For decades, singularities of hypersurfaces have been studied through Hodge and D-module theoretic techniques, namely through the Steenbrink spectrum, Bernstein-Sato polynomial, and more recently, through the Hodge theory of local cohomology modules (equivalently, the theory of Hodge ideals due to Mustata-Popa).
For local complete intersection (LCI) singularities of higher codimension, there are natural analogues of the objects mentioned above. Recently, in joint work with Qianyu Chen, Mircea Mustata and Sebastián Olano, we have shown that much of the same story can be told in the higher codimension LCI setting. I will describe in detail the hypersurface setting, explain the generalizations to the higher codimension case, and describe some of the tools which are used in the proofs, for example, the V-filtration on D-modules and how it relates to the Hodge module structure on local cohomology.

In this talk, we compare a couple of notions of differential form on singular complex algebraic varieties, and relate them to the outermost associated graded spaces of the Hodge filtration of ordinary and intersection cohomology. In particular, we introduce and study singularities, that we call quasi-rational, which are normal and such that for all p, the zeroth cohomology sheaf of the complex of Du Bois p-forms is isomorphic to the direct image of p-forms from a desingularization. This is joint work with Donu Arapura.

A complex variety Z is a rational homology manifold if the local cohomologies at every point are those of a sphere of dimension 2dim(Z). This condition is equivalent to requiring that the constant sheaf is isomorphic to the Intersection Cohomology complex of Z. The Hodge Rational Homology level HRH(Z) is the largest degree at which this isomorphism is partially satisfied with respect to the Hodge filtration. In this talk, we study this invariant when Z is a variety that is a local complete intersection. In the case of hypersurfaces, this degree can be fully understood and related to classical invariants of singularities. We also show how these results extend to local complete intersections and how they relate to new invariants that arise in this context. This is joint work with B. Dirks and D. Raychaudhury.

The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge–Du Bois diamond. For example, the failure of the first nontrivial Poincaré duality is reflected in the defect of factoriality. Based on joint work with Mihnea Popa, I will discuss how local and global conditions on singularities influence the topology of algebraic varieties.

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 rational homology manifolds, 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. In this talk, I will give some general characterizations of this notion via local cohomology and link invariants, and give examples. Work in collaboration with B. Dirks and S. Olano.

We explain how recent progress in higher Du Bois singularities leads to new results about K-regularity, a notion that measures the homotopy invariance of algebraic K-theory.

For a complex algebraic variety X embedded inside a smooth variety Y, the local cohomology sheaves of X in Y carry additional structure of a (mixed) Hodge module. In the hypersurface and the local complete intersection (lci) case, this has been widely leveraged to prove various results about higher Du Bois and higher rational singularities, among other things. We investigate these local cohomology sheaves when X is a toric variety (which is typically non-lci) and give a comprehensive description of their Hodge module structures. As a result, we get interesting consequences regarding the depths of reflexive differentials on a toric variety. Additionally, we describe in further detail two interesting cases, the affine toric varieties associated to (i) the cone over a simplicial polytope, (ii) the cone over a simple polytope. This is based on joint work with Hyunsuk Kim.

We establish appropriate generic vanishing theorems for singular varieties, generalizing the well-known generic vanishing theorem by Green and Lazarsfeld and the generic vanishing theorem of Nakano type by Popa and Schnell. Our theorem explains the counterexample of Hacon and Kovács.