The Néron model of a higher-dimensional Lagrangian fibration

2024/10/28 by Yoon-Joo Kim, Kim, Yoon-Joo · 4 citations
Engineering · #Algebraic Geometry (math.AG) #Elasticity and Material Modeling #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2410.21193

openalex publication_date 2024/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let π: X → 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 B₀ = B ∖ Disc(π), its preimage by X₀ = π^-1(B₀), and the complement of the critical locus by X' = X ∖ Sing(π). Under an assumption that the morphism X' → B is surjective, we construct (1) the Néron model of the abelian fibration π₀ : X₀ → B₀ and (2) the Néron model of its automorphism abelian scheme Aut^∘_π0 → B₀. Contrary to the case of elliptic fibrations, X' may not be the Néron model of X₀; this is precisely because of the existence of flops in higher-dimensional symplectic varieties. Using such techniques, we analyze when X' → B is a torsor under a smooth group scheme and also revisit some known results in the literature.

Cited by

• Torsor Neron models of hyperkahler manifolds
• Higher direct images of dualizing sheaves III
• Hilbert schemes of elliptic surfaces: group actions and derived categories
• The Tate-Shafarevich group of a polarised K3 surface

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