The workshop will focus on the enumerative invariants of moduli spaces of sheaves. Particular topics are Hilbert schemes of points, wall-crossing formulas, and modularity.

Date: Monday, September 18, 2023 to Tuesday, September 19th, 2023

**Speakers:**

- Noah Arbesfeld (Imperial College London)
- Gergely Bérczi (Aarhus)
- Nikolas Kuhn (Oslo)
- Oliver Leigh (Uppsala)
- Denis Nesterov (Vienna)
- David Rydh (KTH)
- Thorsten Schimannek (Paris)
- Maximilian Schimpf (KTH)

**Location: **

KTH Department of Mathematics, Lindstedsvägen 25

Monday: Room 3721

Tuesday: Room 3721 in the morning, Room 3418 in the afternoon.

08:50 Welcome

09:00-10:15 Talk 1: Bérczi

10:45-12:00 Talk 2: Arbesfeld

14:00-15:15 Talk 3: Rydh

15:45-17:00 Talk 4: Kuhn

Tuesday:

09:00-10:15 Talk 5: Schimpf

10:45-12:00 Talk 6: Schimannek

14:00-15:15 Talk 7: Leigh

15:45-17:00 Talk 8: Nesterov

**Title and abstracts**

**Noah Arbesfeld**: *Computing vertical Vafa-Witten invariants*

I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined by integration over moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions.
One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a certain quiver variety, the instanton moduli space of torsion-free framed sheaves on P^2. Using an identity of Kuhn-Leigh-Tanaka, we deduce constraints on Vafa-Witten invariants conjectured by GĂ¶ttsche-Kool-Laarakker. One consequence is a formula for the contribution of the vertical component to refined Vafa-Witten invariants in rank 2.

**Gergely Bérczi**: *Intersection theory of the Hilbert scheme of points*

While we have a solid understanding of the Hilbert scheme of points on surfaces, the Hilbert scheme over manifolds poses a more complex challenge. It exhibits a blend of unusual and unexplored properties, and our understanding of its topology, singularities, and deformation theory is very limited.
We report on recent insights into the distribution of torus fixed points within the components of the Hilbert scheme, and on a novel formula for computing tautological integrals over the main components. We present a Chern-Segre-type positivity conjecture for tautological integrals and discuss applications in enumerative geometry.

**Nikolas Kuhn**: *Degeneration of Sheaves on Fibered surfaces*

Inspired by a question of Donaldson, Gieseker and Li investigated how moduli spaces on a surface behave under a simple degeneration of the surface. While in Gromov-Witten theory, and for Hilbert/Quot schemes of zero-dimensional objects, the analogous problem leads to a degeneration formula, they failed to realize their program for higher rank sheaves. I will explain how, when one restricts to fibered surfaces, and passes to an adapted notion of stability a good theory exists, and one has a degeneration formula. This can be used to compute elliptic genera of moduli spaces of stable sheaves on some elliptic surfaces. I will also point out how this relates to quasi-maps into moduli stacks (as studied by Nesterov), and speculate on some generalization.

**Oliver Leigh**: *The Donaldson-Thomas theory of threefolds and the Elliptic genus of sheaves on surfaces*

Donaldson-Thomas theory is a well-celebrated modern tool for studying Calabi-Yau threefolds. In this theory, one studies weighted Euler characteristics of moduli spaces of sheaves on threefolds. Elliptic genus on the other hand is a refinement of Euler characteristic motivated by a hypothesis of Witten. In this talk I will discuss and present evidence of a surprising relationship between the two. That is, a relationship between the Elliptic genus of sheaves on surfaces and the Donaldson-Thomas theory of elliptically fibred threefolds.

**Denis Nesterov:**: *Unramified Gromov-Witten invariants and Gopakumar-Vafa invariants*

Kim, Kresch and Oh defined moduli spaces of unramified stable maps, which are natural generalisations of Hurwitz spaces for a target of an arbitrary dimension. Just like Hurwitz spaces, which are smooth irreducible varieties after normalisation, moduli spaces of unramified stable maps are 'better' compactifications than moduli spaces of stable maps. Pandharipande conjectured that unramified Gromov-Witten invariants of a threefold are equal to Gopakumar-Vafa (BPS) invariants in the case of Fano classes (classes that intersect negatively with the canonical class). After a gentle introduction to unramified Gromov-Witten theory, we will discuss a work in progress which aims to prove the conjecture for Fano classes and primitive Calabi-Yau classes. The proof is based on a certain wall-crossing technique.

**David Rydh:**: *Derived Kirwan resolution*

Abstract: Kiem, Li and Savvas (2013, 2017) introduced intrinsic blow-ups and a Kirwan resolution procedure for singular G-equivariant schemes and singular Artin stacks with good moduli spaces. Under suitable assumptions, they also endowed the Kirwan resolution with an (almost) perfect obstruction theory leading to an alternative approach to generalized DT invariants in the presence of strictly semi-stable objects. I will describe how their construction has a fully derived interpretation. This is joint work with Edidin, Hekking, Khan and Savvas.

**Thorsten Schimannek**: *Counting curves on non-Kähler Calabi-Yau 3-folds*

The generating function of Gromov-Witten invariants on a smooth projective Calabi-Yau 3-fold arises in physics as the so-called A-model topological string partition function.
Various conjectures relate this to the generating functions of stable pair invariants, Donaldson-Thomas invariants and Gopakumar-Vafa invariants, with all of these invariants providing a different notion of what it means to ``count curves''.
If the Calabi-Yau has singularities that are resolved by torsion curves in a small resolution, the latter is necessarily neither projective nor Kähler.
Nevertheless, physical arguments lead to the notion of torsion refined Gopakumar-Vafa invariants which, as we will argue, can be calculated by combining the information from the A-model topological string partition function on the singular Calabi-Yau itself and on different non-commutative resolutions.
The partition functions can in turn often be obtained, at least for low curve genus, using mirror symmetry and the results agree with predictions from a conjectural mathematical definition of the invariants in cases where the latter admit a direct calculation.
We will illustrate these ideas in a simple example and outline the current status of the mathematical definition of the invariants.
We will then discuss the crucial role of the phenomenon in understanding the modular properties of the topological string partition function on torus fibered Calabi-Yau 3-folds without a section.

**Maximilian Schimpf**: *Stable pairs on local curves and Hilbert schemes on surfaces*

As evidenced by the recent groundbreaking work of John Pardon, local curves (i.e. total spaces of rank 2 vector bundles on curves) are central in the enumerative geometry of threefolds. However, not much is known about their descendent theory besides structure results. This work in progress project aims to rectify this by attempting to find full formulas for all their descendent invariants in PT theory - this has possibly far-reaching consequences for Virasoro constraints, Hilbert schemes of points on surfaces, Nakajima quiver varieties and more.

**Organization:** Georg Oberdieck

**Funding:** The workshop is supported by the starting grant 'Correspondences in enumerative geometry: Hilbert schemes, K3 surfaces and modular forms', No 101041491 of the European Research Council.