impanga 2023/24

We describe when two hyper-Kahler fourfolds of K3^[2]-type of Picard rank 1 with isomorphic transcendental lattices are derived equivalent. Then we present new constructions of pairs of twisted derived equivalent hyper-Kaehler manifolds of Picard rank >1. This is a joint work with Michal Kapustka.

The Grothendieck-Riemann-Roch theorem fails to hold in the case of Deligne-Mumford stacks due to the presence of stabilisers. Several modified constructions, using the inertia stack and related objects, have been proposed by Edidin, Graham, Toën, and others.

In this talk, we will analyse Toën's formulation of the Riemann-Roch theorem from a motivic perspective. More specifically, we will construct an object (associated to every Deligne-Mumford stack) in Voevodsky's triangulated category of motives which we will then use to reformulate (and perhaps even generalise) Toën's Riemann-Roch isomorphism in the language of motivic cohomology. This is joint work with Utsav Choudhury and Amit Hogadi.

(Joint work with T. Keller (Groningen) and Y. Qin (Berkeley))
We consider versions for smooth varieties X over finitely generated fields K in positive characteristic p of several conjectures that can be traced back to Tate, and study their interdependence. In particular, let A/K be an abelian variety. Assuming resolutions of singularities in positive characteristic, I will explain how to relate the BSD-rank conjecture for A to the finiteness of the p-primary part of the Tate-Shafarevich group of A using rigid cohomology. Furthermore, I will discuss what is needed for a generalisation.

A theorem of Esnault states that smooth Fano varieties over finite fields have rational points. What happens if we relax the conditions related to the positivity properties of the anti-canonical class? In this seminar, I will discuss the case of 3-folds with nef anti-canonical class. Specifically, we show that in the case of negative Kodaira dimension, the existence of rational points is established if the cardinality is greater than 19. In the K-trivial case, we prove a similar result, provided that the Albanese morphism is non-trivial. This is joint work with S. Filipazzi.

In this talk I will describe a general philosophy, due to Patimo, for constructing positive combinatorial formulas for Kostka-Foulkes polynomials beyond type A. This amounts to constructing atomic decompositions for crystals as well as swapping functions which allow defining charge statistics. Then I will explain such a construction for crystals of type C₂ and point out future directions to follow. This is joint work with Leonardo Patimo.

The Hilbert scheme of points in the affine complex plane is a smooth variety. Several descriptions of its cohomology groups are known. One may use Białynicki-Birula decomposition, Nakajima creation operators, or the Kirwan map. In the talk I will present a relation between the last two of the mentioned methods. I will describe the action of Nakajima's creation operators on the characteristic classes of the tautological bundle. This is joint work with Magdalena Zielenkiewicz.

The title of the talk recalls some main keywords of much work that Piotr Pragacz did by himself and/or in collaboration with many authors, like e.g. Alain Lascoux, Jan Ratajski, Andrzej Weber, to mention a few. Inspired by the many enlightening conversations I had with Piotr, I will attempt to draw an elementary path connecting the four keywords in the title.

Classical theorems of Kulikov, Persson and Pinkham describe degeneration of one parameter families of K3 surfaces. In the first case it asserts that of a degeneration of K3 surfaces with finite monodromy, there is a semi-stable degeneration with a smooth central fiber. If the central fiber of degeneration admits A-D-E singularities then the theorem follows from the simultanous resolution.
It is known that this does not generalize to higher dimensional Calabi-Yau varieties: local monodromy of odd dimensional A₁ singularity has infinite order. There exists an example of Calabi-Yau threefold degeneration with A₂ singularity, it has local monodromy of order 6 but it has no smooth filling (in algebraic category). On the other case the A₂ singularity has no crepant resolution. I will present an example of a different type: a semi-stable one-parameter family of Calabi-Yau threefolds with trivial local monodromy and central fibers consisting of two components - a smooth, rigid Calabi-Yau manifold and a quadric bundle. I shall briefly discuss geometry of this degeneration.

Joint work with Duco van Straten (Johannes Gutenberg University Mainz).