Events can be found on calendar page, with code DC.
Topic: Derived category, DG categories, motivic methods …
Organizers: Will Donovan, Lin Xun, Zhang Shizhuo
Jan 15 2025
In this talk I will present the results of my recent paper “Categorical resolutions of cuspidal singularities”, see arXiv:2411.19380. I showed that there exists a particularly small (“crepant”) categorical resolution of the derived category of a projective variety with an isolated A_2/cuspidal singularity. More importantly, I explicitly described generators of its kernel. In the special case of an even dimensional variety, I showed that the kernel is generated by two 2-spherical objects, which are related to spinor bundles on a nodal quadric. These objects are particularly interesting because they induce autoequivalences on the categorical resolution.
A similar result was proved last year by A. Kuznetsov and E. Shinder, and simultaneously by another group of mathematicians*, in the case of nodal singularities. I will explain the main differences between nodal and cuspidal singularities in this particular setting.
* W. Cattani, F. Giovenzana, S. Liu, P. Magni, L. Martinelli, L. Pertusi, J. Song.
Jan 19 2022
All derived categories in this talk will be of complexes over group algebras with the group belonging to certain large classes. We will develop a theory of generation operators in the abelian module category and then show how generation properties travel from the abelian category of modules over those group algebras to Gorenstein derived categories and standard derived categories. Much of the work that will be presented here has been published - “R Biswas; Generating derived categories of groups in Kropholler’s hierarchy. Muenster J. Math. (2021)”. The parts about the Gorenstein derived categories are new.
Dec 15 2021
The goal is to discuss recent developments in derived categories and related things. I will start with some surveys on semiorthogonal decompositions.
Jan 12 2022
A conic fibration has an associated sheaf of even Clifford algebra on the base. In this talk, I will discuss the relation between the two spaces associated to a conic fibration over the projective plane: moduli space of modules over even Clifford algebra and Prym variety. In particular, I will describe how to construct a rational map from the moduli spaceof modules over even Clifford algebra to the special subvarieties in Prym varieties and discuss some cases which are birational. As an application, we get an explicit correspondence between instanton bundles ofminimal charge on cubic threefolds and twisted Higgs bundles on curves.
Dec 22 2021
This talk is an introduction to the recent applications of stability conditions in the geometry of Fano threefold. The first part will focus on the properties of classical moduli spaces, such as moduli spaces of sheaves and Hilbert schemes. I will talk about the applications of stability conditions in proving some smoothness and irreducibility results. Then I will explain the birational categorical Torelli theorem for Gushel-Mukai threefolds. If the time is permitted, I will talk about the Brill-Neother reconstruction of Fano threefolds. This talk is based on a series of joint work with Augustinas Jacovskis, Xun Lin, and Shizhuo Zhang.
Jan 5 2022
In this talk, I will discuss the matrix factorization categories associated to the cyclic coverings of projective spaces. I will derive a semi-orthogonal decomposition of the equivariant matrix factorization category of the covering space in terms of matrix factorizations categories of the branched loci.
Jan 26 2022
The residual category (or the Kuznetsov component) of a quadric surface bundles is the non-trivial component in the derived category. It is equivalent to the twisted derived category of a double cover over the base when the quadric surface bundle has simple degeneration (fibers have corank at most 1). I will consider quadric surface bundles with fibers of corank at most 2 and describe their residual categories as (twisted) derived categories of some scheme in two situations: (1) when the bundle has a smooth section; (2) when the base is a surface and both the total space and base are smooth. As an application, I will also describe the residual categories of certain complete intersections of quadrics.
Mar 22 2022
A perverse schober is a categorification of perverse sheaves proposed by Kapranov-Schechtman. In this talk, I would like to construct some examples of perverse schobers on the Riemann sphere using derived categories of Calabi-Yau hypersurfaces and Orlov equivalences.