A Hodge Theoretic generalization of $\mathbb{Q}$-Homology Manifolds

We study a natural Hodge theoretic generalization of rational (or $\mathbb{Q}$-)homology manifolds through an invariant $\mathrm{H}\mathrm{R}\mathrm{H}(\mathrm{Z})$ where $Z$ is a complex algebraic variety. The defining property of this notion encodes the difference between higher Du Bois and higher rational singularities for local complete intersections, which are two classes of singularities that have recently gained much attention. We show that $\mathrm{H}\mathrm{R}\mathrm{H}(\mathrm{Z})$ can be characterized when the variety $Z$ is embedded into a smooth variety using the local cohomology mixed Hodge modules. Near a point, this is also characterized by the local cohomology of $Z$ at the point, and hence, by the cohomology of the link. We give an application to partial Poincaré duality. In the case of local complete intersection subvarieties, we relate $\mathrm{H}\mathrm{R}\mathrm{H}(\mathrm{Z})$ to various invariants. In the hypersurface case it turns out that $\mathrm{H}\mathrm{R}\mathrm{H}(\mathrm{Z})$ can be completely characterized by these invariants. However, for higher codimension subvarieties, the behavior is rather subtle, and in this case we relate $\mathrm{H}\mathrm{R}\mathrm{H}(\mathrm{Z})$ to these invariants through inequalities and give some conditions on when equality holds.