Departement Mathematik

Algebraic geometry and moduli seminar

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Frühjahrssemester 2022

Datum / Zeit Referent:in Titel Ort
* 24. Januar 2022
17:30-18:45 		Prof. Dr. Dimitri Zvonkine
CNRS
Details

Algebraic Geometry and Moduli Seminar

Titel Gromov-Witten invariants of complete intersections
Referent:in, Affiliation Prof. Dr. Dimitri Zvonkine, CNRS
Datum, Zeit 24. Januar 2022, 17:30-18:45
Ort 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
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* 7. Februar 2022
17:30-18:45 		Dr. Tudor Padurariu
Columbia University
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Algebraic Geometry and Moduli Seminar

Titel Hall algebras in Donaldson-Thomas theory
Referent:in, Affiliation Dr. Tudor Padurariu, Columbia University
Datum, Zeit 7. Februar 2022, 17:30-18:45
Ort 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
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* 14. Februar 2022
17:30-18:45 		Prof. Dr. Kai Behrend
UBC
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Algebraic Geometry and Moduli Seminar

Titel Donaldson-Thomas theory of the quantum Fermat quintic
Referent:in, Affiliation Prof. Dr. Kai Behrend, UBC
Datum, Zeit 14. Februar 2022, 17:30-18:45
Ort 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
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* 21. Februar 2022
17:30-18:45 		Prof. Dr. Anders Buch
Rutgers University
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Algebraic Geometry and Moduli Seminar

Titel A Vafa-Intriligator formula for Fano varieties
Referent:in, Affiliation Prof. Dr. Anders Buch, Rutgers University
Datum, Zeit 21. Februar 2022, 17:30-18:45
Ort 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. Februar 2022
17:30-18:45 		Prof. Dr. Tamas Hausel
IST Austria
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Algebraic Geometry and Moduli Seminar

Titel Fixed point scheme as spectrum of equivariant cohomology and Kirillov algebras
Referent:in, Affiliation Prof. Dr. Tamas Hausel, IST Austria
Datum, Zeit 28. Februar 2022, 17:30-18:45
Ort 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
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2. März 2022
13:30-14:45 		Alessio Cela
ETH Zürich
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Algebraic Geometry and Moduli Seminar

Titel Virtual Tevelev degrees and quantum Euler class of Fano complete intersections
Referent:in, Affiliation Alessio Cela, ETH Zürich
Datum, Zeit 2. März 2022, 13:30-14:45
Ort 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. März 2022
16:00-17:15 		Dr. Leonid Monin
MPI Leipzig
Details

Algebraic Geometry and Moduli Seminar

Titel Macaulay inverse systems and computation of cohomology rings
Referent:in, Affiliation Dr. Leonid Monin, MPI Leipzig
Datum, Zeit 4. März 2022, 16:00-17:15
Ort 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. März 2022
17:30-18:45 		Younghan Bae
ETH Zürich
Details

Algebraic Geometry and Moduli Seminar

Titel Counting surfaces on Calabi-Yau fourfolds
Referent:in, Affiliation Younghan Bae, ETH Zürich
Datum, Zeit 7. März 2022, 17:30-18:45
Ort 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
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9. März 2022
13:30-14:45 		Dr. Alessandro Giaccheto
University of Paris-Saclay
Details

Algebraic Geometry and Moduli Seminar

Titel Euler classes and negative powers of the canonical class
Referent:in, Affiliation Dr. Alessandro Giaccheto, University of Paris-Saclay
Datum, Zeit 9. März 2022, 13:30-14:45
Ort 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. März 2022
16:00-17:15 		Dr. Francesca Carocci
EPF Lausanne
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Algebraic Geometry and Moduli Seminar

Titel BPS invariant from non Archimedean integrals
Referent:in, Affiliation Dr. Francesca Carocci, EPF Lausanne
Datum, Zeit 11. März 2022, 16:00-17:15
Ort 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
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Algebraic Geometry and Moduli Seminar

Titel Sheaves, quasimaps, maps, (covers)
Referent:in, Affiliation Denis Nesterov, Universität Bonn
Datum, Zeit 1. April 2022, 16:00-17:15
Ort 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
Details

Algebraic Geometry and Moduli Seminar

Titel Enumeration of tropical curves in abelian surfaces
Referent:in, Affiliation Dr. Thomas Blomme, Université de Genève
Datum, Zeit 6. April 2022, 13:30-14:45
Ort 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
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Algebraic Geometry and Moduli Seminar

Titel On the Gromov-Witten theory of a K3 surface
Referent:in, Affiliation Prof. Dr. Georg Oberdieck, Universität Bonn
Datum, Zeit 8. April 2022, 16:00-17:15
Ort 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
Details

Algebraic Geometry and Moduli Seminar

Titel Virasoro constraints for sheaves and pairs via wall-crossing in vertex algebra
Referent:in, Affiliation Dr. Woonam Lim, ETH Zürich
Datum, Zeit 13. April 2022, 13:30-14:45
Ort 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
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Algebraic Geometry and Moduli Seminar

Titel The birational geometry of the moduli space of pointed hyperelliptic curves
Referent:in, Affiliation Dr. Scott Mullane, Humboldt Universität zu Berlin
Datum, Zeit 29. April 2022, 16:00-17:15
Ort 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. Mai 2022
17:30-18:45 		Prof. Dr. Arend Bayer
University of Edinburgh
Details

Algebraic Geometry and Moduli Seminar

Titel Kuznetsov categories of Fano threefolds
Referent:in, Affiliation Prof. Dr. Arend Bayer, University of Edinburgh
Datum, Zeit 2. Mai 2022, 17:30-18:45
Ort 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
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* 9. Mai 2022
17:30-18:45 		Dr. Navid Nabijou
University of Cambridge
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Algebraic Geometry and Moduli Seminar

Titel From orbifolds to logarithms via birational invariance
Referent:in, Affiliation Dr. Navid Nabijou, University of Cambridge
Datum, Zeit 9. Mai 2022, 17:30-18:45
Ort 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
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* 26. Mai 2022
14:00-16:00 		Dr. Thomas Blomme
Université de Genève
Details

Algebraic Geometry and Moduli Seminar

Titel Enumeration of tropical curves in abelian surfaces (multiple covers)
Referent:in, Affiliation Dr. Thomas Blomme, Université de Genève
Datum, Zeit 26. Mai 2022, 14:00-16:00
Ort HG G 19.1
Enumeration of tropical curves in abelian surfaces (multiple covers)
HG G 19.1
* 20. Juni 2022
17:30-18:45 		Prof. Dr. Marc Levine
Universität Duisburg-Essen
Details

Algebraic Geometry and Moduli Seminar

Titel Equivariant localization in quadratic enumerative geometry
Referent:in, Affiliation Prof. Dr. Marc Levine, Universität Duisburg-Essen
Datum, Zeit 20. Juni 2022, 17:30-18:45
Ort Zoom
Equivariant localization in quadratic enumerative geometry
Zoom

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