Emma Brakkee (University of Amsterdam)
Mukai dual K3 surfaces and L-equivalence
It is expected that there is a connection between derived equivalence and L-equivalence, a relation in the Grothendieck ring of varieties. At the moment however, there are too few examples known to make this connection precise. More examples might be found by studying pairs of K3 surfaces with special geometry, such as Mukai dual K3 surfaces. We will explain what these are and sketch why general Mukai dual K3 surfaces of degree 12 are L-equivalent.
Evgeny Shinder (University of Sheffield)
Jacobians and derived equivalence of elliptic K3 surfaces
I explain a question of Hassett and Tschinkel on whether every Fourier-Mukai partner of an elliptic K3 surface is isomorphic to one of its Jacobians. This question has both geometric and arithmetic significance, in particular it's relevant for the D-equivalence => L-equivalence conjecture and behaviour of rational points under derived equivalence. The answer to the Hassett-Tschinkel question is positive in Picard rank two under a coprimality assumption, and is negative in general. The proofs rely on Mukai's techniques of moduli spaces, Derived Torelli Theorem, Hodge lattices and counting formula for Fourier-Mukai partners. Joint work with Reinder Meinsma.
Andrea Ricolfi (University of Bologna)
A motivic wall-crossing formula
Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are related to Pandharipande-Thomas invariants via a wall-crossing formula known as the DT/PT correspondence, proved by Bridgeland and Toda. The same relation holds for the “local invariants”, those encoding the contribution of a fixed smooth curve in Y. We show how to lift the local DT/PT correspondence to the motivic level, thus proving an explicit formula for the local motivic DT invariants; we do so by exploiting the critical structure on certain Quot schemes acting as our local models. Joint work with Ben Davison.
Marco Rampazzo (University of Bologna)
L-equivalence for Calabi-Yau pairs in generalized Grassmannians
Two varieties X and Y are called L-equivalent if the difference of their classes in the Grothendieck ring of varieties annihilates a power of the Lefschetz motive. This feature is conjectured to be related to the problem of finding an equivalence between their derived categories of coherent sheaves: we consider a class of examples of Calabi—Yau subvarieties of generalized Grassmannians, where L-equivalence arises in a natural way. This is a joint work with Michał Kapustka.
Annalisa Grossi (Technische Universität Chemnitz)
Birational equivalence and derived equivalence for ihs manifolds of OG6 type.
In this talk after a gentle introduction of derived equivalence, K-equivalence, L-equivalence and birational equivalence for ihs manifolds, I will discuss some open questions related to the interplay of these concepts and which equivalences imply of are implied by derived equivalence. Moreover I will recall the notion of M-equivalence introduced by Mongardi Meachan and Yoshioka that allows them to get more examples of the failure of the converse of the Bondal-Orlov conjecture, that states that two ihs manifolds that are birational are derived equivalent. In order to give more examples of derived equivalent manifolds that are not birational I will present a lattice theoretic criterion to determine when a manifold of OG6 type is birational to a moduli space of sheaves on an abelian surface. I will discuss possible further direction of investigation and I will briefly present how to exploit the birational model for OG6 type manifolds due to Mongardi Rapagnetta and Sacc`a and a recent result due to Ploog to find examples of derived equivalent ihs manifolds that are not birational.
Robert Laterveer (University of Strasbourg)
The Y-F(Y) relation
Let Y be a cubic hypersurface, and F(Y) its Fano variety of lines. Galkin and Shinder have discovered a remarkable relation between the classes of Y and F(Y) in the Grothendieck ring of varieties. My talk will review their argument, and give a survey of related work (the Y-F(Y) relation on the level of Chow motives, Voisin’s work on the integral decomposition of the diagonal for cubics, the question whether the Y-F(Y) relation characterizes cubics, …).
Aurelio Carlucci (University of Oxford)
I would like to present an example of explicit coordinates for the moduli scheme of Pandharipade-Thomas (PT) stable pairs. There is a stratification which admits a simple sheaf-theoretic interpretation, but the actual equations are given via a quiver representation approach, following Nagao-Nakajima. The main technical obstacle is that, while stability for the Hilbert scheme (leading to Donaldson-Thomas invariants) translates to a simple algebraic condition, PT pairs involve a case-by-case analysis.
Franco Giovenzana (Chemnitz University of Technology)
O'Grady's Hyperkähler manifolds are constructed as resolutions of some moduli spaces of sheaves on a K3 surface or an abelian surface. Despite the cohomology of the O'Grady's manifolds is well known, the same is not true for the singular varieties. We investigate the Hodge structure of the singular moduli spaces and we draw some conclusion on their singularities. (from joint work with V. Bertini).
Franco Rota (University of Glasgow)
Full exceptional collections for Johnson-Kollar orbisurfaces.
Motivated by the homological mirror symmetry conjecture, we construct explicit exceptional collections for the family of (stacky) log DelPezzo surfaces known as the Johnson-Kollar series. These surfaces have quotient, non-Gorenstein, singularities. Thus, our computation includes on the one hand an application of the GL(2,C) McKay correspondence, and on the other the study of their minimal resolutions, which are birational to a degree 2 del Pezzo surface." This is joint work with Giulia Gugiatti.
Thor Wittich (Heinrich-Heine University Düsseldorf)
The aim of this short talk is to present ongoing work on certain stratifications that allow to control/compute motivic (Igusa) zeta functions/related objects. After very quickly recalling some basics regarding the classical situation of point counting mod prime powers from number theory/algebraic geometry, I would explain the motivic analogue and how the just mentioned stratifications come into play.