Derived FRG - Titles and abstracts

Talks:

Arend Bayer (University of Edinburgh) Mukai bundles on curves, K3 surfaces and Fano threefolds

A central result in the theory of Fano threefolds of index one and their derived categories is the existence of certain simple rigid vector bundles. Unfortunately, the proofs in the literature contain gaps.

I will present work joint with Alexander Kuznetsov and Emanuele Macrì where we address this gap, and extend the result to finite characteristic. The proof is based on extending spherical bundles on K3 hyperplane sections, which in turn relies on uniqueness of vector bundles on curves satisfying a Brill-Noether condition.

Pierrick Bousseau (University of Georgia, Athens): Quivers and curves in higher dimension.

I will describe two correspondences between Donaldson-Thomas invariants of quivers with d-vertices and punctured Gromov-Witten invariants of d-dimensional toric and cluster varieties. This is joint work with Hulya Arguz (arXiv:2302.02068 and work in progress).

Tom Bridgeland (University of Sheffield): Geometry from Donaldson-Thomas invariants

I will give an update on a line of research which aims to use the DT invariants of a CY3 triangulated category to define a geometric structure on its space of stability conditions.

Amin Gholampour (University of Maryland): SL(r)- and PGL(r)-invariants of surfaces

The SL(r)-invariants of a smooth projective complex surface is defined by means of the moduli spaces of rank r torsion free sheaves with fixed determinant. The PGL(r)-invariants are defined similarly using the moduli of twisted sheaves. In a joint work with Dirk Van Bree, Yunfeng Jiang and Martijn Kool we show that under some mild conditions, these two types of invariants coincide. This are some consequences of this including some applications in Vafa-Witten theory.

Woonam Lim (ETH Zurich): Virasoro constraints, vertex operator algebras, and wall-crossing.

In enumerative geometry, Virasoro constraints were first conjectured for the moduli of stable curves (the Witten conjecture) and stable maps. Recently, the analogous constraints were conjectured in several sheaf theoretic contexts; stable pairs on 3-folds and torsion-free sheaves on surfaces. In joint work with A. Bojko and M. Moreira, we generalize and reinterpret Virasoro conjecture in sheaf theory using Joyce’s vertex algebra. A new interpretation makes use of a conformal element and primary states of vertex operator algebras which are classical subjects in representation theory. As an application, we prove the Virasoro constraints for any moduli of torsion-free sheaves on curves and surfaces via Joyce's wall-crossing formulas.

Emanuele Macrì (Université Paris-Saclay): Stability conditions on Hilbert schemes of points on K3 surfaces

I will present joint work with Chuyni Li, Paolo Stellari and Xiaolei Zhao on invariant stability conditions on product varieties. In particular, this provides examples of stability conditions on Hilbert schemes of points on K3 surfaces.

Cristina Manolache (University of Sheffield): Desingularisation of sheaves and reduced Gromov--Witten invariants

Gromov--Witten (GW) invariants of genus g, with g greater than one, do not count curves of genus g in a given space: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called "reduced GW invariants". More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will present a classical construction which allows us to define reduced Gromov--Witten invariants in all genera. This is work with A. Cobos-Rabano, E. Mann and R. Picciotto.

Georg Oberdieck (KTH): Curve counting on the Enriques surface and Borcherds automorphic form

An Enriques surface is the quotient of a K3 surface by a fixed point-free involution. Klemm and Marino conjectured a formula expressing the Gromov-Witten invariants of the local Enriques surface in terms of automorphic forms. In particular, the generating series of elliptic curve counts on the Enriques should be the Fourier expansion of (a certain power of) Borcherds automorphic form on the moduli space of Enriques surfaces. In this talk I will explain a proof of this conjecture.

Laura Pertusi (University of Milan): Title: Non-commutative abelian surfaces and generalized Kummer varieties.

Examples of non-commutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and Gushel-Mukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these non-commutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available.

In this talk we will construct families of non-commutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkahler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.

Alice Rizzardo (University of Liverpool): Truncated A-infinity enhancements of triangulated functors

It is by now well-known that there are many exact functors between derived categories of coherent sheaves on smooth projective varieties that do not admit a lift between corresponding DG enhancements. While the structure of a triangulated functor is insufficient for many constructions in algebraic geometry, it is actually possible to add additional structure to these non-enhanceable functors if one considers the formalism of A_n categories, a truncated version of the A-infinity category axioms.

Ed Segal (University College London): Hybrid models and Serre functors

The Kuznetsov component for a Fano complete intersection can be described as matrix factorizations on a `hybrid model'. I'll review this story, and explain how it helps us understand properties of the Serre functor on the Kuznetsov component. In particular I'll show how we can re-derive some results of Kuznetsov-Perry. This is joint work with Federico Barbacovi.

Junliang Shen (Yale University): Gopakumar-Vafa theory and moduli of sheaves

The physics proposal of Gopakumar-Vafa (GV) connects the geometry of certain moduli of sheaves and counting invatiants associated with Calabi-Yau 3-folds. I will discuss recent developments concerning the GV theory in some examples including moduli of Higgs bundles on a curve and moduli of 1-dimensional sheaves on a surface. I will explain how/why several different branches of mathematics enter the story in the study of these geometries. This is based on joint work with Davesh Maulik, as well as joint work with Yakov Kononov and Weite Pi.

Paolo Stellari (University of Milan): Morita theory for schemes: the interplay between dg and weakly approximable triangulated categories

A celebrated result by Rickard shows that two affine (coherent) schemes are Fourier-Mukai partners if and only if any of the triangulated categories naturally associated to them (e.g. perfect complexes, bounded coherent, unbounded quasi-coherent,...) are equivalent. This is the beginning of the so-called Morita theory for schemes. In this talk we provide a significant generalization of this result which removes the assumption that the schemes are affine. The proof involves a new mixture of techniques coming from dg enhancements and the new theory of weakly approximable triangulated categories. Our approach is based on significant generalizations of the existing results on both sides. This is joint work in progress with A. Canonaco and A. Neeman.

Yukinobu Toda (Kavli IPMU): Quasi-BPS categories for K3 surfaces

In this talk, I will give semiorthogonal decompositions of derived categories of coherent sheaves on moduli stacks of semistable objects on K3 surfaces. An each summand is given by the categorical Hall product of subcategories called quasi-BPS categories, which approximate the categorification of BPS cohomologies for K3 surfaces. When the weight and the Mukai vector is coprime, the quasi-BPS category is shown to be smooth and proper, with trivial Serre functor etale locally on the good moduli space. So it gives a twisted analogue of categorical crepant resolution of the singular symplectic moduli space, and reminiscents categorical analogue of chi-independence phenomena. This is a joint work in progress with Tudor Padurariu.

Rachel Webb (UC Berkeley): Hasset moduli stacks of twisted curves

A stable n-marked curve is a nodal curve with n distinct marked points and finitely many automorphisms. If we choose rational numbers a_1, . . ., a_n in the interval (0, 1], then a weighted stable n-marked curve is a generalization where the marks are allowed to coincide as long as the total weight at any point is at most one. Moduli of weighted stable curves were first constructed by Hassett. On the other hand, a twisted stable n-marked curve is a tame stack whose coarse moduli space is a stable n-marked curve, such that stacky structure is concentrated at nodes and markings and has a specific local description. I will discuss a modification (using log geometry) of the moduli of twisted stable curves where the markings are allowed to coincide, analogous to Hassett's construction for representable curves. This is a joint work with Martin Olsson.

Fei Xie (University of Edinburgh): Tilting theory on singular surfaces

The work of Karmazyn-Kuznetsov-Shinder develops an approach to construct tilting bundles for surfaces with cyclic quotient singularities, which include du Val singularities of type A but not type DE. This provides semiorthogonal decompositions of derived categories of these surfaces with components equivalent to derived categories of local finite dimensional algebras. I will report on how to construct tilting bundles for surfaces with du Val singularities of type D and describe the corresponding finite dimensional dg algebras. This is work in progress.

Posters:

Alberto Cobos-Rabano (University of Sheffield): Degeneration formula for quasimap invariants

One of the main techniques in Gromov-Witten (GW) theory is the degeneration formula. Given a degeneration of a smooth variety $X$ into the union of two smooth varieties $X_1$ and $X_2$ along a smooth divisor $D$, the degeneration formula relates the (absolute) GW invariants of $X$ with the relative GW invariants of the pairs $(X_1,D)$ and $(X_2,D)$. Here the GW invariants of $X$ come from the moduli space $\overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps to $X$.

We are interested in the theory of stable quasimaps to a toric variety $X$. The moduli space of stable quasimaps $\overline{\mathcal{Q}}_{g,n}(X,\beta)$ is another compactification of the moduli space of maps from a smooth curve to $X$ and, as such, it is closely related to the space $\overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps. The difference between $\overline{\mathcal{M}}_{g,n}(X,\beta)$ and $\overline{\mathcal{Q}}_{g,n}(X,\beta)$ is that, not only the underlying curve may become nodal, but also the map itself may become degenerate by acquiring basepoints. The moduli space $\overline{\mathcal{Q}}_{g,n}(X,\beta)$

is used to produce quasimap invariants which are closely related to GW invariants

We present our approach towards a degeneration formula for quasimaps. This requires defining a moduli space of relative quasimaps for a smooth toric pair $(X,D)$, for which we use expansions of $(X,D)$ as in Li's approach for stable maps. This is work in progress with Cristina Manolache and Qaasim Shafi.

Alex Villaro Krüger (NSU Higher School of Economics)

Augustinas Jacovskis (University of Edinburgh): Categorical Torelli Theorems for Fano Threefolds

A lot of geometric information about a variety X can be recovered from its derived category D(X). If the variety is Fano, then X can in fact be reconstructed up to isomorphism from D(X). This begs the question of whether less information than D(X) can determine X up to isomorphism. In this poster, I'll discuss what happens when “less information” means a certain subcategory of D(X) called the Kuznetsov component. In particular, I will discuss joint work with Xun Lin, Zhiyu Liu, and Shizhuo Zhang which describes the situation for index 1 Fano threefolds, and relations to Hodge theory.

Aurelio Carlucci (Oxford): The geometry of certain moduli schemes in Enumerative Geometry

I would like to present an example of explicit coordinates for the moduli scheme of Pandharipade-Thomas (PT) stable pairs. There is a stratification which admits a simple sheaf-theoretic interpretation, but the actual equations are given via a quiver representation approach, following Nagao-Nakajima. The main technical obstacle is that, while stability for the Hilbert scheme (leading to Donaldson-Thomas invariants) translates to a simple algebraic condition, PT pairs involve a case-by-case analysis.

Daniel Halmrast (UC Santa Barbara): A Refined Approach to Stability in the Hyperkahler Setting

Notions of stability in the sense of Bridgeland are explored in the hyperkahler setting. Such

manifolds arise naturally as the target space for an N=(4,4) supersymmetric string theory, and from

this perspective a new category of topological boundary conditions is determined which is closely

related to the derived category of coherent sheaves of various hyperkahler rotations of the manifold.

As in the case of an N=(2,2) supersymmetric model, such boundary conditions are subject to the

usual binding and decay processes, which appear on the categorical level in the structure of a stability

condition. These stability conditions are examined in the specific case of a K3 surface, and these new

categories and structures are compared to the known structure of the derived category of coherent

sheaves on the K3 and its associated stability conditions.

Hannah Dell (University of Edinburgh): Stability conditions on free abelian quotients

We study slope-stable vector bundles and Bridgeland stability conditions on varieties which are a quotient of a smooth projective variety by a finite abelian group $G$ acting freely. We show there is a one-to-one correspondence between $\widehat{G}$-invariant geometric stability conditions on the quotient and $G$-invariant geometric stability conditions on the cover. We apply our results to describe a connected component inside the stability manifolds of free abelian quotients when the cover has finite Albanese morphism. This applies to varieties with non-finite Albanese morphism which are free abelian quotients of varieties with finite Albanese morphism, such as Beauville-type and bielliptic surfaces. This gives a partial answer to a question raised by Lie Fu, Chunyi Li, and Xiaolei Zhao: If a variety $X$ has non-finite Albanese morphism, does there always exist a non-geometric stability condition on $X$? We also give counterexamples to a conjecture of Fu, Li, and Zhao concerning the Le Potier function, which characterises Chern classes of slope-semistable sheaves. As a result of independent interest, we give a description of the set of geometric stability conditions on an arbitrary surface in terms of a refinement of the Le Potier function. This generalises a result of Fu-Li-Zhao from Picard rank one to arbitrary Picard rank.

Hayato Arai (University of Tokyo): Half-spherical twists on derived categories of coherent sheaves

Let $\pi \colon X \to T$ be a flat morphism between smooth quasi-projective varieties and $Y$ be a fiber. We prove that the twist functors on $D^b(X)$ along spherical objects pushed-forward from $D^b(Y)$ induce autoequivalences of $D^b(Y)$. If $X$ is an elliptic surface $S$ and $Y$ is a reducible fiber of type $\rm{I}$, the induced equivalences of $D^b(Y)$ correspond to the half twists on a punctured $2$-torus by homological mirror symmetry. As an application, we describe the subgroup of $\mathrm{Auteq} D^b(S)$ generated by such twist functors in terms of mapping class groups of punctured tori.

Jae Hwang Lee (Colorado State University): A Quantum H∗(G)-module via Quasimap Invariants

For X a smooth variety or Deligne—Mumford stack, the quantum cohomology ring QH^*(X) is a deformation of the usual cohomology ring H^*(X), where the product structure is modified to incorporate /quantum corrections. /These correction terms are defined using Gromov—Witten invariants. For a GIT quotient V//G, the cohomology ring H^*(V//G) also has the structure of a H^*(G)-module. In this work, we use/quasimap invariants with light points/ and a modified version of the WDVV equation to define a quantum deformation of this H^*(G)-module structure. Using localization, we explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the Batyrev ring when the target is a semipositive toric variety.

Jagadish Pine (Chennai Mathematical Institute): A universal moduli space of stable parabolic vector bundles over the moduli space of marked stable curves

We will outline a construction of a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne-Mumford stable curves $\overline{M}_{_{g, n}}$. The moduli space is projective over $\overline{M}_{_{g, n}}$ under certain assumptions. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over nodal curves $([C], x_1, x_2, \cdots, x_n)$ will remain vector bundles which will not only over $C$ but also over certain marked semistable curves whose fixed marked stable model is $([C], x_1, x_2, \cdots, x_n)$. The total space and the fibers over $\overline{M}_{_{g, n}}$ will have good singularities.

Josiah Foster (University of Massachusetts Amherst): The Lefschetz standard conjectures for varieties of generalized Kummer deformation type

We prove the Lefschetz standard conjectures in certain degrees for generalized Kummer varieties and their projective deformations. The proof relies on Markman's description of the monodromy of generalized Kummer varieties, Verbitsky's theory of hyperholomorphic sheaves, and the decomposition theorem.

Lucien Hennecart (University of Edinburgh): BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

2-Calabi-Yau (2CY) categories feature prominently in algebraic geometry and representation theory. In geometry, they appear as categories of semistable sheaves on K3 and Abelian surfaces. In representation theory, representations of (multiplicative) preprojective algebras of quivers and fundamental group algebras of Riemann surfaces give such categories.

We study the Borel-Moore (BM) homology of the stacks of objects in 2CY categories. The moduli spaces can be recovered as the good moduli spaces of these stacks. The advantage of working with stacks is that their Borel-Moore homologies carry an additional algebraic structure known as cohomological Hall algebras (CoHAs). This makes this homology a particularly pleasant object of study and a rich representation theoretic object.

We will present a decomposition result for the Borel-Moore homology of these stacks in terms of the intersection cohomology of the moduli spaces. This decomposition is obtained using CoHAs. We also have a full description, by generators and relations, of the BPS (Lie) algebra (a canonical subalgebra of the CoHA). The consequences are multiple. Considering the stacks of semistable Higgs bundles of fixed slope and representations of the twisted fundamental group algebra, we obtain a nonabelian Hodge isomorphism between the BM homologies of these stacks. Working with quivers and their preprojective algebras, we obtain a proof of the Bozec-Schiffmann positivity conjecture, and a recursive way of computing the intersection Poincaré polynomials of all quiver varieties.

This is based on joint work with Ben Davison and Sebastian Schlegel Mejia.

Luigi Martinelli (Université Paris-Saclay): Towards a categorical resolution of some singular moduli spaces on K3 surfaces

On a projective K3 surface X, fix a Mukai vector of the form v = 2 v_0, with v_0 primitive, and a v-generic polarisation H. Then, the moduli space M of H-semistable sheaves on X with Mukai vector v is singular. If M is the symmetric product of X or belongs to the class that O'Grady found, it admits a crepant resolution of singularities; in this way, we obtain interesting examples of IHS manifolds. In all the other cases (i.e. if the square ⟨v_0^2⟩ of v_0 for the Mukai pairing is at least 4), Kaledin, Lehn and Sorger showed that a crepant resolution does not exist.

Thus, we are led to look for a categorical crepant resolution of M, which should be an example - and motivate a new definition - of IHS category.

This poster deals with the first step in this direction, namely a detailed study of the resolution of M built by O'Grady when ⟨v_0^2⟩ ≥ 4 and of its exceptional divisor.

Luke Naylor (University of Edinburgh): Tighter bounds on ranks of tilt semistabilizers – narrowing down on possibilities quicker

When trying to narrow down the possibilities for the locations of tilt walls, the typical strategy is to find potential semistabilizers who’s Chern characters satisfy a set of inequalities which would be satisfied by genuine ones (including Bogomolov-Gieseker). In 2020, Benjamin Schmidt published a program to compute the possibilities, however the running time suffered from a conservative estimate for a bound on the ranks of tilt semistabilizers. The work I want to present is a set of refinements on this bound, one in the form of a pragmatic formula to calculate by hand which can be as small as a quarter of the original bound. As well as more complicated formulae with extra parameters which are better suited in the context of a newer computer program. This newer program gives instantaneous results in certain examples where the original takes over an hour, and can easily be tried from your browser at https://pseudowalls.gitlab.io/webapp/tilt.sycamore/.

Marina Purri Brant Godinho (University of Glasgow): When do canonical flops induce derived equivalences?

Flops are fundamental birational transformations which, due to the work of Bridgeland, Chen and Van den Bergh, are known to induce derived equivalences between three-folds with mild singularities. However, it is not always true that flops induce derived equivalences in more singular settings. This poster studies of the question of precisely when canonical flops induce derived equivalences between Gorenstein three-folds. We approach this question by studying how the behaviour of the derived category under flops is influenced by properties of the contraction algebra, an invariant introduced by Donovan and Wemyss.

Marta Benozzo (Imperial): On the canonical bundle formula in positive characteristic

An important problem in birational geometry is trying to relate in a meaningful way the canonical bundles of the source and the base of a fibration. Recently, the problem has been approached with techniques from the minimal model program. These methods can be used to prove a canonical bundle formula result in positive characteristic.

Peter Yi Wei (University of Wisconsin-Madison): Green's Conjecture And The Geometric Syzygy Conjecture in Char p

We study the syzygies of canonical curves of genus g>3 over an algebraically closed field F with char(F)=p>0. With a deformation argument on the moduli space of K3 surfaces, we prove the generic Green's conjecture and a special case of the generic Geometric Syzygy Conjecture, under an assumption of a lower bound on p.

Qingyuan Jiang (University of Edinburgh)

Renata Picciotto (University of Angers): The derived moduli of sections and virtual pushforwards

Derived algebraic geometry provides a powerful way to encode the "virtual structures" of moduli problems I will display ideas from a joint work with D. Karn, E. Mann and C. Manolache in which we define a derived enhancement for the moduli space of sections. This enriched space encodes the perfect obstruction theory and virtual structure sheaves of many theories, such as Gromov-Witten and quasimaps theories. To illustrate the potential of this approach, I will show how we use local derived charts to prove a virtual pushforward formula between stable maps and quasimaps without relying on torus localization.

Robert Hanson (Instituto Superior Tecnico)

Shengxuan Liu (University of Warwick): Kernels of categorical resolutions of nodal singularities

Resolution of singularities is a central topic of algebraic geometry. Hironaka showed that the resolution of singularities exists over \mathbb{C}. An analogous definition in derived categories was proposed by Lunts and the existence of categorical resolutions was shown by Kuznetsov and Lunts. One thus considers whether there is a link between categorical resolutions and classical resolutions of singularities. In this talk, I will discuss the case of nodal singularities. I will start with definitions related to categorical resolutions. Then I will mention the property of the kernel generator of one categorical resolution. Also, I will describe this kernel generator explicitly in the case of the Kuznetsov component of a nodal cubic fourfold. This is joint work with W. Cattani, F. Giovenzana, P. Magni, L. Martinelli, L. Pertusi, and J. Song.

Shizhuo Zhang (Max-Planck institute): Categorical Torelli theorem via equivariant Kuznetsov components

Let Y be a smooth cubic threefold. It is known that irs Kuznetsov component determines its isomorphism class, known as categorical Torelli theorem. We consider a special type of cubic threefold with a geometric involution, called non-Eckardt type cubic threefold. We show that the equivariant category of Kuznetsov component under the Z_2 action generated by this involution already reconstruct the isomorphism class under genericity condition. Moreover, an equivariant infnitesimal categorical Torelli theorem is obtained.

It is based on a joint work with Casalaina-Martin Sebastian, Xun Lin and Zheng Zhang.

Tahsin Saffat (UC Berkeley): Langlands Correspondence for tamely ramified Eisenstein Series over P^1

The Langlands correspondence for a smooth curve over a finite field describes the space of functions on the rational points of Bun_G(X) through the K-theory of LocSys(X_C). It is expected that the space of functions admits a gluing description from a degeneration of X to a nodal graph of genus zero curves. This is motivated by the analogy to geometric Langlands correspondence, which is expected to be an equivalence between 4d TQFTs. The gluing description is built out of atomic building blocks corresponding to P^1 with 0,1,2, or 3 points of tame ramification, all of which are understood except for the last. I will present our work on the space of Eisenstein series on P^1, tamely ramified at three points.

Tanguy Vernet (EPF Lausanne): Rational singularities for moment maps of totally negative quivers

Moment maps of quivers give local models of moduli spaces parametrizing objects in several seemingly unrelated categories: local systems on a smooth complex curve, semistable sheaves on a K3 surface, representations of multiplicative preprojective algebras... It is therefore fruitful to study singularities of quiver moment maps in order to gain information on singularities of these moduli spaces. In this talk, I will show that a large class of quiver moment maps have rational singularities: namely moment maps of totally negative quivers. This has interesting arithmetic consequences on counts of jets of these moduli spaces over finite fields.

Tomohiro Karube (University of Tokyo): Stability conditions on degenerated elliptic curves

Bridgeland introduced the concept of stability conditions on triangulated categories, which draws inspiration from the field of string theory. My research focuses on studying stability conditions on reducible Kodaira curves that arise from degenerations of elliptic curves. Specifically, I describe the connected components of the spaces of stability conditions and calculate the groups of deck transformations associated with these components. Additionally, I establish a relationship between the space of stability conditions and the autoequivalence group of degenerated elliptic curves.

Tomoki Yoshida and Yuki Mizuno (Waseda University): The Projectivity of Bridgeland Moduli Spaces of del Pezzo Surface of Picard Rank Three

In this poster, we partially showed the projectivity of the moduli spaces of Bridgeland semi-stable objects on the del Pezzo surface of Picard rank $3$ for any divisorial stability conditions.

Weite Pi (Yale University): Moduli of one-dimensional sheaves on P^2: from cohomology to Gopakumar--Vafa theory

Gopakumar--Vafa invariants are certain curve counting invariants predicted by physicists for Calabi--Yau 3-folds. They are expected to behave nicely (e.g. being integral) and recover other enumerative theories such as Gromov--Witten and Donaldson--Thomas invariants. One mathematical proposal defines these invariants via the cohomology of moduli of one-dimensional sheaves. When we specialize to local P^2, this brings us to the study of the moduli of one-dimensional sheaves on P^2. We present several results on the cohomological and enumerative aspects of this moduli space, including a minimal generation result, a χ-dependence result, and a conjectural product formula for the Gopakumar--Vafa invariants. One key ingredient is to produce certain “Mumford relations” for these moduli spaces. Based on joint work with Y. Kononov, W. Lim, M. Moreira, and J. Shen.

Xiaohan YAN (Sorbonne Université): Quantum K-theory of flag varieties via abelian/non-abelian correspondence

Quantum K-theory studies a K-theoretic analogue of the Gromov-Witten invariants defined as holomorphic Euler characteristics of vector bundles over the moduli spaces of stable maps. In genus-zero case, such invariants are recorded in the generating function commonly known as the big J-function. We reconstruct the big J-function of flag varieties (non-abelian GIT quotients of vector spaces) from the well-understood small J-function of certain toric varieties (the associated abelian quotients), through the $D_q$-module structure on its image cone and fixed point localization on moduli spaces. The same method may be applied to study various twisted quantum K-theories of the flag varieties, leading to demonstration of several properties including a non-abelian quantum Lefschetz theorem and a duality between level structures.

Zhiyu Liu (Zhejiang University): New perspectives on categorical Torelli theorems for del Pezzo threefolds

We apply two different viewpoints to study a del Pezzo threefold Y of Picard rank one (e.g. cubic threefold or quartic double solid) via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component:

(i) Brill-Noether reconstruction. We show that Y can be uniquely recovered as a Brill-Noether locus of Bridgeland stable objects in its Kuznetsov component. As an application, we give a uniform proof of the categorical Torelli theorem when the degree d of Y satisfies degree 2≤d≤4.

(ii) Exact equivalences. We prove that, up to composing with an explicit auto-equivalence, any Fourier-Mukai type exact equivalence of Kuznetsov components of two del Pezzo threefolds of degree 2≤d≤4 can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of exact auto-equivalences of the Kuznetsov component of Y of Fourier-Mukai type.