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Spring Semester 2022

Date / Time Speaker Title Location
* 24 January 2022
17:30-18:45 		Prof. Dr. Dimitri Zvonkine
CNRS
Event Details

Algebraic Geometry and Moduli Seminar

Title Gromov-Witten invariants of complete intersections
Speaker, Affiliation Prof. Dr. Dimitri Zvonkine, CNRS
Date, Time 24 January 2022, 17:30-18:45
Location Zoom
Abstract We show that there is an effective way to compute all Gromov-Witten (GW) invariants of all complete intersections. The main tool is Jun Li's degeneration formula: it allows one to express GW invariants of a complete intersection from GW invariants of simpler complete intersections. The main difficulty is that, in general, the degeneration formula does not apply to primitive cohomology insertions. To circumvent this difficulty we introduce simple nodal GW invariants. These invariants do not involve primitive cohomology classes, but instead make use of imposed nodal degenerations of the source curve. Our work contains two main statements: (i) simple nodal GW invariants can be computed by the degeneration formula, (ii) simple nodal GW invariants determine all GW invariants of a complete intersection. The first statement is geometric; the second uses the invariance of GW invariants under monodromy and some representation theory. In this talk I will spend more time on part (ii) to complete more geometrically oriented talks by my co-authors: Hulya Arguz, Pierrick Bousseau and Rahul Pandharipande.
Gromov-Witten invariants of complete intersectionsread_more
Zoom
* 7 February 2022
17:30-18:45 		Dr. Tudor Padurariu
Columbia University
Event Details

Algebraic Geometry and Moduli Seminar

Title Hall algebras in Donaldson-Thomas theory
Speaker, Affiliation Dr. Tudor Padurariu, Columbia University
Date, Time 7 February 2022, 17:30-18:45
Location Zoom
Abstract Kontsevich-Soibelman defined the cohomological Hall algebra (CoHA) of a quiver with potential. By a result of Davison-Meinhardt, CoHAs are deformations of the universal enveloping algebra of the BPS Lie algebra of the quiver with potential. My plan is to present the analogues stories in the categorical and K-theoretic contexts. I will introduce the categorical and K-theoretic Hall algebras of a quiver with potential and explain how to prove versions of the Davison-Meinhardt theorem in these contexts. These results have applications in categorical Donaldson-Thomas theory and in the study of Hall algebras of surfaces.
Hall algebras in Donaldson-Thomas theoryread_more
Zoom
* 14 February 2022
17:30-18:45 		Prof. Dr. Kai Behrend
UBC
Event Details

Algebraic Geometry and Moduli Seminar

Title Donaldson-Thomas theory of the quantum Fermat quintic
Speaker, Affiliation Prof. Dr. Kai Behrend, UBC
Date, Time 14 February 2022, 17:30-18:45
Location Zoom
Abstract We study non-commutative projective varieties in the sense of Artin-Zhang, which are given by non-commutative homogeneous coordinate rings, which are finite over their centre. We construct moduli spaces of stable modules for these, and construct a symmetric obstruction theory in the CY3-case. This gives deformation invariants of Donaldson-Thomas type. The simplest example is the Fermat quintic in quantum projective space, where the coordinates commute up to carefully chosen 5th roots of unity. We explore the moduli theory of finite length modules, which mixes features of the Hilbert scheme of commutative 3-folds, and the representation theory of quivers with potential. This is mostly work of Yu-Hsiang Liu, with contributions by myself and Atsushi Kanazawa.
Donaldson-Thomas theory of the quantum Fermat quintic read_more
Zoom
* 21 February 2022
17:30-18:45 		Prof. Dr. Anders Buch
Rutgers University
Event Details

Algebraic Geometry and Moduli Seminar

Title A Vafa-Intriligator formula for Fano varieties
Speaker, Affiliation Prof. Dr. Anders Buch, Rutgers University
Date, Time 21 February 2022, 17:30-18:45
Location Zoom
Abstract I will speak about a generalization of the Vafa-Intriligator formula that expresses the Gromov-Witten invariants of a Fano variety, for fixed markings and insertions of even degrees, in terms of eigenvalues of multiplication operators on the small quantum cohomology ring. This generalizes earlier known formulas for cominuscule flag varieties, including Grassmannians of type A and Grassmannians of maximal isotropic subspaces. In the Grassmannian cases, the relevant eigenvalues are explicitly known from the work of Rietsch and Cheong. The eigenvalues can also be described explicitly for Fano complete intersections. Applications include simple formulas for Tevelev degrees. This is joint work with Rahul Pandharipande.
A Vafa-Intriligator formula for Fano varietiesread_more
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* 28 February 2022
17:30-18:45 		Prof. Dr. Tamas Hausel
IST Austria
Event Details

Algebraic Geometry and Moduli Seminar

Title Fixed point scheme as spectrum of equivariant cohomology and Kirillov algebras
Speaker, Affiliation Prof. Dr. Tamas Hausel, IST Austria
Date, Time 28 February 2022, 17:30-18:45
Location Zoom
Abstract As an example of the multiplicity algebras of the Arnold school we will look at the equivariant cohomology of a Grassmannian and notice that it is inscribed in its regular fixed point scheme generalising observations of Brion-Carrell. In turn, we will describe it explicitely as the classical Kirillov algebra of a fundamental representation of SL_n, originally observed by Panyushev. These observations are motivated by mirror symmetry considerations originating in recent work with Hitchin on the mirror of very stable upward flows in the Hitchin system, which we will briefly indicate.
Fixed point scheme as spectrum of equivariant cohomology and Kirillov algebrasread_more
Zoom
2 March 2022
13:30-14:45 		Alessio Cela
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Virtual Tevelev degrees and quantum Euler class of Fano complete intersections
Speaker, Affiliation Alessio Cela, ETH Zürich
Date, Time 2 March 2022, 13:30-14:45
Location HG G 43
Abstract I will speak about geometric and virtual Tevelev degrees of Fano complete intersections, discuss when they agree and how to compute them. The notion of quantum Euler class will play a central role in the discussion. I will explain how to compute this class in the case of our interest.
Virtual Tevelev degrees and quantum Euler class of Fano complete intersectionsread_more
HG G 43
4 March 2022
16:00-17:15 		Dr. Leonid Monin
MPI Leipzig
Event Details

Algebraic Geometry and Moduli Seminar

Title Macaulay inverse systems and computation of cohomology rings
Speaker, Affiliation Dr. Leonid Monin, MPI Leipzig
Date, Time 4 March 2022, 16:00-17:15
Location HG G 43
Abstract t was observed by Pukhlikov and Khovanskii that the BKK theorem implies that the volume polynomial on the space of polytopes is the Macaulay generator of the cohomology ring of a smooth projective toric variety. This provides a way to express the cohomology ring of toric variety as a quotient of the ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. The crucial ingredient of this observation is an explicit expression for the Macaulay generator of graded Gorenstein algebras generated in degree 1. In my talk I will explain this construction in detail, then I will tell about recent results on explicit expression for the Macauley generator of an arbitrary algebra with Gorenstein duality. Finally, if time permits, I will show how these results yield to the computation of the cohomology rings of more general classes of algebraic varieties.
Macaulay inverse systems and computation of cohomology ringsread_more
HG G 43
* 7 March 2022
17:30-18:45 		Younghan Bae
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Counting surfaces on Calabi-Yau fourfolds
Speaker, Affiliation Younghan Bae, ETH Zürich
Date, Time 7 March 2022, 17:30-18:45
Location Zoom
Abstract I will describe a sheaf theoretic way to enumerate surfaces on a smooth quasi-projective Calabi-Yau fourfold. By Borisov-Joyce/Oh-Thomas, a moduli space of sheaves or complexes on a Calabi-Yau fourfold has a virtual class. When we count two dimensional objects, this theory has to be modified to get nontrivial invariants. I will explain how to reduce the theory and deformation invariance along the Hodge loci. This can be related to the variational Hodge conjecture in some examples. Next I will explain two ways to compactly stable pairs, which we call PT0, PT1 pairs. Several conjectural correspondences between DT, PT0 and PT1 invariants will be discussed with evidences. This is a joint work in progress with Martijn Kool and Hyeonjun Park.
Counting surfaces on Calabi-Yau fourfoldsread_more
Zoom
9 March 2022
13:30-14:45 		Dr. Alessandro Giaccheto
University of Paris-Saclay
Event Details

Algebraic Geometry and Moduli Seminar

Title Euler classes and negative powers of the canonical class
Speaker, Affiliation Dr. Alessandro Giaccheto, University of Paris-Saclay
Date, Time 9 March 2022, 13:30-14:45
Location HG G 43
Abstract In 2008, Chiodo extended Mumford’s formula for the Chern character of the Hodge bundle to the direct image of universal r-th roots of powers of the canonical class. Until recently, only positive powers found direct applications. In this talk, I will report on two recent results where negative powers are considered. The first one is the computation of the Euler characteristic of the moduli space of curves, which provides an intersection-theoretic proof of the Harer–Zagier formula. The second one is a deformation of Norbury’s class, which provides tautological relations in terms of kappa classes (recently proposed by Kazarian–Norbury) and a proof of Norbury’s conjecture. Based on joint works with D. Lewański, P. Norbury and (in progress) with N. Chidambaram and E. Garcia-Failde.
Euler classes and negative powers of the canonical classread_more
HG G 43
11 March 2022
16:00-17:15 		Dr. Francesca Carocci
EPF Lausanne
Event Details

Algebraic Geometry and Moduli Seminar

Title BPS invariant from non Archimedean integrals
Speaker, Affiliation Dr. Francesca Carocci, EPF Lausanne
Date, Time 11 March 2022, 16:00-17:15
Location HG G 43
Abstract We consider moduli spaces M(ß,χ) of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on M(ß,χ) with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles. Recent results of Maulik-Shen and Kinjo-Coseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles. This is a joint work with Giulio Orecchia and Dimitri Wyss.
BPS invariant from non Archimedean integralsread_more
HG G 43
1 April 2022
16:00-17:15 		Denis Nesterov
Universität Bonn
Event Details

Algebraic Geometry and Moduli Seminar

Title Sheaves, quasimaps, maps, (covers)
Speaker, Affiliation Denis Nesterov, Universität Bonn
Date, Time 1 April 2022, 16:00-17:15
Location HG G 43
Abstract We will discuss a wall-crossing between Donaldson-Thomas theory of a threefold Surface x Curve and Gromov-Witten theory of a moduli space of sheaves on the Surface. The wall-crossing is provided by the notion of a quasimap to a moduli space of sheaves and Yang Zhou's theory of calibrated tails. The geometry behind this kind of wall-crossings seems to be responsible for many correspondences between different enumerative theories centred around threefolds of the type Surface x Curve.
Sheaves, quasimaps, maps, (covers)read_more
HG G 43
6 April 2022
13:30-14:45 		Dr. Thomas Blomme
Université de Genève
Event Details

Algebraic Geometry and Moduli Seminar

Title Enumeration of tropical curves in abelian surfaces
Speaker, Affiliation Dr. Thomas Blomme, Université de Genève
Date, Time 6 April 2022, 13:30-14:45
Location HG G 43
Abstract Tropical geometry is a powerful tool that allows one to compute enumerative algebraic invariants through the use of some correspondence theorem, transforming an algebraic problem into a combinatorial problem. Moreover, the tropical approach also allows one to twist definitions to introduce mysterious refined invariants, obtained by counting tropical curves with polynomial multiplicities. So far, this correspondence has mainly been implemented in toric varieties. In this talk we will study enumeration of curves in abelian surfaces and line bundles over an elliptic curve.
Enumeration of tropical curves in abelian surfacesread_more
HG G 43
8 April 2022
16:00-17:15 		Prof. Dr. Georg Oberdieck
Universität Bonn
Event Details

Algebraic Geometry and Moduli Seminar

Title On the Gromov-Witten theory of a K3 surface
Speaker, Affiliation Prof. Dr. Georg Oberdieck, Universität Bonn
Date, Time 8 April 2022, 16:00-17:15
Location HG G 43
Abstract I will discuss certain analogies between Gromov-Witten and Nekrasov theory of a K3 surface, in particular regarding quasi-modularity and special evaluations. Nekrasov theory is defined by integration over the Hilbert scheme of points on the K3 surface. In the second half I will explain how the two theories are related. This is work in progress.
On the Gromov-Witten theory of a K3 surfaceread_more
HG G 43
13 April 2022
13:30-14:45 		Dr. Woonam Lim
ETH Zürich
Event Details

Algebraic Geometry and Moduli Seminar

Title Virasoro constraints for sheaves and pairs via wall-crossing in vertex algebra
Speaker, Affiliation Dr. Woonam Lim, ETH Zürich
Date, Time 13 April 2022, 13:30-14:45
Location HG G 43
Abstract Virasoro constraints in Gromov-Witten theory have a long history going back to Witten's conjecture in 1990. Only recently, Virasoro constraints have been introduced in the context of sheaf theory via GW/PT correspondence on certain threefolds. In this talk, we provide a different interpretation of the Virasoro constraints for sheaves and pairs without reference to Gromov-Witten theory. This requires vertex algebra structure on the moduli stack of sheaves which was introduced by Joyce to study wall-crossing. Using the new formulation, we prove compatibility between Virasoro constraints and wall-crossing. We apply this to study Virasoro constraints of sheaves on surfaces with only (p,p) cohomology. This is a joint work in progress with A. Bojko and M. Moreira.
Virasoro constraints for sheaves and pairs via wall-crossing in vertex algebraread_more
HG G 43
29 April 2022
16:00-17:15 		Dr. Scott Mullane
Humboldt Universität zu Berlin
Event Details

Algebraic Geometry and Moduli Seminar

Title The birational geometry of the moduli space of pointed hyperelliptic curves
Speaker, Affiliation Dr. Scott Mullane, Humboldt Universität zu Berlin
Date, Time 29 April 2022, 16:00-17:15
Location HG G 43
Abstract The moduli space of pointed hyperelliptic curves is a seemingly simple object with perhaps unexpectedly interesting geometry. I will report on joint work with Ignacio Barros completing the classification of both the Kodaira dimension and the structure of the effective cone of these moduli spaces.
The birational geometry of the moduli space of pointed hyperelliptic curvesread_more
HG G 43
* 2 May 2022
17:30-18:45 		Prof. Dr. Arend Bayer
University of Edinburgh
Event Details

Algebraic Geometry and Moduli Seminar

Title Kuznetsov categories of Fano threefolds
Speaker, Affiliation Prof. Dr. Arend Bayer, University of Edinburgh
Date, Time 2 May 2022, 17:30-18:45
Location Zoom
Abstract I will give a survey of recent results on Kuznetsov categories of Fano threefolds of Picard rank one. These results give additional structure to their classification, and their moduli spaces. The techniques involved include moduli spaces of Bridgeland-stable objects, Brill-Noether statements, and equivariant categories (spiced with a pinch of derived algebraic geometry).
Kuznetsov categories of Fano threefoldsread_more
Zoom
* 9 May 2022
17:30-18:45 		Dr. Navid Nabijou
University of Cambridge
Event Details

Algebraic Geometry and Moduli Seminar

Title From orbifolds to logarithms via birational invariance
Speaker, Affiliation Dr. Navid Nabijou, University of Cambridge
Date, Time 9 May 2022, 17:30-18:45
Location Zoom
Abstract Logarithmic and orbifold structures provide two different paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under toroidal blowups of the target, while the orbifold theory is not. By identifying a suitable system of “slope-sensitive” blowups, we define a “limit orbifold theory” and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood.
From orbifolds to logarithms via birational invarianceread_more
Zoom
* 26 May 2022
14:00-16:00 		Dr. Thomas Blomme
Université de Genève
Event Details

Algebraic Geometry and Moduli Seminar

Title Enumeration of tropical curves in abelian surfaces (multiple covers)
Speaker, Affiliation Dr. Thomas Blomme, Université de Genève
Date, Time 26 May 2022, 14:00-16:00
Location HG G 19.1
Enumeration of tropical curves in abelian surfaces (multiple covers)
HG G 19.1
* 20 June 2022
17:30-18:45 		Prof. Dr. Marc Levine
Universität Duisburg-Essen
Event Details

Algebraic Geometry and Moduli Seminar

Title Equivariant localization in quadratic enumerative geometry
Speaker, Affiliation Prof. Dr. Marc Levine, Universität Duisburg-Essen
Date, Time 20 June 2022, 17:30-18:45
Location Zoom
Equivariant localization in quadratic enumerative geometry
Zoom

Notes: red marked events are important and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

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