Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kahler geometry

Abstract

Let $M$ be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration $π$ with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure $(I, g, ω)$ and a symplectic flat connection $\nabla$ such that the metric $g$ is locally the Hessian of a function. We prove that any Lagrangian subvariety $Z\subset M$ which intersects smooth fibers of $π$ and smoothly projects to $π(Z)$ is a toric fibration over its image $π(Z)$ in $B$, and this image is also special Kähler. This answers a question of N. Hitchin related to Kapustin-Witten BBB/BAA duality.