Arkadij Bojko (slides)
Title: Wall-crossing for Calabi-Yau fourfolds and applications
Abstract: Joyce's vertex algebras are a powerful new ingredient added to the existing theory of wall-crossing for sheaves on surfaces. My work focuses on proving wall-crossing in two dimensions higher - for Calabi-Yau fourfolds. It is desirable that the end result can have many concrete applications to existing conjectures. For this purpose, I introduce yet another new structure into the picture - formal families of vertex algebras. Apart from being a natural extension of the theory, they allow to wall-cross with insertions instead of the plain virtual fundamental classes. To make the whole machinery work with (polynomial) Bridgeland stability conditions and sheaf-counting classes for fourfolds, I require a different approach compared to the surface case. In the talk, I will discuss the main difficulties that I encountered, and I will present examples using the complete package.
Tom Bridgeland (Minicourse) (notes)
Title: Geometric structures defined by Donaldson-Thomas invariants
Abstract: The genus 0 Gromov-Witten invariants of a variety V can be encoded in a Frobenius structure on a small (possibly formal) ball in the cohomology space H^*(V,C). I will explain an analogous, but so far rather speculative, story in which the Donaldson-Thomas invariants of a CY3 triangulated category are encoded in a geometric structure on its space of stability conditions. The relevant structure has been christened a Joyce structure; the key difference from a Frobenius structure is that instead of pencils of linear connections one considers pencils of non-linear, symplectic connections. Another key difference between the two stories is in the way the enumerative invariants enter: in the GW case the connection 1-form is given directly by the third partial derivatives of the genus 0 GW generating function, whereas in the DT case the connections are given indirectly by their Stokes data.
In the first lecture I will explain Stokes data in the linear setting and then explain why DT invariants can be viewed as defining non-linear Stokes data. In the second lecture I will explain the definition of a Joyce structure (so that Dominic can go home and tell his friends). In the last lecture I will talk about whatever seems relevant by that stage, possibly how to go from DT invariants to Joyce structures (Riemann-Hilbert problems) and/or a nice class of examples of Joyce structures relating to the geometry of the Hitchin system.
Francesca Carocci
Title: BPS invariant from p-adic integrals
Abstract: We consider moduli spaces of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on the moduli spaces with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles.
Pierre Descombes
Title: Cohomological and K theoretic DT invariants
Abstract: The numerical DT invariant, introduced by Richard Thomas, provides a kind of virtual Euler characteristic for moduli space of sheaves on CY3. Two kinds of refining of this theory were then developed: a cohomological refining, using vanishing cycle and Hodge theory, was suggested by Kontsevich and Soibelman and developed by Joyce et al. A second approach, in the presence of a group action scaling the CY3, using equivariant K theoretic class, was defined by Nekrasov and Okounkov. It was suggested that in this case K theoretic DT invariants would provide the virtual chi y genus of the cohomological one, but a proof was missing. We will sketch this proof during this talk.
Augustinas Jacovskis
Title: Categorical Torelli for double covers
Abstract: Consider a threefold double cover X of (weighted) projective space, ramified in a canonically polarised surface Z. In this talk I'll describe a semiorthogonal decomposition of the mu_2-equivariant Kuznetsov component of X, and show that it contains a copy of Db(Z). This gives a relationship between the K-theory of the equivariant Kuznetsov component, and the primitive cohomology of Z. Using this relationship and classical Torelli theorems for hypersurfaces in (weighted) projective space, I'll show that for certain classes of prime Fano threefolds, an equivalence of Kuznetsov components implies that they're isomorphic. This is joint work with Hannah Dell and Franco Rota.
Dominic Joyce (slides)
Title: The structure of invariants counting coherent sheaves on complex surfaces
Abstract: Let X be a complex projective surface with geometric genus pg > 0. We can form moduli spaces M(r,a,k)st ⊂ M(r,a,k)ss of Gieseker (semi)stable coherent sheaves on X with Chern character (r,a,k), where we take the rank r to be positive. In the case in which stable = semistable, there is a (reduced) perfect obstruction theory on M(r,a,k)ss, giving a virtual class [M(r,a,k)ss]virt in homology.
By integrating universal cohomology classes over this virtual class, one can define enumerative invariants counting semistable coherent sheaves on X. These have been studied by many authors, and include Donaldson invariants, K-theoretic Donaldson invariants, Segre and Verlinde invariants, part of Vafa-Witten invariants, and so on. In my paper https://arxiv.org/abs/2111.04694, in a more general context, I extended the definition of the virtual class [M(r,a,k)ss]virt to allow strictly semistables, proved wall-crossing formulae for these classes and associated “pair invariants”, and gave an algorithm to compute the invariants [M(r,a,k)ss]virt by induction on the rank r, starting from data in rank 1, which is the Seiberg-Witten invariants of X and fundamental classes of Hilbert schemes of points on X. This is an algebro-geometric version of the construction of Donaldson invariants from Seiberg-Witten invariants; it builds on work of Mochizuki 2008.
This talk will report on a project to implement this algorithm, and actually compute the invariants [M(r,a,k)ss]virt for all ranks r > 0. I prove that the [M(r,a,k)ss]virt for fixed r and all a,k with a fixed mod r can be encoded in a generating function involving the Seiberg-Witten invariants and universal functions in infinitely many variables. I will spend most of the talk explaining the structure of this generating function, and what we can say about the universal functions, the Galois theory and algebraic numbers involved, and so on. This proves several conjectures in the literature by Lothar Göttsche, Martijn Kool, and others, and tells us, for example, the structure of U(r) and SU(r) Donaldson invariants of surfaces with b2+ > 1 for any rank r ≥ 2.
Martijn Kool
Title: K-theoretic DT/PT correspondence for surfaces on Calabi-Yau fourfolds
Abstract: We introduce K-theoretic virtual invariants for certain moduli spaces of "surfaces on Calabi-Yau fourfolds". The most obvious moduli space to consider is the Hilbert scheme of 2-dimensional subschemes. However, unlike the case of "curves on Calabi-Yau threefolds", there are now two types of stable pairs moduli spaces. We discuss how these three moduli spaces are related by varying GIT stability and how they parametrize objects in the derived category. For toric Calabi-Yau fourfolds, and one of the two stable pairs theories, we introduce a conjectural vertex DT/PT correspondence which we can verify in examples. Joint work with Y. Bae and H. Park.
Naoki Koseki
Title: Gopakumar-Vafa invariants on local curves
Abstract: Gopakumar and Vafa proposed an ideal way to count curves in a Calabi-Yau 3-fold, which we call Gopakumar-Vafa (GV) theory. GV theory is conjecturally equivalent to other curve counting theories such as Gromov-Witten theory (GV=GW).
I will explain a joint work with Tasuki Kinjo, in which we proved the so-called chi-independence property in GV theory for a particular class of Calabi-Yau 3-folds: local curves. This gives a strong evidence of the GV=GW conjecture in this case.
Chunyi Li (slides)
Title: Counting Stable Spherical Bundles on a K3 Surface
Abstract: For a K3 surface with higher Picard number, Huybretchs asks if every spherical bundle is semistable with respect to some polarization and if there is a `counting theory' for spherical bundles. Unfortunately, both problems fail in a naive way. More precisely,
1. there exists an example of a spherical vector bundle that is never semistable;
2. there exists an example of K3 surface and infinitely many spherical vector bundles with the same spherical Mukai vector v. Moreover, each of the vector bundles is stable with respect to some polarization.
However, we may put some assumptions on S so that the `counting theory' can make sense. In particular, when Nef(S) is rational polyhedral, we can do the actual counting. I will discuss more details of spherical bundles and relevant questions in the talk. This is a joint work with Shengxuan Liu.
Margarida Melo
Title: Compactified and Tropicalized Universal Jacobians
Abstract: Moduli spaces of tropical objects can often be obtained as tropicalization of suitable compactifications of algebro-geometric objects. In the talk we will start by giving an overview of the construction of universal compactified Jacobians along with their tropical counterparts. We then show how these can be obtained by tropicalizing two versions of the algebraic universal Jacobian: the logarithmic universal Jacobian and the non archimedian universal Jacobian. We then discuss, within this picture, new stability conditions for (universal) compactified Jacobians.
The talk will be based on joint work of the speaker with S. Molcho, M. Ulirsch, F. Viviani and J. Wise.
Miguel Moreira
Title: The cohomology ring of moduli spaces of 1-dimensional sheaves on P2
Abstract: This talk will be about the cohomology ring of moduli spaces and moduli stacks of semistable 1-dimensional sheaves on P^2. I will explain an approach to describing these rings in terms of generators and relations which allowed us to completely determine the cohomology rings of moduli spaces up to degree 5 and of moduli stacks up to degree 4. One useful new idea that we use is the existence of a representation of half of the Virasoro algebra on the cohomology of the stacks. I will also discuss a conjecture relating the perverse filtration (which carries important curve counting information) and the generators we use, which is an analogue of the P=W conjecture in this compact and Fano setting. This is joint work with Yakov Kononov, Woonam Lim and Weite Pi.
Georg Oberdieck
Title: Quantum cohomology of the Hilbert scheme of points on an elliptic surface
Abstract: Quantum cohomology is a deformation of the classical cup product that encodes the 3-pointed Gromov-Witten invariants of a space. In this talk I will explain the proof of a formula for the quantum product with divisor classes on the Hilbert scheme of points on an elliptic surface S with respect to fiber curve classes. By the work of Hu-Li-Qin this determines the full quantum product with divisor classes whenever p_g(S)>0. Joint work with Aaron Pixton.
Tudor Pădurariu
Title: Quasi-BPS categories for K3 surfaces
Abstract: BPS invariants and cohomology are central objects in the study of (Kontsevich-Soibelman) Hall algebras or in enumerative geometry of Calabi-Yau 3-folds.
In joint work with Yukinobu Toda, a categorical version of BPS cohomology for local K3 surfaces, called quasi-BPS categories. For a generic stability condition, we construct semi orthogonal decompositions of Hall algebra of K3 surfaces in products of quasi-BPS categories of K3 surfaces. When the weight and the Mukai vector are coprime, the quasi-BPS category is smooth, proper, and with trivial Serre functor etale locally on the good moduli space. Thus quasi-BPS categories provide (twisted) categorical (etale locally) crepant resolutions of the moduli space of semistable sheaves on a K3 surface for generic stability condition and a general Mukai vector. Time permitting, I will also discuss a categorical version of the \chi-independence phenomenon for BPS invariants.
Franco Rota
Title: Non-commutative deformations and contractibility of rational curves
Abstract: When can we contract a rational curve C? The situation is much more complicated for threefolds than it is for surfaces: Jimenez gives examples of (-3,1)-rational curves neither contract nor move. Their behaviour is controlled by the functor of non-commutative deformations of C, which conjecturally controls exactly their contractibility.
I will report on work in progress with M. Wemyss, and reinterpret some of Jimenez's examples in terms of non-commutative deformations.