Integrable systems from intermediate Jacobians of 5-folds

TL;DR

This paper offers a straightforward Hodge-theoretic proof that the relative intermediate Jacobian of a universal family of cubic 5-folds extending a given cubic 4-fold forms a Lagrangian fibration, linking algebraic geometry and integrable systems.

Contribution

It provides a new, simplified proof of a result connecting intermediate Jacobians of cubic 5-folds to integrable systems, expanding understanding of their geometric structure.

Findings

The relative intermediate Jacobian of the universal family is a Lagrangian fibration.

The proof is based on Hodge theory, simplifying previous approaches.

Establishes a link between cubic 4-folds and integrable systems.

Abstract

Given a cubic 4-fold $Y$, we provide an easy Hodge-theoretic proof of the following result of Iliev--Manivel: the relative intermediate Jacobian of the universal family of cubic 5-folds $Z$ extending $Y$ is a Lagrangian fibration.