Schedule and Abstracts

Titles and Abstracts:

Evgeny Shinder: Homological Bondal-Orlov localization conjecture

An old conjecture going back to Bondal and Orlov predicts a relation between the derived categories of a variety with rational singularities and its resolution of singularities. I will explain the proof of the surjectivity part of this conjecture. The proof is based on an argument from Hodge theory. This is joint work with Mirko Mauri.

Tyler Kelly: Open Mirror Symmetry for LG models.

In this talk I will explain recent work joint with Mark Gross and Ran Tessler that establishes a form of mirror symmetry between certain Landau-Ginzburg models using open enumerative invariants. This involves relating a new open version of FJRW theory with the mirror's Saito-Givental theory. I'll try to mention some relations to categories, considering the audience.

Eleonore Faber: Line singularities and Grassmannian cluster structures

This talk is about a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules over the commutative ring $\mathbb{C}[x,y]/(x^k)$. This yields an infinite rank analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su.

In the special case k=2, $\mathrm{Spec}(\mathbb{C}[x,y]/(x^2))$ is a type $A_{\infty}$-curve singularity and the ungraded version of our category has been studied by Buchweitz, Greuel, and Schreyer in the 1980s. We show that this Frobenius category has infinite type A cluster combinatorics, in particular, that it has cluster-tilting subcategories modelled by certain triangulations of the (completed) infinity-gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed infinity-gon. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

Nina Morishige: Gopakumar-Vafa invariants of multi-Banana configurations

The multi-Banana configuration is a smooth local Calabi-Yau threefold of Schoen type. It can be constructed as a conifold resolution of the fiber product of two elliptic surfaces over a formal disc, each of which has an Iv, respectively Iw, singularity on the central fiber. We discuss the genus 0 Gopakumar-Vafa invariants of certain fiber curve classes of the multi-Banana configuration, and illustrate the computation explicitly for the cases v=1 and v=w=2. The resulting partition function can be expressed in terms of elliptic genera of the complex plane, or classical theta functions, respectively.

Agnieszka Bodzenta: Reconstruction of a surface from reflexive sheaves

Consider the category D of normal surfaces and open embeddings with complement of codimension two. I will show that any connected component of D contains a unique codim-2-saturated surface, i.e. such X that any morphism in D with domain X is an isomorphism. I will call such X the codim-2-saturated model of any surface X_0 in its connected component.

The category Ref of reflexive sheaves is constant on a connected component of D. I will recover X from Ref(X_0). The main tool in the reconstruction will be the right abelian envelope of Ref(X_0) with its canonical exact structure. It will allow us to pass from Ref(X_0) to Ref(U) for any open U in X_0. A characterization of quasi-affine surfaces in terms of their categories of reflexive sheaves will finally allow us to define X as the colimit of spectra of centers of Ref(U) over all quasi-affine open U in X. This is based on a joint work with A. Bondal.

Fei Xie: Quadric bundles of relative even dimension

I will discuss derived categories of families of even dimensional quadric hypersurfaces, more precisely, their non-trivial semi-orthogonal components called residual categories. I will show that the residual category of this family with fibres of corank at most 2 is the derived category of some scheme after an étale base change. This generalises the well known result of such a family with simple degeneration (fibres have corank at most 1). This is achieved by working with Clifford ideals and hyperbolic reduction.

Martin Kalck: Derived categories of singular projective varieties and finite dimensional algebras

We will describe recent progress on describing derived categories of coherent sheaves for certain singular projective varieties in terms of derived categories of finite dimensional algebras, which are typically noncommutative. This is based on ongoing joint works with Yujiro Kawamata & Nebojsa Pavic and with Carlo Klapproth, Nebojsa Pavic & Evgeny Shinder.

Ben Davison: Nonabelian Hodge theory for singular moduli stacks

The classical nonabelian Hodge isomorphism provides a homeomorphism between different coarse moduli spaces appearing in nonabelian Hodge theory, in particular the moduli space of semistable Higgs bindles of fixed rank and degree on a smooth projective curve C, and certain twisted representations of the fundamental group of C. In the case in which the rank and the degree are coprime, this yields an isomorphism in cohomology between the corresponding moduli stacks.

In the case of non-coprime rank and degree, the stacks appearing in nonabelian Hodge theory are singular, and much more stacky, and it appears to be hopeless to write down a homeomorphism between them. However, using local neighbourhood theorems, cohomological Hall algebras, and some cohomological DT theory, it nonetheless turns out to be possible to write down canonical isomorphisms in Borel-Moore homology relating them. This is joint work with Lucien Hennecart and Sebastian Schlegel Mejia.

CONTRIBUTED TALKS:

Ananyo Dan: McKay-type correspondence for non-quotient singularities

Given a quotient surface singularity X, McKay correspondence gives a 1-1 correspondence between irreducible components of the minimal resolution of X and the non-trivial indecomposable reflexive sheaves on X. Such a correspondence has been generalized to certain cases of higher dimensional quotient singularities by Ito and Reid, where they give a correspondence between crepant divisors on a resolution and maximal Cohen Macaulay modules. Using the language of derived categories these results have been extended to certain other quotient singularities by King-Reid-Bridgeland. In this talk, we will focus on the classical correspondence given by Artin-Verdier and observe how it generalizes to non-quotient surface singularities. I will give ideas how this can be extended to the higher dimension case. If time permits, I will say a couple of words on how to apply this to prove the Drozd-Greuel-Kashuba conjecture. This is joint work with J. F. de Bobadilla and A. R. Velazquez.

Matt Habermann: Homological Berglund-Hübsch-Henningson mirror symmetry for curve singularities

Invertible polynomials are a class of hypersurface singularities which are defined from square matrices with non-negative integer coefficients. Berglund—Hübsch mirror symmetry posits that the polynomials defined by a matrix and its transpose should be mirror as Landau—Ginzburg models, and an extension of this idea due to Berglund and Henningson postulates that this equivalence should respect equivariant structures. In this talk, I will explain my recent work on establishing homological Berglund—Hübsch—Henningson mirror symmetry in the first non-trivial dimension; that of curves. The key input, inspired by the derived McKay correspondence, is a model for the orbifold Fukaya—Seidel category in this context.

Hannah Dell: Geometric stability conditions on quotients

A Bridgeland stability condition on a smooth projective variety X is called geometric if all skyscraper sheaves are stable. We are interested in such stability conditions, as X can be recovered from the corresponding moduli spaces of stable objects. In this talk, I will discuss two ways of relating geometric stability conditions on a variety X with any free quotient of X by a finite abelian group. One method uses Deligne’s notion of group actions on triangulated categories. The second method involves Le Potier functions (which give constraints on the Chern characters of semistable sheaves).

Augustinas Jakovski: The period map for Gushel--Mukai threefolds

The intermediate Jacobian of a variety X carries a lot of geometric information about X, e.g. rationality, whether Torelli statements hold, etc. Let X be a Gushel--Mukai threefold. I will explain how the Kuznetsov component of X and the intermediate Jacobian of X are related, and how in this case they conjecturally carry the same information. This is based on joint work with Xun Lin, Zhiyu Liu, and Shizhuo Zhang.

Aimeric Malter: The use of VGIT to study derived categories of mirror constructions

Abstract: In this talk I aim to show how methods of VGIT can be used to compare the derived categories associated to different mirror constructions. On the example of the Libgober-Teitelbaum and Batyrev-Borisov mirror constructions, I will discuss how to use VGIT and partial compactifications to provide equivalences on the level of derived categories. Afterwards, I will talk about some direct generalisations of the Libgober-Teitelbaum construction and how the previous methods can be adapted to provide some categorical resolutions.

Dave Murphy: Grothendeick Groups of a Completion of Discrete Cluster Categories.

Abstract: Discrete cluster categories of Dynkin type A were introduced by Igusa and Todorov as the stable Frobenius category of certain cyclic posets, and a completion of them was introduced by Paquette and Yildirim in the last few years. These completions are defined combinatorically and have recently appeared via a natural construction in algebraic geometry.

In this talk, we briefly look at the completions due to Paquette and Yildirim, and compute their triangulated Grothendieck groups.

Karoline Van Gemst: Frobenius manifold mirror symmetry and Saito discriminant strata

Frobenius manifolds are complex manifolds with a certain algebraic structure on their tangent bundle. They were first formally defined by Boris Dubrovin in the 1990’s in order to describe the geometry of two-dimensional topological field theories, but it turns out that such manifolds arise in a-priori very distinct fields such as enumerative geometry, singularity theory and integrable systems, and are intimately related to the phenomenon of mirror symmetry. In this short talk I will highlight some recent results (published jointly with Andrea Brini) on Frobenius manifold mirror symmetry, and a work-in-progress application generalising a result by Georgios Antoniou, Misha Feigin and Ian Strachan.