A Hodge Theoretic generalization of $\mathbb{Q}$-Homology Manifolds I: General Case

We study a natural Hodge-theoretic generalization of rational (or $\mathbb{Q}$-)homology manifolds through an invariant $\HRH(Z)$ attached to a complex algebraic variety $Z$. 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 $\HRH(Z)$ 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 Poincar\'{e} duality. We also introduce the generic local cohomological defect ${\rm lcdef}_{\textrm{gen}}(Z)$ and relate it to $\HRH(Z)$. Various examples are discussed at the end.