Twists of intermediate Jacobian fibrations

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.