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

Abstract: Let $M$ be a holomorphic symplectic K\"ahler manifold equipped with a Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is equipped with a special K\"ahler structure, that is, a K\"ahler structure $(I, g, \omega)$ 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 $\pi$ and smoothly projects to $\pi(Z)$ is a toric fibration over its image $\pi(Z)$ in $B$, and this image is also special K\"ahler. This answers a question of N. Hitchin related to Kapustin-Witten BBB/BAA duality.