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

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 ∇ such that the metric g is locally the Hessian of a function. We prove that any Lagrangian subvariety Z⊂ M which intersects smooth fibers of π and smoothly projects to π(Z) is a torus fibration over its image π(Z) in B, and this image is also special Kähler. This answers a question of Nigel Hitchin related to Kapustin–Witten BBB/BAA duality.