Pieter Belmans—seminar

In Luxembourg

Below is the current schedule for the algebraic geometry seminar. Talks take place in MNO, first floor.

In weeks without an external speaker we will sometimes set up internal events, besides the ongoing reading groups, which are not explictly mentioned on this page.

November 13 (11h00)

Chiara Sava

∞-Dold-Kan correspondence via representation theory.

In this talk we give a purely derivator theoretical reformulation and proof of a classic result of Happel and Ladkani, showing that it occurs uniformly across stable derivators and it is then independent of coefficients. The resulting equivalence provides a bridge between homotopy theory and representation theory: indeed, we explain how our result is a derivator-theoretic version of the ∞-Dold-Kan correspondence for bounded cochain complexes.

November 6 (11h00)

Claudio Bartocci

Moduli spaces of quiver representations

I will recall some results about quiver varieties representing moduli spaces of framed sheaves on algebraic surfaces, with special attention to Hirzebruch surfaces. Some results concerning the existence of Poisson structures on these varieties will be discussed.

October 30 (13h00)

Paolo Tomasini

Equivariant elliptic cohomology and mapping stacks

In this talk we will present a new construction of equivariant elliptic cohomology via mapping stacks. All necessary background on elliptic cohomology will be given throughout the talk. Then we will introduce a suitable notion of elliptic Hochschild homology in terms of functions on a stack of quasi-constant maps from an elliptic curve, which allows to compute (rationalized) equivariant elliptic cohomology.

May 23 (16h30)

Raymond Cheng

Double quartic fivefolds

Double covers of projective 5-space branched along a quartic fourfold are Fano varieties with a CY3 category for its Kuznetsov component. The purpose of this talk is to examine, in relation to Kuznetsov’s rationality philosophy and also a non-commutative version of Reid’s Fantasy, two special families of such varieties in which the Kuznetsov component is twisted-geometric. This is joint work with Alex Perry and Xiaolei Zhao.

April 10 (15h00)

Max Briest

On the bounded derived category of coherent sheaves of the orthogonal Grassmannian $\mathrm{OGr}(3,2n+1)$

We set the orthogonal Grassmannian $X = \operatorname{OGr}(3,2n+1)$ -- i.e. the $3$-dimensional isotropic subspaces in a $2n+1$-dimensional vector space -- and we are interested in the bounded derived category of coherent sheaves on $X$, namely $\mathbf{D}^{\mathrm{b}}(X)$. This highly interesting geometric invariant $\mathbf{D}^{\mathrm{b}}(X)$ admits two exceptional Lefschetz subcollections which can be merged to a single big collection. We outline the shape of these two subcollections and discuss small cases of $n$; e.g. $X = \operatorname{OGr}(3,9)$.

March 6 (16h00)

Sebastian Torres

Windows and the BGMN conjecture

Let $C$ be a smooth projective curve of genus at least 2, and let $N$ be the moduli space of semistable rank-two vector bundles of odd degree on $C$. We construct a semi-orthogonal decomposition in the derived category of $N$ conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form $\mathbf{D}^{\mathrm{b}}(C_d)$ where $C_d$ are $d$-th symmetric powers of $C$, and the semi-orthogonal complement to these blocks is conjecturally trivial. In order to prove our result, we use the moduli spaces of stable pairs over $C$. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall. This is a joint work with J. Tevelev.

February 5 (15h30)

Hannah Dell

Stability conditions on free abelian quotients

The space of Bridgeland stability conditions on a given triangulated category is a complex manifold. This gives us a way to extract geometry from homological algebra. In this talk, I will describe how stability conditions behave under actions of finite abelian groups. We will apply this to derived categories of surfaces that are free quotients by finite abelian groups. When the cover has finite Albanese morphism, this produces a connected component of so-called geometric stability conditions inside the stability manifold of the quotient. A consequence of this is a disproof of the expectation that surfaces with irregularity 0 always admit a wall of the geometric chamber. If time permits, I will discuss joint work in progress with Edmund Heng and Tony Licata to extend this to non-abelian group actions.

November 15 (15h00)

Sebastian Schlegel Mejia

BPS Algebras and Generalised Kac—Moody algebras from 2-Calabi—Yau categories

Associated to a 2-Calabi–Yau (2CY) abelian category – informally, a noncommutative symplectic surface – is its cohomological Hall algebra (CoHA), an algebra with underlying vector space given by the Borel–Moore homology of the moduli of objects in the category. CoHAs were originally defined by Kontsevich and Soibelman as a mathematical incarnations of Harvey and Moore’s algebras of BPS states. Following the approach of Davison and Meinhardt, BPS Lie algebras are supposed to be smaller and manageable objects which “control” the CoHA. From the moduli theorist's point of view, BPS Lie algebras tell you how to compute the Borel—Moore homology of moduli stacks in the presence of strictly semistables, in terms of cohomological invariants of the stable loci.

Making this precise, I will present a Poincaré–Birkhoff–Witt type theorem for the CoHA of a 2CY category in terms of the BPS Lie algebra. A crucial step is to identify the BPS algebra as (the positive half of) a generalised Kac–Moody Lie algebra modelled on the intersection cohomology of moduli spaces. This could be interpreted as a mathematical incarnation of observations made by Harvey and Moore relating algebras of BPS states to generalised Kac–Moody algebras. Time permitting, I will mention applications to the cohomology of Nakajima quiver varieties.

The talk is based on joint work with Ben Davison and Lucien Hennecart.

October 18 (15h00)

Augustinas Jacovskis

Categorical Torelli for $X_2$

Consider a threefold double cover $X$ of projective space, ramified in a general type surface $Z$. In this talk I'll describe a semiorthogonal decomposition of the $\mathbb{Z}/2\mathbb{Z}$-equivariant Kuznetsov component of $X$, and show that it contains a copy of $\mathbf{D}^{\mathrm{b}}(Z)$. 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 projective space, I'll show that an equivalence of Kuznetsov components of Picard rank 1, index 1, genus 2 Fano threefolds implies that they're isomorphic. This is joint work with Hannah Dell and Franco Rota.

May 10 (14h00)

S. Paul Smith

The symplectic leaves for the elliptic Poisson bracket on projective space defined by Feigin-Odesskii and Polishchuk

In 1998, Feigin and Odesskii, and independently Polishchuk, defined a Poisson structure on the complex projective space of dimension $n-1$. Given a Poisson manifold $(M,\Pi)$ there is a unique decomposition of $M$ into symplectic leaves, the leaves being the submanifolds on which the 2-form $\Pi$ restricts to a symplectic form. This talk, based on joint work with Alex Chirvasitu and Ryo Kanda (arXiv 2210.13042), describes the symplectic leaves.

The Poisson bracket is determined by a degree-$n$ holomorphic line bundle on a complex elliptic curve (a 1-dimensional complex torus, or compact Riemann surface of genus one) and can be thought of as being determined by the elliptic curve alone (because the choice of line bundle does not matter). Equivalently, the Poisson structure is determined by an elliptic curve embedded in the projective space as a degree-$n$ normal curve, i.e., the curve does not lie on any hyperplane. A formula for the Poisson bracket involves theta functions. The geometry of such embedded elliptic curves has been studied since the early 1800's, and is extraordinarily rich. The symplectic leaves can be described in terms of the secant varieties for (i.e., linear subspaces spanned by points on) the embedded elliptic curve.

April 26, 2023 (14h00)

Severin Barmeier

A_∞ deformations of extended Khovanov arc algebras and Stroppel's conjecture

Extended Khovanov arc algebras are finite-dimensional graded algebras which naturally appear in a variety of contexts such as link homology, representation theory and symplectic geometry. These algebras are defined via a diagrammatic calculus obtained from a 2d TQFT which lies at the heart Khovanov's categorification of the Jones polynomial of knots and links. After reviewing the diagrammatic description, I will explain how to describe these algebras and their Koszul duals via quivers with relations which allows us to compute their Hochschild cohomology groups, settling a conjecture by C. Stroppel. As a consequence we obtain explicit A_∞ deformations of extended Khovanov arc algebras which can also be viewed as A_∞ deformations of Fukaya-Seidel categories associated to Hilbert schemes of points on type A Milnor fibres. This talk is based on arxiv:2211.03354.

April 20, 2023 (10h00)

Hans Franzen

Towards a degree formula for quiver moduli

It is known that the Chow ring of a quiver moduli space is generated by Chern classes of universal bundles and presented by tautological relations. This allows to find a linear basis of the Chow ring in terms of monomials in Chern classes. In order to compute the degree of a line bundle, it is necessary to express the class of a point as a linear combination of these monomials. We express the point class as a degeneracy locus of a bundle. This determines a recursive procedure for obtaining the degree. We present some solutions which were obtained with the help of SAGE. The talk is based on ongoing joint work with Pieter Belmans.

April 13, 2023 (10h00)

Louis Ioos

A Riemann-Roch formula for singular symplectic reductions

Given a Hamiltonian action of a Lie group G on a symplectic manifold, the quantization commutes with reduction principle ([Q,R]=0) of Guillemin-Sternberg states that the space of $G$-invariants of the quantization of this manifold coincides with the quantization of its symplectic reduction by $G$. This principle provides in particular a geometric approach to the study of the representation theory of $G$. In this talk, I will consider the case where $G$ is a circle and where the symplectic reduction is a compact singular symplectic space, then present an approach to establish this principle based on the Berline-Vergne formula and the asymptotics of the Witten integral. This talk is based on a joint work in collaboration with Benjamin Delarue and Pablo Ramacher.

March 22, 2023 (15h30)

Pedro Núñez

Adapted differentials on Campana orbifolds

In this talk we introduce Campana orbifolds and discuss differential forms on them. A common way to define these differentials is by passing to a suitably ramified cover of the underlying variety. We explain how to give a choice-free definition as a presheaf on the category of schemes over the underlying variety, and show that this presheaf is a sheaf with respect to Voevodsky's qfh-topology. In particular it is an étale sheaf with transfers, and we briefly discuss the relevance of these notions and their relation to motivic cohomology.

March 22, 2023 (17h00)

Andreas Demleitner

Rigidity and classifying hyperelliptic manifolds

Hyperelliptic surfaces occur in the Enriques-Kodaira classification of compact complex surfaces and were classified by the Italian geometers in the beginning of the 20th century. In the talk, we will introduce higher-dimensional analogs called "hyperelliptic manifolds". Using the classical theorems of Bieberbach, we explain how to classify hyperelliptic manifolds in higher dimensions. Applying the classification algorithm in a special case, we obtain the existence of diffeomorphic rigid hyperelliptic fourfolds, which are not biholomorphic. Joint work with Christian Gleissner.

March 15, 2023 (exceptionally in MNO 0.010)

Nicolas Perrin

Cohomology of hyperplane sections of adjoint varieties

Given a reductive group $G$, I will first classify all pairs $(X,V)$ where $V$ is a $G$-representation and $X$ is a projective rational homogeneous $G$-variety with Picard rank 1 equivariantly embedded in $\mathbb{P}(V)$ for which the general hyperplane section $Y$ is stable under a maximal torus of $G$.

For these hyperplane sections $Y$, I will then describe some aspects of the geometry of $Y$ with a special focus on cohomology and quantum cohomology. (joint work with Vladimiro Benedetti)

November 30, 2022

Raf Bocklandt

Local systems of derived categories of gentle algebras

We will show how you can construct local systems of derived categories of gentle algebras by viewing them as Fukaya categories of surfaces and explore the connection with the work of Spenko and Van den Bergh on perverse schobers coming from noncommutative crepant resolutions of toric quotients.

November 9, 2022

Shengxuan Liu

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 basic definitions and notations in derived categories and singularities. Then I will show how to use the Lefschetz decomposition of a quadric to give a categorical resolution of a nodal variety and the property of the kernel generator of this categorical resolution. If time permits, I will describe a categorical resolution 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.

October 26, 2022

Fei Xie

Residual categories of quadric surface bundles

I will discuss the relation between the relative Hilbert scheme of lines of a quadric surface bundle and its residual category, the non-trivial semiorthogonal component in its derived category. For a smooth 4-fold with the structure of a quadric surface bundle over a smooth surface, there is a finite number of fibres that are quadrics of corank 2 (unions of two projective planes). This relation can be used to give a nice description of the derived category of the smooth 4-fold. This result applies to cubic 4-folds containing a plane and even-dimensional Fano varieties of dimension at most 10 that are smooth complete intersections of three quadrics. Time permitting, I will also discuss how this result can be generalised to quadric surface bundles over a general base.

October 5, 2022

Andreas Krug

Compactified Jacobians of non-integral curves and Lagrangian fibrations

I report on joint work with Adam Czaplinski, Manfred Lehn, and Sönke Rollenske. We describe moduli spaces of stable sheaves, which are generically line bundles, on a certain class of non-integral curves, which we call extended ADE curves. This class of curves generalises the non-integral fibres of elliptic fibrations. The main motivation is that our moduli spaces occur as general singular fibres of Lagrangian fibrations.

September 28, 2022

Maxim Smirnov

Quantum cohomology and derived categories of coadjoint varieties

We will discuss properties of quantum cohomology, both small and big, of coadjoint varieties of simple algebraic groups and how they relate to the structure of Lefschetz collections in the derived categories of these varieties. Some general conjectures pertaining to this will be formulated. The talk is based on the joint works with Alexander Kuznetsov and Nicolas Perrin.

November 4, 2021

Hans Franzen

Positivity properties of moduli spaces of quiver representations

I will report on joint work in progress with P. Belmans, C. Damiolini, V. Hoskins, S. Makarova, and T. Tajakka.

The talk will be about (parts of) a geometric proof, using dimension estimates, that a certain line bundle on a moduli space of semi-stable quiver representations is ample, provided that the quiver is acyclic. This re-proves a known result on semi-invariants of quivers. We believe that our approach can be used to provide estimates for the lowest degree in which semi-invariant functions exist.