A hyper-K\"ahler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold

TL;DR

This paper constructs a smooth, compact hyper-K"ahler manifold from the intermediate Jacobian fibration of a cubic fourfold, confirming a conjecture and linking it to known hyper-K"ahler examples.

Contribution

It provides a new compactification of the intermediate Jacobian fibration, resulting in a hyper-K"ahler manifold deformation equivalent to O'Grady's example.

Findings

Constructed a smooth projective compactification of the intermediate Jacobian fibration.

Proved the resulting manifold is deformation equivalent to O'Grady's hyper-K"ahler manifold.

Confirmed a conjecture by Markushevich regarding the geometry of cubic fourfolds.

Abstract

For a general cubic fourfold, it was observed by Donagi and Markman that the relative intermediate Jacobian fibration associated to the family of its hyperplane sections carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification of the intermediate Jacobian fibration giving a ten-dimensional compact hyper-K\"ahler manifold, which we then show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Markushevich.