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 for IHS fourfolds
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abstract
In this paper we study the Lagrangian fibrations for projective irreducible symplectic fourfolds and exclude the case of non-smooth base. Our method could be extended to the higher-dimensional cases.
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Conditional proof of semiampleness for nef non-big line bundles on hyperkahler manifolds via deformation invariance, established through a new Teichmuller space parametrizing pairs (M,L) and a global Torelli theorem for it.
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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.
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The abundance and SYZ conjectures in families of hyperkahler manifolds
Conditional proof of semiampleness for nef non-big line bundles on hyperkahler manifolds via deformation invariance, established through a new Teichmuller space parametrizing pairs (M,L) and a global Torelli theorem for it.