Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Abstract

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold $X$, and let $\varpi :\hat{\mathcal{A}}\to M$ be the relative Albanese over $M$. We prove that $\hat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\hat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\hat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.

Final version