Symmetries and vanishing theorems for symplectic varieties

We describe the local and Steenbrink vanishing problems for singular symplectic varieties with isolated singularities. We do this by constructing a morphism $${\mathbb{D}}_X({\underset{‾}{\mathrm{\Omega }}}_X^{n+p})\to {\underset{‾}{\mathrm{\Omega }}}_X^{n+p}$$ for a symplectic variety $X$ of dimension $2n$ for $\frac{1}{2}{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}}_X(X_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}})<p$, where ${\underset{‾}{\mathrm{\Omega }}}_X^k$ 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.