Schedule
Monday [06.06.2022]
09:15 - 10:30 Huybrechts
11:00 - 12:15 Szendroi
Lunch at 12:30
16:30 - 18:15 Feyzbakhsh
Tuesday [07.06.2022]
09:15 - 10:30 Hausel
11:00 - 12:15 Trapeznikova
Lunch at 12:30
16:30 - 18:15 Davison
Wednesday [08.06.2022]
09:15 - 10:30 Göttsche
11:00 - 12:15 Joyce [slides]
Lunch at 12:30
16:30 - 18:15 Oberdieck
Thursday [09.06.2022]
09:15 - 10:30 Moreira
11:00 - 12:15 Bojko [notes]
Lunch at 12:30
16:30 - 18:15 Ranganathan
Friday [10.06.2022]
09:30 - 10:45 Molcho
Titles and Abstracts
A. Bojko
Virasoro for moduli of sheaves: the wall-crossing
Joyce recently introduced a new wall-crossing framework for sheaf-counting theories which lead to many open questions. To formulate it, he constructs a vertex algebra structure coming from geometric data which contains a natural choice of something called a Virasoro element. A completely independent work of Moreira--Oblomkov--0kounkov--Pandharipande has lead to transporting the Virasoro constraints from the Gromov--Witten side to stable pairs on 3-folds and eventually reducing it to Hilbert schemes of points on surfaces. The latter describes an example of a geometric realization of the former and can therefore be extended to more general moduli spaces including higher rank stable sheaves on surfaces and curves as was observed by van Bree and discussed by Miguel Moreira.
After explaining the vertex algebra construction of Joyce, I will formulate the geometric Virasoro constraints in a language compatible with it. The first result that this allows to prove is Virasoro constraints for the Grassmannian and flag variety, where the Virasoro vector does the full job. Moving on to one dimension higher and studying moduli spaces of sheaves and their pairs on curves introduces new complications, but still leads to a full proof. I will finish with some speculations of what is required in higher dimensions. This is joint work with M. Moreira and W. Lim and I will rely on the language and examples introduced by Miguel in his talk.
B. Davison
Nonabelian Hodge isomorphism for moduli stacks
If C is a smooth projective complex curve, then by classical nonabelian Hodge theory there is a diffeomorphism between the coarse moduli space of representations of the fundamental group of C, and the coarse moduli space of degree zero semistable Higgs bundles on C. In particular, the Borel-Moore homology of these two moduli spaces is isomorphic.
In this talk I will construct an isomorphism between the Borel-Moore homologies of the full stack of representations of the fundamental group and the full stack of degree zero semistable Higgs bundles. Rather than proceeding via any kind of isomorphism between these two stacks, the isomorphism in BM homology takes a roundabout route, via an isomorphism of the "BPS cohomology" of the two moduli problems. This in turn is provided by the classical nonabelian Hodge theory, along with a freeness result regarding the BPS Lie algebra on both sides of nonabelian Hodge theory. This is joint work in progress with Lucien Hennecart and Sebastain Schlegel-Mejia.
S. Feyzbakhsh
Rank r DT theory from rank 1
Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macri-Toda, such as the quintic 3-fold. I will first explain work with Richard Thomas which proves the existence of a universal formula expressing Joyce’s generalised DT invariants counting Gieseker semistable sheaves of any rank r on X in terms of those counting sheaves of rank 1. By the MNOP conjecture, the latter are determined by the Gromov-Witten invariants of X. In the second part of my talk, I will show our formula can be made explicit for rank r=0 or 2 when X is of Picard rank one.
L. Göttsche
Segre and Verlinde numbers of Hilbert schemes of points
This is joint work with Anton Mellit.
We give the complete generating functions for the Segre and Verlinde
numbers of Hilbert schemes of points.
The formula is proven for surfaces with K_S^2=0, and conjectured in
general. Without restriction on K_S^2 we prove the conjectured
Verlinde-Segre correspondence relating Segre and Verlinde
numbers of Hilbert schemes. Finally we find a generating function for
finer invariants, which specialize to both the Segre and Verlinde
numbers, giving some kind of explanation of the Verlinde-Segre
correspondence.
T. Hausel
Enhanced mirror symmetry for Langlands dual Hitchin systems
I will schematically recall mirror symmetry ideas between Langlands dual Hitchin systems. Enhancements were suggested by Kapustin-Witten, by considering hyperkahler branes and
certain Hecke and Wilson operators. We will apply them in the simplest case on the
structure sheaf of the moduli spaces. What results is a circle of conjectures about the mirrors
of certain upward flows in the Hitchin system, and bundle of algebra structures on the universal
Higgs bundle in a corresponding representation along the Langlands dual Hitchin section.
We model them as spectra of equivariant cohomology of affine Schubert varieties - like
Grassmannians- and Kirillov algebras of irreducible representations of the Langlands
dual group.
D. Huybrechts
Nodal quintic surfaces and the Fano varieties of lines
I will survey old and new results concerning this classical class of
surfaces (Hodge, Chow, ...) and discuss further open questions.
D. Joyce
Counting semistable coherent sheaves on surfaces
In arXiv:2111.04694 I set out a programme which gives a common universal structure to many theories of enumerative invariants counting semistable objects in abelian or derived categories in Algebraic Geometry, for example, counting coherent sheaves on curves, surfaces, Fano 3-folds, Calabi-Yau 3- or 4-folds, or representations of quivers (with relations). I will outline the general programme briefly, and go into detail on the case of invariants counting (Gieseker or \mu-)semistable sheaves on complex projective surfaces.
Let X be a complex projective surface, and M the moduli stack of objects in D^b coh(X), and M^{pl} the "projective linear” of objects in D^b coh(X) modulo "projective linear” isomorphisms (quotient by multiples of identity morphisms).
My theory makes the homology H_*(M) into a vertex algebra, and H_*(M^{pl}) into a Lie algebra. Work of my student Jacob Gross allows us to describe H_*(M) and H_*(M^{pl}) and their algebraic structures completely explicitly when X is a surface: H_*(M) is the tensor product of a group algebra and a superpolynomial algebra in infinitely many variables, with a super-lattice vertex algebra structure, and H_*(M^{pl}) is the quotient of this by a certain ideal. (This is why I work with moduli of objects in D^b coh(X), and not just in coh(X).)
Let t be Gieseker or \mu-semistability on X with respect to a real Kahler class. Then for each Chern character \alpha we can define the moduli stacks M^{st}_\alpha(t) \subseteq M^{ss}_\alpha(t) of t-(semi)stable coherent sheaves in class \alpha, regarded as open substacks of M^{pl}. If stable = semistable then M^{ss}_\alpha(t) is a proper moduli scheme with a Behrend-Fantechi obstruction theory, so has a virtual class [M^{ss}_\alpha(t)]_\virt, which we regard as living in the homology H_*(M^{pl}).
For any \alpha, my theory defines an invariant [M^{ss}_\alpha(t)]_\inv in H_*(M^{pl}), equal to the virtual class if stable = semistable. These invariants satisfy a wall-crossing formula under change of stability condition t, written using the Lie bracket on H_*(M^{pl}). If the geometric genus p_g is positive the invariants are “reduced” for rank \alpha > 0.
Because the invariants [M^{ss}_\alpha(t)]_\inv lie in a polynomial ring in infinitely many variables, they contain a lot of information. In particular, basically every invariant of coherent sheaves on surfaces that people study — virtual Euler characteristics, virtual \chi_y-genera, Donaldson invariants, K-theoretic Donaldson invariants, Segre integrals, Verlinde integrals, … — is an integral of some universal cohomology class over the invariants [M^{ss}_\alpha(t)]_\inv, so if we can compute [M^{ss}_\alpha(t)]_\inv, we compute all these other invariants too.
For projective surfaces X we can also define invariants in an auxiliary abelian category A of "pairs” whose objects are triples (V,E,\phi) of a vector space V, a coherent sheaf E and a morphism \phi: V \otimes L —> E for L a fixed line bundle on X. By using wall-crossing formulae in A we can write higher rank invariants in terms of rank 1 invariants. In this way we can write higher-rank sheaf counting invariants in terms of invariants counting rank 1 pairs (roughly, algebraic Seiberg-Witten invariants) and rank 1 sheaves (roughly, Hilbert schemes of points on X). This is an algebraic analogue of computing Donaldson invariants from Seiberg-Witten invariants. The method already appears in Mochizuki 2009, but my new Lie algebra and wall crossing formula ideas allow me to be more explicit.
In work in progress, I am trying to give a general formula for the generating function of invariants [M^{ss}_\alpha(t)]_\inv in any rank r > 0, in terms of certain universal functions in infinitely many variables, at least when p_g > 0. Completing this programme would prove a number of conjectures by Gottsche-Kool and others.
S. Molcho
The double ramification cycle via degeneracy loci
The double ramification (DR) cycle is a class of special interest in the Chow ring of the moduli space of curves. Roughly, fixing integers a_1,a_2,...a_n summing up to 0, the DR cycle is the cycle of curves admitting a rational function with zeros and poles given according to the a_i. A remarkable formula, called Pixton's formula, expresses the DR cycle in terms of tautological classes and provides the most thorough way to understand it to date. However, to make progress -- for instance, to approach the analogous problem in higher dimensions -- one needs to study a certain refinement of the DR cycle, called the logarithmic double ramification (logDR) cycle, for which such a formula is unavailable. In this talk, I want to discuss two approaches to obtain such a formula: one based on recent work with Holmes, Pandharipande, Pixton and Schmitt, producing a refinement of Pixton's formula, and one based on work in progress with Abreu-Pagani, which uses more elementary techniques and produces a different formula. Both approaches rely on the geometry of compactified Jacobians, and the latter suggests the definition of new cycles in the tautological ring, related to Brill-Noether theory.
M. Moreira
Virasoro for moduli of sheaves: the geometry
This talk will be the first of two reporting on joint work with A. Bojko and W. Lim on Virasoro constraints for moduli spaces of sheaves. The discovery of Virasoro constraints for stable pairs via the GW/PT correspondence led to a new study of such constraints for integrals of descendents in different moduli of sheaves, and it appears to be a general phenomenon. In our work we fit the Virasoro operators in the vertex algebra Joyce recently introduced to study wall-crossing and use that to show compatibility between wall-crossing and the constraints.
In my talk I will give a general introduction to descendent invariants on moduli of sheaves, state our conjectures and sketch the path to prove Virasoro for moduli of stable sheaves on curves via rank reduction.
G. Oberdieck
Holomorphic anomaly equations for the Hilbert schemes of points
of a K3 surface
Holomorphic anomaly equations are structural properties predicted by
physics for the Gromov-Witten theory of Calabi-Yau manifolds. In this
talk I will explain the conjectural form of these equations for the
Hilbert scheme of points of a K3 surface, and explain how to prove them
for genus 0 and up to three markings. As a corollary, the (reduced)
quantum cohomology of Hilb^n K3 is determined up to finitely many
coefficients. The results also imply that the DT theory of CHL
Calabi-Yau threefolds is governed by Jacobi forms. If time permits I
will discuss holomorphic anomaly equations in a more general setting.
D. Ranganathan
Gromov-Witten theory via roots and logarithms
I will discuss the relationship between the Gromov-Witten theory of simple normal crossings pairs and the Gromov-Witten theory of orbifolds obtained by root constructions along the divisor. When the divisor is smooth, the comparison is a rich computational tool, for example giving rise to Pixton’s formula for the double ramification cycle. When the divisor is simple normal crossings, I will explain a new comparison result, proved in recent work with Battistella and Nabijou, and discuss its consequences. In the second half of the talk, I will work in progress concerning a parallel result in the “negative contact” regime, and its potential implications for mirror symmetry and symplectic cohomology.
B. Szendroi
ADE singularities, Quot schemes and generating functions
Starting with an ADE singularity C^2/Gamma for Gamma a finite subgroup of SL(2,C), one can build various moduli spaces of geometric and representation-theoretic interest as Nakajima quiver varieties. These spaces depend in particular on a stability parameter; quiver varieties at both generic and non-generic stability are of geometric interest. We will explain some of these connections, focusing in particular on generating functions of Euler characteristics at different points in stability space. Based on joint papers and projects with Craw, Gammelgaard, Gyenge, and Nemethi.
O. Trapeznikova
The Verlinde formula and K-theory of the moduli space of parabolic vector
bundles
The Verlinde formula, an expression for the Hilbert function of the moduli spaces of parabolic vector bundles on Riemann surfaces, is one of the most beautiful results in enumerative geometry. In this talk, I will present a new proof of this formula (joint work with Andras Szenes) based on a wall-crossing technique and the tautological Hecke correspondence. A more general problem is the calculation of Euler characteristics of universal vector bundles on moduli spaces, and I will explain how our approach can be used to deduce explicit formulas in this case.
