Twists of intermediate Jacobian fibrations

TL;DR

This paper investigates the structure and twists of hyperk"ahler Lagrangian fibrations linked to cubic fourfolds, introducing an analytic Jacobian sheaf and computing the Tate--Shafarevich group using advanced cohomological tools.

Contribution

It introduces the analytic relative Jacobian sheaf for hyperk"ahler fibrations and computes the Tate--Shafarevich group in terms of analytic Brauer groups, connecting Hodge theory and algebraic geometry.

Findings

Tate--Shafarevich group is isomorphic to the first cohomology of the Jacobian sheaf.

Primitive Hodge lattice of cubic fourfold relates to the cohomology of OG10 hyperk"ahler manifold.

Established a link between Hodge lattices and second cohomology via isometry.

Abstract

We study the sections, Tate--Shafarevich twists, and the period for an OG10 hyperk\"ahler Lagrangian associated to a cubic fourfold. To do so, we introduce the analytic relative Jacobian sheaf for a Lagrangian fibration of a hyperk\"ahler variety. The Tate--Shafarevich group parameterizing twists is isomorphic to the first cohomology group of this sheaf and we compute it in terms of certain analytic Brauer groups associated to the cubic fourfold. We prove that the primitive Hodge lattice of the cubic fourfold is, up to a sign, isometric to a distinguished sublattice of the second cohomology group of the associated OG10 hyperk\"ahler manifold. Among the main tools we use are intersection complexes with integral coefficients, Decomposition Theorem, Hodge modules and Deligne cohomology.