Instytut Matematyczny Polskiej Akademii Nauk
Time: Friday, November 09, 2018, 14:30
Serre-Tate theory for Calabi-Yau varieties
Classical SerreTate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the moduli space with a Frobenius lifting and canonical multiplicative coordinates. I will discuss a project, joint with Maciej Zdanowicz (EPFL), whose aim is to construct canonical liftings modulo p2 of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two.
Freie Universität Berlin
Time: Thursday, November 08, 2018, 16:00
Moduli of hypersurfaces in weighted projective space
The moduli space of smooth hypersurfaces in projective space was constructed by Mumford in the 60’s using his newly developed classical (a.k.a. reductive) Geometric Invariant Theory. I wish to generalise this construction to hypersurfaces in weighted projective space (or more generally orbifold toric varieties). The automorphism group of a toric variety is in general non-reductive and I will use new results in non-reductive GIT, developed by F. Kirwan et al., to construct a moduli space of quasismooth hypersurfaces. I will give geometric characterisations of notions of stability arising from non-reductive GIT.
Time: Thursday, November 08, 2018, 13:00
Enumerative geometry and the holomorphic anomaly equation
I will give a (gentle) introduction to the role of the holomorphic anomaly equation in curve counting problems starting with maps to elliptic curves, proceeding to local toric geometries, and concluding with the famous quintic 3-fold. Though I will discuss joint work with Hyenho Lho, much of what I will say is work of others (Pixton, Oberdieck-Pixton, Chen-Guo-Janda-Ruan, Jun Li and collaborators).
MPI MiS, Leipzig
Time: Thursday, November 08, 2018, 14:30
The anticanonical complex - a combinatorial tool for Fano varieties
Toric Fano varieties are in one to one correspondence with certain lattice polytopes, the so called Fano polytopes. Moreover classification of toric Fano varieties with respect to their singularity type turns out to be purely combinatorial: the position of lattice points in the Fano polytope determines the singularity type. The anticanonical complex has been introduced as a natural generalisation of the toric Fano polytope and so far has been succesfully used for the study of varieties with a torus action of complexity one. We enlarge the area of application of the anticanonical complex to Fano varieties with torus action of arbitrary complexity, for example arrangement varieties. In particular, we show that the possibility to apply the anticanonical complex to these varieties is connected to certain properties of their quotients.
Time: Friday, November 09, 2018, 11:30
Derived categories of fibrations of quintic del Pezzo surfaces
I will provide two approaches for fining a semiorthogonal decomposition of the derived category of fibrations of quintic del Pezzo surfaces with RDP singularities. Similar to Kuznetsov’s work on sextic del Pezzo surfaces, the components of the semiorthogonal decomposition can be interpreted as the moduli spaces of semistable sheaves on fibers with fixed Hilbert polynomials. Alternatively, there is a rank 2 vector bundle on a quintic del Pezzo surface with RDP singularities that embeds the surface as a linear section of a Grassmannian. The semiorthogonal decomposition can be obtained by applying Homological Projective Duality. This is work in progress.
Leibniz Universität Hannover
Time: Friday, November 09, 2018, 10:20
Formality conjecture and moduli spaces of sheaves on K3 surfaces
The formality conjecture for K3 surfaces, formulated by D.Kaledin and M.Lehn, states that on a complex projective K3 surface, the differential graded algebra RHom(F,F) is formal for any coherent sheaf F polystable with respect to an ample line bundle. In this talk, I will explain how to combine techniques from twistor spaces, dg categories and Fourier-Mukai transforms to prove this conjecture, and how to generalize it to derived objects. Based on joint work with Nero Budur.