Shafarevich-Tate groups of holomorphic Lagrangian fibrations II
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917112709120000 |
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| author | Abasheva, Anna |
| author_facet | Abasheva, Anna |
| contents | Let $X$ be a compact hyperkähler manifold with a Lagrangian fibration $π\colon X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration $π^φ\colon X^φ\to B$ which is isomorphic to $π$ locally over the base. In particular, $π^φ$ has the same fibers as $π$. A twist $X^φ$ corresponds to an element $φ$ in the Shafarevich-Tate group of $X$. We show that $X^φ$ is Kähler when a multiple of $φ$ lies in the connected component of unity of the Shafarevich-Tate group and give a necessary condition for $X^φ$ to be bimeromorphic to a Kähler manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09178 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shafarevich-Tate groups of holomorphic Lagrangian fibrations II Abasheva, Anna Algebraic Geometry Complex Variables 14J42, 14F25, 32Q15, 32J25 Let $X$ be a compact hyperkähler manifold with a Lagrangian fibration $π\colon X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration $π^φ\colon X^φ\to B$ which is isomorphic to $π$ locally over the base. In particular, $π^φ$ has the same fibers as $π$. A twist $X^φ$ corresponds to an element $φ$ in the Shafarevich-Tate group of $X$. We show that $X^φ$ is Kähler when a multiple of $φ$ lies in the connected component of unity of the Shafarevich-Tate group and give a necessary condition for $X^φ$ to be bimeromorphic to a Kähler manifold. |
| title | Shafarevich-Tate groups of holomorphic Lagrangian fibrations II |
| topic | Algebraic Geometry Complex Variables 14J42, 14F25, 32Q15, 32J25 |
| url | https://arxiv.org/abs/2407.09178 |