Compactifying Lagrangian fibrations

TL;DR

This paper develops a framework for compactifying Lagrangian fibrations of geometric origin using holomorphic symplectic varieties, providing criteria and applications to various fibrations and their algebraic structures.

Contribution

It introduces a general compactification framework for quasi-projective Lagrangian fibrations, including a criterion and applications to algebraic and geometric structures.

Findings

Existence of holomorphic symplectic compactifications for certain Lagrangian fibrations.

Application of the framework to fibrations with local sections over open subsets.

Construction of smooth commutative algebraic groups acting on the fibrations.

Abstract

We suggest a general framework for compactifing quasi-projective Lagrangian fibrations of geometric origin by holomorphic symplectic varieties. This framework includes a compactification criterion, which we then apply to various fibrations of geometric origin, and a discussion on holomorphic forms that are defined via correspondences in geometric examples. As application, we show that given a Lagrangian fibration $X \to B$ admitting local sections over an open subset $V$ with codimension $\ge 2$ complement, there exists a (possibly singular) holomorphic symplectic compactification of the Albanese fibration $A \to V$ (which we show exists as a smooth commutative algebraic group with connected fibers acting on $X_{V}$), as well as of any other torsor over $A$, or over any smooth commutative group scheme over $B$ with connected fibers that is isogenous to $A$.