$ is a compact hyperkahler manifold of maximal holonomy, and the general fiber of the Lagrangian projection $π$ is primitive (that is, not divisible) in integer homology. We also assume that $π$ has reduced fibers in codimension 1. Then $M$ has a degenerate twistor deformation $M'$ such that the Lagrangian projection $π:\; M' \to X$ admits a meromorphic section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sections of Lagrangian fibrations on holomorphic symplectic manifolds Bogomolov, Fedor Kamenova, Ljudmila Verbitsky, Misha Algebraic Geometry Complex Variables Differential Geometry 53C26, 14J42 Let $M$ be a holomorphically symplectic manifold, equipped with a Lagrangian fibration $π:\; M \to X$. A degenerate twistor deformation (sometimes also called ``a Tate-Shafarevich twist'') is a family of holomorphically symplectic structures on $M$ parametrized by $H^{1,1}(X)$. All members of this family are equipped with a holomorphic Lagrangian projection to $X$, and their fibers are isomorphic to the fibers of $π$. Assume that $M$ is a compact hyperkahler manifold of maximal holonomy, and the general fiber of the Lagrangian projection $π$ is primitive (that is, not divisible) in integer homology. We also assume that $π$ has reduced fibers in codimension 1. Then $M$ has a degenerate twistor deformation $M'$ such that the Lagrangian projection $π:\; M' \to X$ admits a meromorphic section. |
| title | Sections of Lagrangian fibrations on holomorphic symplectic manifolds |
| topic | Algebraic Geometry Complex Variables Differential Geometry 53C26, 14J42 |
| url | https://arxiv.org/abs/2407.07877 |