In this talk, I will discuss a conjectural characteristic-class version of the Hodge index theorem for singular complex algebraic varieties, formulated by Brasselet, Schuermann, and Yokura, which expresses the Goresky-MacPherson homology L-classes in terms of suitable Hodge-theoretic L-classes. I will survey recent results on spaces for which the conjecture has been proved.

In this talk, I will discuss the minimal model program for the Hilbert scheme of points on the blowup of the affine plane at a point. I will give explicit descriptions of the resulting birational models for this space and the wall-crossing phenomena relating them. This talk is based on joint work with Eugene Gorsky, Alexei Oblomkov, and Joshua Turner.

Suppose we are given an arbitrary line bundle \( \mathcal{L} \) on a complex algebraic variety \( X \). For a positive integer \( N \geq 0 \), suppose there exists a global section \( s \in \Gamma(X, \mathcal{L}^{N}) \) that defines an effective Cartier divisor \( D \). If we denote by \( \pi: Y \rightarrow X \) the cyclic covering of the divisor \( D \) resulting from the global section \( s \), I will describe the cyclic covering in terms of Saito's theory of mixed Hodge modules.

Toric variety is a normal variety containing an algebraic torus as an open dense subset whose action on itself extends to the whole space. They provide a fruitful interplay between algebraic geometry and convex geometry since properties on one side (e.g. smoothness, compactness) can be translated into properties involving discrete objects (e.g. cones, fans, polytopes). I will talk about singularities of these varieties from a Hodge theoretic point of view, with applications towards local cohomology and singular cohomology, based on joint works with Sridhar Venkatesh.

A polynomial map \( f: \mathbb{C}^{n+1} \to \mathbb{C} \) is known to be a topologically locally trivial fibration outside the preimage of a finite set of bifurcation values. The topology of its fibers is influenced both by affine singularities and by the singularities of \( f \) "at infinity". We will survey classical results and present new ones concerning the topology of general and atypical fibers, with particular emphasis on the role of singularities at infinity. Joint work with Laurentiu Maxim.

We discuss a work in progress introducing an alternative notion of higher Du Bois and higher rational singularities beyond the locally complete intersection setting. These notions, which we call (for the moment) k-cohomologically Du Bois and k-cohomologically rational, are defined via natural filtrations on local cohomology modules together with recently introduced singularity invariants, and agree with the established notions for lci singularities. We discuss highly desirable properties of these classes, including inversion of adjunction, constancy of Hodge numbers in flat projective families, and cohomological insignificance of smoothings, while providing new proofs of classical results along the way.

The Hodge conjecture gives a description of algebraic cycles on a smooth projective variety in terms of Hodge structures. However, it is well known that Hodge structures are not sufficient to describe algebraic cycles on open or possibly singular varieties. In previous joint work with M. Nori and V. Srinivas, we showed that one can naturally enrich the notion of a mixed Hodge structure. Moreover, it was shown that links of complete varieties have natural enriched Hodge structures. In this talk, I will survey these constructions and more recent results on analogs on K-theory, cycles, and enriched versions of Deligne-Beilinson cohomology in this context as well as some applications. This is based on joint work with M. Nori and V. Srinivas.

Secant varieties are classical objects in algebraic geometry. Given a smooth projective variety inside a projective space, its secant variety is by definition the closure of the union of secant lines. It is almost always singular and sits inside the same projective space by its construction. In this talk, we will discuss the singularities of secant varieties. In particular, we will study the Du Bois complex of secant varieties and will also discuss its local cohomology. The results are obtained in various collaborations with Q. Chen, B. Dirks, S. Olano and L. Song.