Torsion Tate-Shafarevich twists of projective Lagrangian fibrations induce isomorphisms on rational cohomology that preserve Hodge structures and pairings, and the spaces are deformation-equivalent when the base is smooth and sections exist.
Lagrangian fibrations of hyperkähler fourfolds.J
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Topology of projective Tate-Shafarevich twists
Torsion Tate-Shafarevich twists of projective Lagrangian fibrations induce isomorphisms on rational cohomology that preserve Hodge structures and pairings, and the spaces are deformation-equivalent when the base is smooth and sections exist.