Hyper-Kaehler compactification of the intermediate Jacobian fibration of a cubic fourfold : the twisted case

TL;DR

This paper extends previous work on hyper-Kaehler compactifications of intermediate Jacobian fibrations of cubic fourfolds by constructing a twisted case, resulting in new hyper-Kaehler manifolds that are isogenous but not isomorphic.

Contribution

It introduces a twisted version of the hyper-Kaehler compactification construction, expanding the class of known hyper-Kaehler manifolds related to cubic fourfolds.

Findings

Construction of twisted hyper-Kaehler manifolds from cubic fourfolds

Identification of isogenous but not isomorphic or birational manifolds

Complementary to previous deformation and compactification results

Abstract

The starting point of this note is our recent paper with Laza and Sacc\`a on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-K\"ahler manifolds.