Programme

Abstracts

Enrico Fatighenti (Sapienza)

Fano varieties of K3 type and their properties.

Fano varieties of K3 type are a special class of Fano varieties, which are usually studied for their link with hyperkaehler geometry, rationality properties, and much more. In this talk, we will recap some recent results, obtained jointly with Bernardara, Manivel, Mongardi, and Tanturri, that focus on the explicit construction of examples and the study of their Hodge-theoretical properties.

Emma Brakkee (Amsterdam)

Moduli of twisted K3 surfaces with a view towards counting Brauer classes

We construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order, by mimicking the construction of the moduli of non-twisted K3 surfaces as a quotient of a bounded symmetric domain. We discuss how moduli of twisted K3 surfaces could be used to count non-constant Brauer classes on K3 surfaces with high Picard number over

number fields.

François Charles (Paris-Saclay) (online)
Arithmetic aspects of the twistor construction

We will survey results and questions on the role of the twistor construction in the arithmetic of K3 surfaces, and will explain how to use the global geometry of positive-characteristic analogues of the twistor space to prove results on the geometry of supersingular K3 surfaces.

Laure Flapan (Michigan State)

Product identities in the Chow rings of hyperkähler manifolds

For a moduli space M of sheaves on a K3 surface, we propose a series of conjectural identities in the Chow ring of self-products of M which generalize the classic Beauville-Voisin identity for K3 surfaces. We then verify these identities in the case that M is a Hilbert scheme of points on a K3 surface. This is joint work with I. Barros, A. Marian, and R. Silversmith.

Soheyla Feyzbakhsh (Imperial College London) (online)

Rank r DT theory from rank 1

Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macri-Toda, such as the quintic 3-fold. After a brief introduction of weak Bridgeland stability conditions, I will describe work with Richard Thomas which expresses Joyce’s generalised DT invariants counting Gieseker semistable sheaves of any rank r on X in terms of those counting sheaves of rank 1. By the MNOP conjecture, the latter are determined by the Gromov-Witten invariants of X. Finally, I will show our result gives an explicit formula for rank r=0 or 2 when X is of Picard rank one.

Lars Halvard Halle (Bologna)

On log good reduction of K3 surfaces admitting a triple-point-free model

Let X be a proper smooth variety over a complete discretely valued field K. One says that X has log good reduction if it admits a log smooth and proper model over the ring of integers of K. In general, it is quite non-obvious whether or not such a model exists.

If X is a curve, there is a neat criterion (due to T. Saito and J. Stix) in terms of the geometry of a suitable normal crossings model of X. In my talk, I will present a geometric criterion for log good reduction when X is a K3 surface admitting a triple-point-free model. This is joint work in progress with J. Nicaise.

Annalisa Grossi (Chemnitz) (online)

Ihs varieties as symplectic quotients of ihs manifolds.

We aim at constructing ihs varieties with trivial algebraic regular fundamental group starting from an ihs manifold X and a finite group G of symplectic actions on X. I will present some work in progress in the case where X is a generalized Kummer fourfold or an O’Grady’s sixfold. The case of generalized Kummer fourfolds is the content of a joint work with Bertini, Capasso, Mauri and Mazzon. Moreover I will present some classification results about symplectic birational transformations of manifolds of OG6 type (joint work with Onorati and Veniani) and how these results could be exploit for our purpose.

Margherita Lelli-Chiesa (Roma Tre) (online)

Irreducibility of Severi varieties on K3 surfaces

Let (S,L) be a general K3 surface of genus g. I will prove that the closure in |L| of the Severi variety parametrizing curves in |L| of geometric genus h is connected for h>=1 and irreducible for h>=4, as predicted by a well known conjecture. This is joint work with Andrea Bruno.

Chunyi Li (Warwick)

Bridgeland stability conditions on the Hilbert scheme of K3 surfaces.

Let E be an elliptic curve and S be the Kummer K3 surface associated with E\times E. We construct a family of stability conditions on Hilb^nS with the central charges -ch_{2n}^{b+ia} parameterized by a>0 and b. In the talk, I will explain the background of the construction of stability conditions and some recent techniques introduced in the project. This is an ongoing project joint with Emanuele Macrì, Paolo Stellari, and Xiaolei Zhao.

Emanuele Macrì (Paris-Saclay) (online)

Lagrangian fibrations on hyper-Kähler fourfolds

We will present joint work in progress with Olivier Debarre, Daniel Huybrechts and Claire Voisin on the SYZ hyper-Kähler conjecture for fourfolds under certain topological assumptions.

Howard Nuer (Technion - Israel Institute of Technology)

Higher rank Brill-Noether theory: Picard rank one K3’s and other surfaces with (numerically) trivial canonical bundle

We will outline in this talk a technique for studying the Brill-Noether problem for sheaves on surfaces of numerically trivial canonical bundle. Using Bridgeland stability conditions, Fourier-Mukai transforms, and wall-crossing, we reduce both the weak Brill-Noether problem (i.e. studying the cohomology groups of the generic sheaf) and the question of generic global generation to a computational exercise of solving a number of inequalities. The shape of these inequalities has a number of beautiful and immediate consequences. For example, we derive from these inequalities that given any positive rank r, only finitely many Mukai vectors of rank r fail weak Brill-Noether over all K3 surfaces of Picard rank one. In fact, for a Mukai vector v=(r,dH,a), with H the ample generator of Pic(X), we give precise bounds on d, a, and H^2 that guarantee that weak Brill-Noether and generic global generation hold. Another consequence we derive from these inequalities is another proof of the classification of Ulrich bundles on K3 surfaces of Picard rank one. Time permitting we will discuss how our technique provides an algorithm for classifying the counterexamples to weak Brill-Noether wih a number of concrete examples. This reports on previous joint work with Izzet Coskun and Kota Yoshioka and ongoing work with these coauthors + Jack Huizenga.

Georg Oberdieck (Bonn)

A tale of three counting theories on a K3 surface

I will discuss the relationship between three different counting theories associated to a K3 surface S:

(i) Gromov-Witten theory of the Hilbert scheme of points of S with complex structure of the source curve fixed to be an elliptic curve E,

(ii) Donaldson-Thomas theory of S×E,

(iii) Virtual Euler characteristics of Quot schemes of stable sheaves on the K3 surfaces.

This leads to new evaluations of these virtual Euler numbers, and to multiple cover formulas for all three theories.

Laura Pertusi (Milan) (online)

Serre-invariant stability conditions and cubic threefolds

Stability conditions on the Kuznetsov component of a Fano threefold of Picard rank 1, index 1 and 2 have been constructed by Bayer, Lahoz, Macrì and Stellari, making possible to study moduli spaces of stable objects and their geometric properties. In this talk we investigate the action of the Serre functor on these stability conditions. In the index 2 case and in the case of GM threefolds, we show that they are Serre-invariant. Then we prove a general criterion which ensures the existence of a unique Serre-invariant stability condition and applies to some of these Fano threefolds. Finally, we apply these results to the study of moduli spaces in the case of a cubic threefold X. In particular, we prove the smoothness of moduli spaces of stable objects in the Kuznetsov component of X and the irreducibility of the moduli space of stable Ulrich bundles on X. These results come from joint works with Song Yang, with Soheyla Feyzbakhsh and with Ethan Robinett.

Giovanni Staglianò (Catania)

Explicit unirationality of some moduli spaces of K3 surfaces.

Abstract: The $19$-dimensional moduli space $\mathcal F_g$ of polarized K3 surfaces of genus $g$ (and degree $2g-2$) is known to be unirational for some low values of $g$, by results of Mukai, Nuer, Farkas and Verra. However, only for very few values of $g$ the construction of

unirationality provides a computer-implementable algorithm to determine the equations of the general member of $\mathcal F_g$. We describe a procedure to determine explicitly the equations of the general K3 surface of genus $g$ as a function of a number of specific independent variables. This procedure is implemented in \emph{Macaulay2} and in particular it yields the explicit unirationality of $\mathcal F_g$ for $g=11,14,20,22$. This is based on joint works with Michael Hoff and Francesco Russo.

Lenny Taelman (Amsterdam)

Deformations of ordinary Calabi-Yau varieties

I will talk about joint work with Lukas Brantner.

Let X be a variety with trivial canonical bundle over a perfect field k of characteristic p. We show that if X is ordinary, then its deformation space is a formal torus over the Witt vectors of k. In particular, deformations are unobstructed, and these varieties have a canonical lift to characteristic zero. We also show that this canonical lift is algebraizable.

Special cases of this are known by results of Serre-Tate, Deligne, Nygaard, Ward, and Achinger-Zdanowicz. Our proof uses derived deformation theory in an essential way, and one aim of the talk is to showcase how techniques from derived algebraic geometry can be used in proving results in classical deformation theory.