The N\'eron model of a higher-dimensional Lagrangian fibration

TL;DR

This paper constructs Néron models for higher-dimensional Lagrangian fibrations in symplectic varieties, extending classical concepts from elliptic fibrations and analyzing their automorphism schemes.

Contribution

It introduces the construction of Néron models for abelian fibrations in higher dimensions, addressing complexities due to flops and automorphisms.

Findings

Constructed Néron models for abelian fibrations over higher-dimensional bases.

Identified conditions under which the fibration is a torsor under a smooth group scheme.

Revisited and extended known results in the context of symplectic varieties.

Abstract

Let $\pi : X \to B$ be a projective Lagrangian fibration of a smooth symplectic variety $X$ to a smooth variety $B$. Denote the complement of the discriminant locus by $B_0 = B \setminus \operatorname{Disc}(\pi)$, its preimage by $X_0 = \pi^{-1}(B_0)$, and the complement of the critical locus by $X' = X \setminus \operatorname{Sing}(\pi)$. Under an assumption that the morphism $X' \to B$ is surjective, we construct (1) the N\'eron model of the abelian fibration $\pi_0 : X_0 \to B_0$ and (2) the N\'eron model of its automorphism abelian scheme $\operatorname{Aut}^{\circ}_{\pi_0} \to B_0$. Contrary to the case of elliptic fibrations, $X'$ may not be the N\'eron 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.