Events can be found on calendar page, with code EG.
Topic: Curve counting invariants, birational geometry, moduli spaces, Hall algebras
Organizers: Will Donovan, Zhang Nantao, Hao Zhang
Some talks are posted on researchseminars.org.
Dec 1 2021
3-fold flopping contractions form a fundamental building block of the higher-dimensional Minimal Model Program. They exhibit extremely rich geometry, which has been investigated by many people over the past half-century. I will present an elegant and visually-pleasing relationship between enumerative invariants of flopping contractions and certain hyperplane arrangements constructed combinatorially from root system data. I will discuss both Gopakumar-Vafa (GV) and Gromov-Witten (GW) invariants, explaining how these are related to one another and how they are encoded in finite and infinite arrangements, respectively. Finally, I will discuss wall-crossing: our combinatorial approach allows us to explicitly construct flops from root system data, leading to a new “direct” proof of the Crepant Transformation Conjecture, with a very explicit formulation. This is joint work with Michael Wemyss.
Feb 24 2022
This talk is about the paper “Motivic degree zero Donaldson-Thomas invariants” by Kai Behrend, Jim Bryan, Balázs Szendröi. Arxiv/DOI
I will talk about definition of motivic weights, definition of motivic Donaldson-Thomas invariants and sketch of calculation of virtual motive of Hilbert scheme of points.
Mar 17 2022
Donaldson–Thomas theory was conceived as a method of counting certain sheaves in Calabi-Yau threefolds, which are supposed to encode ‘BPS numbers’ in string theory. More recent developments have led to broader, refined versions of this theory, which produce motivic or cohomological invariants from moduli spaces of semistable objects in the derived category. In this talk I will focus on DT theory for crepant resolutions of compound Du-Val singularities, which include threefold flops, as well as some divisor-to-curve contractions and quotient singularities. I will explain how one can determine the moduli of semistable objects in this setting via a tilting method that is governed by Dynkin diagram combinatorics. Using this, I will show that the motivic incarnations of the BPS numbers vanish for K-theory classes outside an associated root lattice, and exhibit additional symmetries among these invariants. To make this explicit, I will use the example of a dihedral quotient singularity, for which the invariants can be fully calculated.
Mar 24 2022
We will discuss a wall-crossing between Donaldson-Thomas theory of a threefold Surface x Curve and Gromov-Witten theory of a moduli space of sheaves on the Surface. The wall-crossing is provided by the notion of a quasimap to a moduli space of sheaves and Yang Zhou’s theory of calibrated tails. The geometry behind this kind of wall-crossings seems to be responsible for many correspondences between different enumerative theories centred around threefolds of the type Surface x Curve.
Mar 31 2022
Quasimaps to Hilbert schemes of surfaces S resemble the Donaldson-Thomas theory of S times a curve. This correspondence can be made precise for the appropriate DT stability chamber, namely the so-called Bryan-Steinberg stable pairs. I will explain why BS pairs and quasimaps are equivalent whenever they are comparable. Quasimaps have been used recently to study 3d mirror symmetry, which when pushed through this equivalence has implications for some aspects of sheaf-counting theories, including the (DT) crepant resolution conjecture.
Apr 7 2022
We will consider the following problem: if (C,x_1,…,x_n) is a fixed general pointed curve, and X is a fixed target variety with general points y_1,…,y_n, then how many maps f:C -> X in a given homology class are there, such that f(x_i)=y_i? When considered virtually in Gromov-Witten theory, the answer may be expressed in terms of the quantum cohomology of X, leading to explicit formulas in some cases (Buch-Pandharipande). The geometric question is more subtle, though in the presence of sufficient positivity, it is expected that the virtual answers are enumerative. I will give an overview of recent progress on various aspects of this problem, including joint work with Farkas, Pandharipande, and Cela, as well as work of other authors.
May 19 2022
Abstract: Gromov-Witten invariants of holomorphic symplectic 4-folds vanish and one can consider the corresponding reduced theory. In this talk, we will explain a definition of Gopakumar-Vafa type invariants for such a reduced theory. These invariants are conjectured to be integers and have alternative interpretations using sheaf theoretic moduli spaces. Our conjecture is proved for the product of two K3 surfaces, which naturally leads to a closed formula of Fujiki constants of Chern classes of tangent bundles of Hilbert schemes of points on K3 surfaces. On a very general holomorphic symplectic 4-folds of K3^[2] type, our conjecture provides a Yau-Zaslow type formula for the number of isolated genus 2 curves of minimal degree. Based on joint works with Georg Oberdieck and Yukinobu Toda.
May 26 2022
Let X be a Calabi-Yau 3-fold containing a ruled surface W and let B be the homology class of the lines in the ruling. Physics suggests that curve counting on X should satisfy some symmetry relating curves in classes β and β’=β+(W.β)B. In this talk I’ll explain how to make such a symmetry precise with a new rationality result for the Pandharipande-Thomas invariants of X. Mathematically, the symmetry is explained by a certain involution of the derived category of X constructed using a particular spherical functor; our proof is an instance of the general principle that automorphisms of the derived category should constrain enumerative invariants. This is joint work with Tim Buelles and it is highly inspired in the proof of rationality for the PT generating series of an orbifold by Beentjes-Calabrese-Rennemo.