Twists of intermediate Jacobian fibrations

Abstract

We study the sections, Tate--Shafarevich twists, and the period for an OG10 hyperkähler Lagrangian associated to a cubic fourfold. To do so, we introduce the analytic relative Jacobian sheaf for a Lagrangian fibration of a hyperkähler 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 hyperkähler manifold. Among the main tools we use are intersection complexes with integral coefficients, Decomposition Theorem, Hodge modules and Deligne cohomology.

arXiv commentNew results: computation of Tate-Shafarevich groups for arbitrary smooth cubics (Sections 3 and 6), and Hodge isometry between primitive cohomology groups of OG10 fibrations and cubics (Theorem D and Section 4). Several results are generalized and the paper is reorganized accordingly. Comments are welcome