Classification of terminalizations of symplectic quotients of K3^{[n]} and generalized Kummer varieties yields at least nine new deformation types of irreducible symplectic varieties of dimension four.
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Quasi-F^e-splitting for all e implies numerically log canonical for numerically Q-Gorenstein normal singularities, with converse in dim 2 when p does not divide the Gorenstein index, plus a classification of 2D quasi-F-split cases.
Under a codimension assumption on the singular locus, isomorphism of the m-th differential sheaf implies isomorphisms for all lower i on complex hypersurfaces, with a positive characteristic analogue discussed.
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Terminalizations of quotients of compact hyperk\"ahler manifolds by induced symplectic automorphisms
Classification of terminalizations of symplectic quotients of K3^{[n]} and generalized Kummer varieties yields at least nine new deformation types of irreducible symplectic varieties of dimension four.
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Quasi-$F$-splitting versus log canonicity
Quasi-F^e-splitting for all e implies numerically log canonical for numerically Q-Gorenstein normal singularities, with converse in dim 2 when p does not divide the Gorenstein index, plus a classification of 2D quasi-F-split cases.
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Higher singularities for hypersurfaces
Under a codimension assumption on the singular locus, isomorphism of the m-th differential sheaf implies isomorphisms for all lower i on complex hypersurfaces, with a positive characteristic analogue discussed.