Complex Lagrangians in a hyperKahler manifold and the relative Albanese

Abstract

Let M be the moduli space of complex Lagrangian submanifolds of a hyperKahler manifold X, and let omega : (A) over cap -> M be the relative Albanese over M. We prove that (A) over cap has a natural holomorphic symplectic structure. The projection omega defines a completely integrable structure on the symplectic manifold (A) over cap. In particular, the fibers of omega are complex Lagrangians with respect to the symplectic form on (A) over cap. We also prove analogous results for the relative Picard over M.