The Néron model of a higher-dimensional Lagrangian fibration

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 $B_0=B\setminus \mathrm{Disc}(\pi )$, its preimage by $X_0={\pi }^{-1}(B_0)$, and the complement of the critical locus by $X^{\prime }=X\setminus \mathrm{Sing}(\pi )$. Under an assumption that the morphism $X^{\prime }\to B$ is surjective, we construct (1) the Néron model of the abelian fibration ${\pi }_0:X_0\to B_0$ and (2) the Néron model of its automorphism abelian scheme ${\mathrm{Aut}}_{{\pi }_0}^{\circ }\to B_0$. Contrary to the case of elliptic fibrations, $X^{\prime }$ 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 $X^{\prime }\to B$ is a torsor under a smooth group scheme and also revisit some known results in the literature.

86 pages; minor revision