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.
Compactifying Lagrangian fibrations
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Constructs irreducible symplectic varieties of dimension 2n (2≤n≤10) with 16≤b2≤24 as non-trivial terminalisations of finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers.
Constructs Poincaré sheaf on compactified Prym varieties for singular curves and proves Fourier-Mukai autoequivalence, with applications to motivic decomposition and perverse filtrations.
Lecture notes summarizing recent progress on hyper-Kähler varieties via Lagrangian fibrations, atomic sheaves, and derived categories.
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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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Irreducible symplectic varieties via K3-del Pezzo double covers
Constructs irreducible symplectic varieties of dimension 2n (2≤n≤10) with 16≤b2≤24 as non-trivial terminalisations of finite symplectic quotients of Beauville-Mukai systems on very general K3-del Pezzo double covers.
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Autoduality of compactified Pryms for \'etale double covers of curves with planar singularities
Constructs Poincaré sheaf on compactified Prym varieties for singular curves and proves Fourier-Mukai autoequivalence, with applications to motivic decomposition and perverse filtrations.
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Hyper-K\"ahler varieties: Lagrangian fibrations, atomic sheaves, and categories
Lecture notes summarizing recent progress on hyper-Kähler varieties via Lagrangian fibrations, atomic sheaves, and derived categories.