On a generalization of rational homology manifolds

A complex variety Z is called a rational homology manifold if the homology of the link of each singularity of Z is the same as that of a sphere. While smooth varieties are rationally smooth, there are several examples of singular varieties that satisfy this condition. These varieties exhibit interesting geometric properties, including Poincaré duality. We study a natural weakening of this notion, which gives rise to the notion of k-Hodge rational homology varieties. This notion captures the difference between higher Du Bois and higher rational singularities, two classes of singularities that have recently attracted significant interest. Work in collaboration with B. Dirks and S. Olano.