Symmetries and vanishing theorems for symplectic varieties

TL;DR

This paper investigates vanishing theorems and symmetries in singular symplectic varieties, constructing specific morphisms related to Du Bois complexes and duality, with applications to symplectic germs and 4-folds.

Contribution

It introduces a morphism linking Du Bois complexes and duality for singular symplectic varieties, revealing symmetries and vanishing properties, and applies these to specific geometric contexts.

Findings

The morphism is a quasi-isomorphism for p = n-1.

Symmetry descends to the Hodge filtration on intersection Hodge modules.

Applications include properties of symplectic germs and cohomology of primitive symplectic 4-folds.

Abstract

We describe the local and Steenbrink vanishing problems for singular symplectic varieties with isolated singularities. We do this by constructing a morphism $$\mathbb D_X(\underline \Omega_X^{n+p}) \to \underline \Omega_X^{n+p}$$ for a symplectic variety $X$ of dimension $2n$ for $\frac{1}{2}\mathrm{codim}_X(X_{\mathrm{sing}}) < p$, where $\underline \Omega_X^k$ 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.