Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

TL;DR

This paper demonstrates that the relative Albanese and Picard over the moduli space of complex Lagrangian submanifolds in a hyperKähler manifold naturally acquire holomorphic symplectic structures, revealing integrable systems.

Contribution

It establishes the existence of natural holomorphic symplectic structures on the relative Albanese and Picard over the moduli space, and shows these define integrable systems.

Findings

The relative Albanese has a natural holomorphic symplectic structure.

The projection to the moduli space defines a completely integrable system.

Fibers of the projection are complex Lagrangian submanifolds.

Abstract

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