Symmetries and vanishing theorems for symplectic varieties

2024/10/10 by Benjamin Tighe, Tighe, Benjamin
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2410.07515

Abstract

We describe the local and Steenbrink vanishing problems for singular symplectic varieties with isolated singularities. We do this by constructing a morphism \mathbb D_X(\underline Ω_X^n+p) → \underline Ω_X^n+p for a symplectic variety X of dimension 2n for (1)/(2)codim_X(X_sing) < p, where \underline Ω_X^k is the k^th-graded piece of the Du Bois complex and \mathbb D_X is the Grothendieck duality functor. We show this morphism is a quasi-isomorphism when p = n-1 and that this symmetry descends to the Hodge filtration on the intersection Hodge module. As applications, we describe the higher Du Bois and higher rational properties for symplectic germs and the cohomology of primitive symplectic 4-folds.

Related

• Local vanishing and Hodge filtration for rational singularities
• Differential Forms and Hodge Structures on Singular Varieties
• Vanishing cycles in complex symplectic geometry
• Extension of 2-forms and symplectic varieties
• Pointwise purity, derived Satake, and Symplectic duality