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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.21193 |
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| _version_ | 1866915286870917120 |
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| author | Kim, Yoon-Joo |
| author_facet | Kim, Yoon-Joo |
| contents | Let $π: X \to B$ be a projective Lagrangian fibration of a smooth symplectic variety $X$ to a smooth variety $B$. Denote the complement of the discriminant locus by $B_0 = B \setminus \operatorname{Disc}(π)$, its preimage by $X_0 = π^{-1}(B_0)$, and the complement of the critical locus by $X' = X \setminus \operatorname{Sing}(π)$. Under an assumption that the morphism $X' \to B$ is surjective, we construct (1) the Néron model of the abelian fibration $π_0 : X_0 \to B_0$ and (2) the Néron model of its automorphism abelian scheme $\operatorname{Aut}^{\circ}_{π_0} \to B_0$. Contrary to the case of elliptic fibrations, $X'$ may not be the Néron model of $X_0$; this is precisely because of the existence of flops in higher-dimensional symplectic varieties. Using such techniques, we analyze when $X' \to B$ is a torsor under a smooth group scheme and also revisit some known results in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Néron model of a higher-dimensional Lagrangian fibration Kim, Yoon-Joo Algebraic Geometry Let $π: X \to B$ be a projective Lagrangian fibration of a smooth symplectic variety $X$ to a smooth variety $B$. Denote the complement of the discriminant locus by $B_0 = B \setminus \operatorname{Disc}(π)$, its preimage by $X_0 = π^{-1}(B_0)$, and the complement of the critical locus by $X' = X \setminus \operatorname{Sing}(π)$. Under an assumption that the morphism $X' \to B$ is surjective, we construct (1) the Néron model of the abelian fibration $π_0 : X_0 \to B_0$ and (2) the Néron model of its automorphism abelian scheme $\operatorname{Aut}^{\circ}_{π_0} \to B_0$. Contrary to the case of elliptic fibrations, $X'$ may not be the Néron model of $X_0$; this is precisely because of the existence of flops in higher-dimensional symplectic varieties. Using such techniques, we analyze when $X' \to B$ is a torsor under a smooth group scheme and also revisit some known results in the literature. |
| title | The Néron model of a higher-dimensional Lagrangian fibration |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.21193 |