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Working paper

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

Verbitsky M., Kamenova L.
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 toπ(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.