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

Let M be a holomorphic symplectic Kahler manifold equipped with a Lagrangian fibration $$\pi $$ with compact fibers. The base of this manifold is equipped with a special Kahler structure, that is, a Kahler 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 torus fibration over its image $$\pi (Z)$$ in B, and this image is also special Kahler. This answers a question of Nigel Hitchin related to Kapustin–Witten BBB/BAA duality.