**IMPANGA** is an algebraic geometry seminar organized by Piotr Achinger, Jarosław Buczyński, Michał Kapustka, and Jakub Koncki.
In the academic year 2023/24, the seminar meets twice per month for a one day session on **Friday**, with two 60 min talks separated by a lunch break **(11:00-12:00 and 13:30-14:30)**.
IMPANGA meets in **Room 403 at IMPAN** (unless stated otherwise).

To receive notifications about upcoming seminars join

impanga-mailing-list@impan.pl using google account, or contact one of the organizers.

IMPANGA was founded at IMPAN in 2000 by late Piotr Pragacz.
See here for information on former meetings of IMPANGA

## Upcoming meeting

## Future meetings

- Dec 15: Noemie Combe (MIMUW) and Kamil Rychlewicz (ISTA)
- Jan 12:
- Jan 26:
- Mar 1:
- Mar 15:
- Apr 12:
- Apr 26:
- May 10:
- May 24:
- Jun 7:

## Past meetings (2023/24 term)

### December 1 (impanga 449)

#### Constructions of derived equivalent hyper-Kaehler fourfolds

**Speaker:** Grzegorz Kapustka (Jagiellonian University)
11:00–12:00, Kraków

Abstract

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.

#### A motivic Riemann-Roch theorem for Deligne-Mumford stacks

**Speaker:** Neeraj Deshmukh (IMPAN)
13:30–14:30, Kraków

Abstract

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.

### November 17 (impanga 448)

#### Conjectures on L-functions for varieties over function fields and their relations

**Speaker:** Veronika Ertl (IMPAN)
11:00–12:00, IMPAN 403

Abstract

(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.

#### Rational points on 3-folds with nef anti-canonical class over finite fields

**Speaker:** Fabio Bernasconi (University of Basel)
13:30–14:30, IMPAN 321

Abstract

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.

### October 20 (impanga 447)

#### Charge and Atoms beyond type A

**Speaker:** Jacinta Torres (UJ)
11:00–12:00, IMPAN 403

Abstract

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 C_{2} and point out future directions
to follow. This is joint work with Leonardo Patimo.

#### Nakajima's creation operators and the Kirwan map

**Speaker:** Jakub Koncki (IMPAN)
13:30–14:30, IMPAN 403

Abstract

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.

### October 06 (impanga 446): special meeting in honor of Piotr Pragacz

#### Degeneracy loci, Schur Functions, Schubert Calculus and Wronskians

**Speaker:** Letterio Gatto (Politecnico di Torino)
11:00–12:00, IMPAN 321

Abstract

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.

#### A special Calabi–Yau degeneration with trivial monodromy

**Speaker:** Sławomir Cynk (UJ)
13:30–14:30, IMPAN 321

Abstract

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 A_{1} singularity
has infinite order. There exists an example of Calabi-Yau threefold degeneration with A_{2} singularity, it has local monodromy of order 6 but it has no smooth filling (in algebraic category). On the other case the A_{2} 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).

Contact:

*jakubkoncki@impan.pl*
Mailing list:

https://groups.google.com/a/impan.pl/g/impanga-mailing-list