Hitchin and Calabi-Yau integrable systems via variations of Hodge structures

Understand

A complex integrable system determines a family of complex tori over a Zariski-open and dense subset in its base.

• This family in turn yields an integral variation of Hodge structures of weight $\pm 1$.
• In this paper, we study the converse of this procedure.
• Starting from an integral variation of Hodge structures of weight $\pm 1$, we give a criterion for when its associated family of complex tori carries a Lagrangian structure, i.e.