Department of MathematicsSingular fibers in minimal elliptic fibrations were classified by Kodaira and Néron in the 1960s. In his proof, Néron constructed and systematically used a special group scheme acting on an elliptic fibration. This group scheme is now called the Néron model.
A Lagrangian fibration is a higher-dimensional generalization of an elliptic fibration. Néron’s theory is restricted to 1-dimensional bases, so one cannot use Néron’s original approach to study higher-dimensional Lagrangian fibrations. The higher-dimensional analog of Néron’s definition was recently proposed by David Holmes. Quite unfortunately, Holmes also showed that such a generalized Néron model often fails to exist, even in simple cases.
In this talk, we show that Holmes’s generalized Néron model does exist for an arbitrary projective Lagrangian fibration of a smooth symplectic variety, under a single assumption that the Lagrangian fibration has no fully-nonreduced fibers. This generalizes Néron’s result to many higher-dimensional Lagrangian fibrations. Such a construction has several applications. First, it extends Ngô's results on Hitchin fibrations to many Lagrangian fibrations. Second, it allows Lagrangian fibrations to be considered as a minimal model-compactification of a smooth commutative group scheme-torsor. Third, it provides a tool to study birational behaviors of Lagrangian fibrations. Finally, the notion of a Tate-Shafarevich twist can be understood via the Néron model.