In a series of extremely influential papers that appeared 40 years ago, Kollár showed several surprising properties of higher direct images of dualizing sheaves via projective morphisms. In this talk I will report on recent joint work with Kollár that should be considered a sequel to his original papers (hence the counter in the title). We show that for a flat morphism between varieties with rational singularities even stronger statements are true. In particular, Kollár’s splitting theorem from 40 years ago holds for the higher direct images of the relative dualizing sheaf and some other results. For instance we obtain that the higher direct images of the structure sheaf are locally free. As an application in another direction, using an argument of Y. Kim, this leads to the statement that the identity (or neutral) component of the relative Picard scheme of a flat morphism between varieties with rational singularities is a smooth algebraic group scheme.
Recently K-stability has been used to construct well-behaved moduli spaces for Fano varieties. Smooth Fano varieties (in characteristic zero) are unobstructed, so they give smooth(-ish) points on moduli spaces. In this talk I will report on recent work (joint with Alice Villa) about the construction of K-stable Fano varieties with obstructed deformations.
Moduli spaces of semistable vector bundles of fixed rank and degree on curves are fundamental objects in algebraic geometry. When rank and degree are coprime, the stability and semistability conditions coincide and the corresponding moduli space is a smooth projective variety whose geometry, topology and intersection theory have been studied extensively over the past several decades. When we drop the coprimality assumption, singularities appear and this has historically posed significant challenges for computing intersection pairings. In this talk we present a recent work in collaboration with Olga Trapeznikova, which introduces a novel and more accessible methodology for tackling this problem. The core strategy involves realizing the intersection cohomology of the singular moduli space as a canonical subspace of the cohomology of a smooth moduli space of parabolic bundles. We will show how this parabolic approach enables the calculation of intersection pairings by leveraging the Hecke correspondence and the Jeffrey-Kirwan residue formula. By comparing this methodology to the standard Jeffrey-Kirwan-Kiem-Woolf blow-up construction, we will discuss how this framework not only provides a computationally efficient alternative but also offers a clear geometric interpretation for arbitrary rank r.
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The conference "WINTER MEETING IN ALGEBRA AND GEOMETRY 2026" is a Winter meeting in Algebra and Geometry in Rome that takes place in the rione Testaccio. For more info here is the Link to the Official Webpage
Christmas Break
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The seminar is usually held every Thursday at 14:15–15:45 in Aula M1. This year seminars will be held in person, but you can email one of the organizers if you are interested in attending remotely through Microsoft Teams.
The seminar is organized by Luca Schaffler and Amos Turchet and maintained by the Geometry Group and the Number Theory Group of the Department of Mathematics and Physics at Roma Tre University.
We acknowledge the support of the grants PRIN2022: Moduli spaces and birational geometry and PRIN2022: Semiabelian varieties, Galois representations and related Diophantine problems, the “Programma per Giovani Ricercatori Rita Levi Montalcini”, and the Department of Mathematics and Physics at Roma Tre University.
