Algebraische Geometrie

	Monday	Tuesday	Wednesday	Thursday	Friday
Times
9:00-10:00	Caldararu I	Caldararu II	Caldararu III	Caldararu IV	B5

10:15-11:15	Perutz I	Perutz II	A5	Perutz III	C3

11:30-12:00	B1 (+15)	A3	B3 (+30)	C1	C4 (+30)

14:30-15:30	A1	B2	/	C2	/

15:30-16:30	A2	A4	/	B4	/

Speakers:

A1a) Meazzini, Francesco: section 1 in https://arxiv.org/abs/q-alg/9709040

A1b) Poliakova, Dasha: section 3 in https://arxiv.org/abs/q-alg/9709040

A2a) Marchetti, Giovanni: L-infinity definition and equivalent coalgebra point of view https://arxiv.org/abs/math/0405485

A2b) Kaushal, Tanya: section 4 in https://arxiv.org/abs/q-alg/9709040

A3) Simi, Luca: L-infinity and formality criteria https://arxiv.org/abs/hep-th/9406095

A4) Filip, Matej: chapter 1-5 of https://arxiv.org/abs/alg-geom/9710032

A5) Arkhipov, Sergey: chapter 6-8 of https://arxiv.org/abs/alg-geom/9710032

B1) Barbieri, Anna: Classical Geometry of the Calabi-Yau moduli space (45min), chaper 2 in http://math.bu.edu/people/sili/thesis_SiLi.pdf

B2a) Buring, Ricardo: Kontsevich quantization and Feynman diagrams, section 3.1 plus background from Kontsevich's work

B2b) Tirelli, Andrea: Batalin-Vilkovisky formalism, section 3.2

B3a) Genovese, Francesco: Effective Field theory, Renormalization group flow and Chern Simons theory, section 3.3+3.4 until and including CS example 3.22

B3b) Belmans, Pieter: Quantum master equation, deformation obstruction complex and theory on the complex plane 3.4.2 to end of 3.5

B4a) Sutherland, Tom: Classical BCOV theory and its quantization, section 4

B4b) Melani, Valerio: Quantization of BCOV on the elliptic curve I, existence and uniqueness, section 5.1 until middle of page 98 (to the end of the proof of 5.12), i.e. section 5.1.1-5.3.1

B5a) Bousseau, Pierrick: Quantization of BCOV on the elliptic curve II, proving the dilaton axiom and holomorphicity, section 5.3.2-5.4

B5b) Ruddat, Helge: Proving higher genus mirror symmetry for the elliptic curve, section 6

C1) Anno, Rina: "Mumford toric degenerations", section 1.1 and 1.2 of https://arxiv.org/pdf/0808.2749.pdf

C2a) Felten, Simon: "From a toric degeneration to (B,P,phi)"

C2b) Logvinenko, Timothy: "Scattering I"

C3a) Nikolaev, Nikita: "Scattering II"

C3b) Zhi, Jin: "Overview of the Gross-Hacking-Keel construction"

C4a) Kelly, Tyler: "Broken lines"

C4b) van Garrel, Michel: "Generalized theta functions"

Andrei Caldararu: "Categorical Gromov-Witten Invariants"

Caldararu I: What are categorical Gromov-Witten invariants?

I will start my lecture by reviewing the construction of classical Gromov-Witten invariants, and working through a standard example, the g=1, n=1 GW invariants of an elliptic curve. Then I will outline how we can hope to obtain these invariants from the Fukaya category by an abstract categorical construction. In the case of an arbitrary abstract category I will discuss what properties the category should have, what replaces the quantum cohomology ring, etc. I will conclude with a discussion of potential applications of the construction of categorical Gromov-Witten invariants and a general discussion of how mirror symmetry is supposed to work.

Caldararu II: Chains on moduli space of curves, ribbon graphs, and A_infinity algebras

I will begin by reviewing the main algebraic structure on the space of chains on the moduli spaces of curves, namely that they form a Batalin-Vilkovisky algebra. We will study the Quantum Master Equation in such chains; by work of Costello its solution gives rise to the string vertices of Zwiebach and Sen. I will follow with an introduction to ribbon graphs and how they give a model for the chains on M_{g,n}. I will conclude with a quick review of A_infinity algebras and a description of the original Kontsevich partition function which pairs ribbon graphs with cyclic A_infinity algebras.

Caldararu III: Algebraic structures on the periodic cyclic complex

I will introduce the Hochschild chain complex of an A_infinity algebra, and its two main differentials, the Hochschild differential b and the Connes differential B, leading to the construction of the different variants of cyclic homology (periodic, negative, positive). Using this construction I will outline Kontsevich-Soibelman's construction of the action of the PROP of chains on M_{g,n} on the cyclic complex of an A_infinity algebra. Following Costello, part of this action allows us to regard the periodic cyclic complex as a symplectic vector space with a Lagrangian inside it. This leads to the construction of a Weyl algebra and Fock module associated with this structure. Costello constructs a deformation of this Fock module induced using a choice of string vertices. This leads to the construction of an abstract Gromov-Witten potential (a line in a Fock space).

Caldararu IV: Hodge-de Rham degeneration and applications

The periodic cyclic homology carries a natural filtration, the Hodge filtration. I will outline its construction, and explain how a choice of a splitting of this filtration allows us to regard the abstract Gromov-Witten potential previously constructed as an actual categorical Gromov-Witten potential. Everything will be put together into the calculation of the categorical B-model potential of the family of elliptic curves (joint work with Junwu Tu). I will conclude with a list of potential directions in which the whole construction can be extended, and expected applications and conjectures.

Tim Perutz: "From homological to Hodge-theoretic mirror symmetry"

(Based on work of S. Ganatra, N. Sheridan and myself, and various subsets of this trio.)

Perutz I: Fukaya categories, open-closed maps, and pairings.

I will review some of the mathematics of Fukaya categories of symplectic manifolds. These categories come with important structure: the closed-open map CO, which is a map of algebras from quantum cohomology to Hochschild cohomology of the category, and the open-closed map OC, from Hochschild homology to quantum cohomology, which is an isometry with respect to a “Mukai pairing” on Hochschild homology and a cup-product pairing on quantum cohomology. (The B-model analog of this structure was studied by Andrei Caldararu several years ago, and an abstract of OC is key to Costello’s construction, as discussed in Andrei’s lectures.) Abouzaid has shown that a subcategory generates the Fukaya category if OC is surjective on that subcategory.

Perutz II: Smoothness, automatic generation, and homological mirror symmetry.

Symplectic topologists have recently been learning a principle which Kontsevich has advocated for some time: the importance of categorical smoothness. The prototypical example of a smooth DG category is the derived category of a smooth algebraic variety. In the setting of homological mirror symmetry (HMS), the Fukaya category of the mirror symplectic manifold will then also be smooth. From smoothness many things flow: sharp versions of generation for the Fukaya category; that OC and CO are isomorphisms; that OC coincides with Costello’s abstract version, hence that his formal constructions are in fact geometric; and that HMS implies closed-string mirror symmetry.

Perutz III: The cyclic open-closed map, the categorical Gauss—Manin connection, and the mirror map.

The cyclic homology of an A-infinity category, defined over (say) a formal punctured disc, carries a Gauss—Manin connection, as constructed by Getzler. In the case of the derived category of a smooth algebraic variety over the punctured disc, this is expected (but not actually proven) to coincide with the classical GM connection in algebraic de Rham cohomology. I will explain that, in the case of a smooth Fukaya category, it coincides with the quantum differential operator, a connection in quantum cohomology. Under the assumption of HMS, the consequences are as follows: (i) that with an undetermined mirror map, the mirror map can be characterized Hodge-theoretically; and (ii) that the quantum differential equation matches the (derived) algebro-geometric GM connection. Statement (ii) is (genus 0) Hodge-theoretic mirror symmetry. It encompasses the famous rational curve-counts on the quintic 3-fold.

Spring School: Enumerative Invariants from Differential Graded Lie Algebras and Categories

Spring School for Master & PhD students and Postdocs
March 25 - 31, 2018 in Italy http://www.montegufoni.it/
Senior Speakers:
Andrei Caldararu
Tim Perutz
Organizers:
Helge Ruddat (ruddat)
Tom Sutherland (sutherland)
Estelle Bonmann (ebonmann)
(respectively @uni-mainz.de)

TIMETABLE

Accepted participants will be provided shared rooms in the castle. We might be able to modestly subsidize travel costs by a lump sum per person.We will organize and prepare meals ourselves.Each participant will be asked to give a talk presenting material that we assign together beforehand. This will consist of works of Barannikov-Kontsevich, Costello-Li, Gross-Siebert and others.Sorry, the event has reached its participant capacity limit.Literature:

Barannikov, Sergey; Kontsevich, Maxim: Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields.
Caldararu, Andrei; Tu, Junwu: Computing a categorical Gromov-Witten invariant.
Costello, Kevin J.; Li, Si: Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model
Ganatra, Sheel; Perutz, Timothy; Sheridan, Nick: Mirror symmetry - from categories to curve Counts.
Mark Gross, Bernd Siebert: Intrinsic mirror symmetry and punctured Gromov-Witten invariants.

This school is funded by the DFG Emmy-Noether grant RU 1629/4-1.

Elementare Algebraische Geometrie

Sommersemester 2017

Das Seminar findet an den wie in der folgenden Tabelle angegebenen Terminen statt. Montags von 12-14 Uhr sind wir im Raum 04-432 und für die zusätzlichen Sitzungen um 10-12 Uhr, sind wir im Raum 04-230 für diese Uhrzeit.

Vortrag	Datum	Vortrags- kapitel	Vortragende(r)	zu bearbeitende Übungsaufgaben
1	24.4. 10-12 Uhr	§1	Anna Katharina Pnischeck	1.1, 1.11, 3.1, 3.14, 4.11, 5.3
2	24.4. 12-14 Uhr	§1	Lennart Kahl	1.10, 2.12, 3.2, 3.15, 4.12, 5.1
3	08.5.	§2	Patricia Müller	0.1, 2.3, 3.4, 3.16, 5.13, 6.1
4	15.5.	§2	Maximilian Oischinger	0.2, 2.11, 3.6, 4.1, 5.12, 6.3
5	22.5. 10-12 Uhr	§3	Jakob Werner	0.3, 2.10, 3.3, 4.2, 5.11, 6.4
6	22.5. 12-14 Uhr	§3	Lars Hofmann	1.8, 2.2, 3.12, 4.9, 5.5, 5.7
7	29.5.	§4	Ramona Hirschfelder	1.3, 2.8, 3.7, 4.5, 5.9, 6.6
8	12.6.	§4	Helge Ruddat
9	19.6.	§5	Janine Scholtes	1.9, 2.1, 3.13, 4.4, 4.10, 5.4
10	26.6.	§5	Charlotte Eckert	1.6, 2.5, 3.10, 4.7, 5.2, 6.7
11	03.7.	§6	Peter Lang	1.7, 2.4, 3.11, 4.8, 5.6, 6.2

siehe auch: Seminartermine u. -themen

Die Tabelle zeigt die Aufteilung der Seminarthemen und Seminartermine sowie Übungsaufgaben pro Student(in). Sie bezieht sich auf das unten angegebene Buch.

Jeder Vortrag soll nicht viel länger als eine Stunde gehen und danach noch um das Vortragen der Übungsaufgabe zu dem Kapitel (von dem/der jeweils Vortragenden, ca. 15min) ergänzt werden. Außerdem sollte Zeit für Fragen eingeplant werden. (Gutes Zeitmanagement gehört zu einem guten Vortrag)

Jede(r) Vortragende teilt zu Beginn ihres/seines Vortrags einen Handzettel aus, auf dem die wesentliche Begriffe, Definitionen und Resultate des Vortrags zusammengefasst sind. Bitte zur Erstellung davon Latex verwenden. Übungsaufgaben sind bis eine Woche nach dem letzten Vortrag zum jeweiligen Thema einzureichen (auch per Latex-pdf) per Email an elementare.alg.geo_ät_web.de.
Achtet bitte darauf, dass ihr auch die Aufgabenstellung mit aufschreibt und die Aufgaben in ein einziges pdf Dokument kompiliert (jeweils um jede neue Aufgabe ergänzen, bzw gegebenenfalls auch Korrekturen der alten Aufgaben machen, jede neue Aufgabe beginnt auf einer neuen Seite).
Bei mindestens einer Aufgabe ist ein Computerprogramm zu schreiben, schreibt dafür einfach die Beschreibung des Algorithmus auf. In die Seminarnote fließen der eigene Vortrag, die allgemeine Beteiligung im Seminar und die Bearbeitung der Aufgaben ein.

Bitte lest auch das (recht kurze) Kapitel Null im Buch, weil wir darüber keinen Vortrag haben werden, es aber als Allgemeinbildung und Hintergrundwissen hilfreich ist. Auch zum Appendix von Kapitel eins haben wir keinen Vortrag, bitte ebenso bei Zeiten eigenständig anschauen.
Je früher ihr mit der Materialsichtung anfangt, desto besser!

Literatur

Miles Reid: "Undergraduate Algebraic Geometry", Cambridge University Press

Emmy Noether Kolloquium

Topics around Calabi-Yau and Fano manifolds, deformation theory and mirror symmetry

This colloquium invites mostly external speakers. It is loosely associated with the Emmy Noether
grant "Degenerations of Calabi-Yau Manifolds and Related Geometries".

Organizers are: Simon Felten, Matej Filip, Andrea Petracci, Helge Ruddat

Regular times: Tuesday 5-6 pm & Wednesday 9-10 am (German time zone)

Access to the life stream is provided here: https://researchseminars.org/seminar/EmmyKolloq

Upcoming talks (please find the abstracts below):

12.05.2021 09:00 Ziming Ma
SYZ Mirror Symmetry and Maurer-Cartan Equation

19.05.2021 08:30 Taro Sano
Construction of non-Kähler Calabi-Yau manifolds by log deformations

26.05.2021 09:00 Yuto Yamamoto
Tropical contractions to integral affine manifolds with singularities

09.06.2021 09:00 Taro Fujisawa
Geometric polarized log Hodge structures on the standard log point

15.06.2021 17:00 Pieter Belmans
Hochschild cohomology of Fano 3-folds

23.06.2021 09:00 Wei Hong
BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties

07.07.2021 09:00 Lawrence Barrott - cancelled due to illness - postphoned until october
Deforming Calabi-Yau subvarieties

20.07.2021 17:00 Benjamin Gammage
Homological mirror symmetry over the SYZ base

Abstracts

12.05. 2021 09:00
Ziming Ma - SYZ Mirror Symmetry and Maurer-Cartan Equation

The Strominger-Yau-Zaslow conjecture for understanding Mirror Symmetry geometrically, leads to the Fukaya's conjectural reconstruction of mirror manifolds which solves Maurer-Cartan equation near large limits using quantum corrections. In this talk, we will discuss progesses of the Fukaya's conjecture and the formulation of the Maurer-Cartan equation near large structure limits by constructing a dgBV algebra PV *(X), a generalized version of the Kodaira-Spencer dgLa, associated to possibly degenerate Calabi-Yau variety X equipped with local thickening data. This talk is based on joint works with Kwokwai Chan, Conan Leung and Yat-Hin Suen.

19.05.2021 08:30
Taro Sano - Construction of non-Kähler Calabi-Yau manifolds by log deformations

Calabi-Yau manifolds (in the strict sense) form an important class in the classification of algebraic varieties. One can also consider its generalisation by removing the projectivity assumption. Clemens and Friedman constructed infinitely many topological types of non-Kähler Calabi-Yau 3-folds whose 2nd Betti numbers are zero. In this talk, I will present examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers. The construction is by smoothing normal crossing varieties. The key tools of the construction are some isomorphisms between general rational elliptic surfaces which induce isomorphisms between Calabi-Yau manifolds of Schoen type.

26.05.2021 09:00
Yuto Yamamoto - Tropical contractions to integral affine manifolds with singularities

We construct contraction maps from tropical Calabi -Yau varieties to the integral affine maifolds with singularities that arise as the dual intersection complexes of toric degenerations of Calabi-Yau varieties in the Gross-Siebert program. We show that the contractions preserve tropical cohomology groups, and send the eigenwaves to the radiance obstructions. As an application, we also prove the Poincaré-Verdier duality for integral affine manifolds with singularities.

09.06.2021 09:00
Taro Fujisawa - Geometric polarized log Hodge structures on the standard log point

I will talk about the following fact: a projective vertical exact log smooth morphism over the standard log point yields polarized log Hodge structures on the base. In the proof of this fact, the case of a strict log deformation is essential. So, I will mainly talk about this case, and explain how to relate my previous results on the mixed Hodge structures to log Hodge structures for a projective strict log deformation. If the time remaines, I will discuss a generalization to the case of a general base point. This talk is based on a joint work with C. Nakayama.

15.06.2021 17:00
Pieter Belmans - Hochschild cohomology of Fano 3-folds (notes/script)

The Hochschild-Kostant-Rosenberg decomposition gives a description of the Hochschild cohomology of a smooth projective variety in terms of the sheaf cohomology of exterior powers of the tangent bundle. In all but a few cases it is a non-trivial task to compute this decomposition, and understand the extra algebraic structure which exists on Hochschild cohomology. I will give a general introduction to Hochschild cohomology and this decomposition, and explain what it looks like for Fano 3-folds (joint work with Enrico Fatighenti and Fabio Tanturri), and time permitting also for partial flag varieties (joint work with Maxim Smirnov).

23.06.2021 09:00
Wei Hong - BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties

The vector space of holomorphic polyvector fields on any complex manifold has a natural Gerstenhaber algebra structure. In this paper, we study BV operators of the Gersten- haber algebras of holomorphic polyvector fields on smooth compact toric varieties. We give a necessary and sufficient condition for the existence of BV operators of the Gerstenhaber algebra of holomorphic polyvector fields on any smooth compact toric variety

postphoned until October 2021 09:00
Lawrence Barrott - Deforming Calabi-Yau subvarieties

The Doran-Harder-Thompson conjecture is a duality on geometric structures under mirror symmetry. On one side we have smooth degenerations of Calabi-Yau's (CY's) to unions of normal crossings components, and on the other we have fibrations of the mirror CY's by CY subvarieties. In the simplest case it proposes that CY's with a Tyurin degeneration should be mirror to CY's fibred over P1. I will explain how some of the recent machinery of deformation theory for singular CY's of Chan-Leung-Ma and Felton-Filip-Ruddat, together with the mirror construction of the Gross-Siebert program leads to a proof of one direction of this conjecture in classes of examples. If time permits I will sketch how this relates to the period calculations appearing in other papers (Doran-Kostiuk-You) via more recent techniques in the Gross-Siebert program. This is based on joint work with Chuck Doran.

20.07.2021 17:00
Benjamin Gammage - Homological mirror symmetry over the SYZ base

The Gross-Siebert program suggests that mirror symmetry is mediated by the combinatorial data of a dual pair of integral affine manifolds with singularities and polyhedral decomposition. Much is now understood about the passage from the combinatorial data to complex spaces "near the large complex structure limit" - a toric degeneration and its smoothing. In this talk, we discuss the mirror procedure for moving from the combinatorial data to symplectic spaces "near the large volume limit" - a Weinstein symplectic manifold and its compactification -- and we will explain a proof of homological mirror symmetry between the complex and symplectic manifold associated to local pieces of the combinatorial data. This is part of a program with Vivek Shende to prove homological mirror symmetry globally over the SYZ base.

Abstracts of former talks

Abstracts

[toc title="Abstracts of former talks"]

12.05. 2021 Ziming Ma

19.05.2021 Taro Sano

26.05.2021 Yuto Yamamoto

09.06.2021 Taro Fujisawa

15.06.2021 Pieter Belmans

23.06.2021 Wei Hong

22.01.2020 Thomas Prince

21.06.2018 Michel van Garrel

08.06.2018 Ilia Zharkov

18.10.2017 Robin Guilbot

18.10.2017 Valentin Tonita

16.10.2017 Matej Filip

27.06.2017 Martin Ulirsch (University of Michigan, USA)

02.05.2017 Michel van Garrel (Universität Hamburg)

03.11.2016 Tom Sutherland (Università degli Studi di Pavia)

19.07.2016 Tim Kirschner (Universität Duisburg-Essen)

Finite quotients of three-dimensional complex tori

14.07.2016 James Pascaleff (University of Illinois)

26.04.2016 Mohammad Akhtar (Imperial College London)

07.01.2016 Andreas Gross (Universität Kaiserslautern)

17.12.2015 Sara Filippini (Universität Zürich)

29.10.2015 Aleksey Zinger (MPIM Bonn)

25.06.2015 Peter Overholser (Leuven / Imperial College London)

12.05. 2021 Ziming Ma

SYZ Mirror Symmetry and Maurer-Cartan Equation

19.05.2021 Taro Sano

Construction of non-Kähler Calabi-Yau manifolds by log deformations

26.05.2021 Yuto Yamamoto

Tropical contractions to integral affine manifolds with singularities

09.06.2021 Taro Fujisawa

15.06.2021 Pieter Belmans

23.06.2021 Wei Hong

BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties

21.10.2020 Katharina Hübner

Logarithmic differentials on adic spaces

The object of interest in this talk is a certain subsheaf Omega_X of the sheaf of differentials Omega_X of a discretely ringed adic space X over a field k. The first part will be dedicated to an introduction to discretely ringed adic spaces. We will then define $\Omega^+_X$ using K\"ahler seminorms and establish a relation with logarithmic differentials. Finally we study the case where $X = Spa(U,Y)$ for a scheme $Y$ over $k$ and a subscheme $U$ such that the corresponding log structure on $Y$ is log smooth. It turns out that $\Omega^+_X(X)$ equals $\Omega^{log}_{(U,Y)}(U,Y)$.

22.01.2020 Thomas Prince

Calabi-Yau toric hypersurfaces using the Gross-Siebert algorithm

We explain how to form a novel dataset of simply connected Calabi-Yau threefolds via the Gross-Siebert algorithm. These are related to Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated) singularities by a family over a possibly reducible base. Specifically, we explain how to smooth the boundary of a class of 4-dimensional reflexive polytopes to obtain a polarised tropical manifolds. We also describe how to compute the Betti numbers of these Calabi-Yau threefolds. We also explain an extension of this project generalising work of Batyrev-Kreuzer on conifold transitions.

21.06.2018 Michel van Garrel

Number of rational curves in log versus local geometries

Let X be a smooth projective variety and let D be a nef divisor. It is well known that D corresponds to a line bundle O(-D), which leads one to consider two geometries associated to D. On one hand, there is the logarithmic geometry of the pair (X,D). On the other hand, there is the local geometry of the total space of
O(-D). In this collaboration with Tom Graber and Helge Ruddat, we show that in an appropriate sense (in terms of log and local Gromov-Witten invariants), the number of log rational curves of (X,D) equals (up to a factor) the number of rational curves of O(-D).

08.06.2018 Ilia Zharkov

Topological model for affine hypersurfaces

Given an affine complex hypersurface I will define a phase tropical hypersurface and show that is homeomorphic to the complex one. I will also describe some immersed spheres which suppose to represent Lagrangian objects generating the Fukaya category of the hypersurface.

18.10.2017 Robin Guilbot

On embedded Mirror Symmetry

A wide majority of the known instances of Mirror Symmetry between families of Calabi-Yau varieties are realized as complete intersections in toric varieties.
In these examples the features of Mirror Symmetry are more or less direct consequences of convex-combinatorial dualities. But the elegance of these constructions
is somehow balanced by their peculiarity : toric complete intersections are expected to form a small minority of all the Calabi-Yau varieties.
I will Review the most famous toric mirror constructions, describe a generalization of the hypersurface case introduced in joint work with M. Artebani and P. Comparin,
and sketch the foundations of a new construction for non-complete intersections based on embedded toric degenerations, following J. Böhm’s PhD thesis.

18.10.2017 Valentin Tonita

K-theoretic mirror formulae

Permutation equivariant K-theoretic Gromov Witten invariants, introduced by Givental, are
certain Euler characteristics on the moduli spaces of stable maps to a (smooth, projective)
variety X. I will define the invariants and show how to write K-theoretic I-functions for large classes of varieties
(e.g. toric, certain complete intersections), i.e. certain q-hypergeometric series which are
generating series of these invariants in genus zero . Time permitting, I will discuss the ideas behind
the proofs of these results.

16.10.2017 Matej Filip

Hochschild cohomology and Deformation quantization of affine toric varieties

For an affine toric variety we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Using this description we prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization. Restricting to the commutative case, recent developments in constructing the versal deformation of an affine toric variety will be explained.

27.06.2017 Martin Ulirsch (University of Michigan, USA)

A moduli stack of tropical curves

In this talk I am going to give an introduction to these fascinating moduli spaces and discuss recent work with Renzo Cavalieri, Melody Chan, and Jonathan Wise (arXiv 1704.03806), where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this 2-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Moreover, I will propose a way of describing the process of tropicalization via logarithmic geometry in the sense of Kato-Illusie using the theory of Artin fans. Finally, given time, I will also report on an ongoing follow-up project (joint with Margarida Melo, Filippo Viviani, and Jonathan Wise) that uses these techniques to construct the universal Picard variety in logarithmic and tropical geometry.

02.05.2017 Michel van Garrel (Universität Hamburg)

Rational curves in log K3 surfaces

In this talk, we address the basic question of how to count rational curves in log K3 surfaces. We will present partial results in that direction and give a full conjectural description. This is baded on two joints works, one with T. Graber and H. Ruddat, and the other one with J. Choi, N. Takahashi ad S. Katz.

03.11.2016 Tom Sutherland (Università degli Studi di Pavia)

Stability conditions from periods of elliptic curves

I will describe spaces of stability conditions on some Calabi-Yau-3 categories with a simple combinatorial presentation through the study of the period map of meromorphic differentials on associated families of elliptic curves.

19.07.2016 Tim Kirschner (Universität Duisburg-Essen)

Finite quotients of three-dimensional complex tori

I will report on a current project with Patrick Graf (Bayreuth). Using Graf's recent results about the algebraic approximation of Kähler threefolds of Kodaira dimension zero, we show that a three-dimensional compact, connected Kähler space X with isolated canonical singularities is the finite quotient of a complex torus if and only if the first and second Chern classes of X vanish. This brings together an old theorem of Yau (where X is smooth) and a theorem of Shepherd-Barron and Wilson (where X is projective).

14.07.2016 James Pascaleff (University of Illinois)

Symplectic Cohomology and Wall Crossing

In this talk I will describe a way that certain wall-crossing formulae can be seen in terms of symplectic cohomology, which is a Floer theoretic invariant of non-compact symplectic manifolds. In the case of log Calabi-Yau manifolds, this invariant is supposed to be mirror-dual to the poly-vector fields. I will draw connections to the theory of cluster varieties as studied by Gross-Hacking-Keel-Kontsevich. This is partially based on discussions with Dmitry Tonkonog (Cambridge).

26.04.2016 Mohammad Akhtar (Imperial College London)

Mirror symmetry and the classification of Fano varieties

The classification of Fano varieties is an important long-standing problem in algebraic geometry. A new approach to this problem via mirror symmetry was recently proposed by Coates-Corti-Galkin-Golyshev-Kasprzyk. Their philosophy was that Fano varieties can be classified by studying their Laurent polynomial mirrors. This talk will survey the results of a collaborative effort to apply this philosophy to the classification of Fano orbifold surfaces. We will describe a conjectural picture which suggests that classifying suitable deformation classes of certain Fano orbifold surfaces is equivalent to classifying Fano lattice polygons up to an appropriate notion of equivalence. Central to this framework is the notion of mirror duality (between a Fano orbifold surface and a Laurent polynomial) and the closely related operations of algebraic and combinatorial mutations. We will also discuss how combinatorial mutations allow us to find mirror dual Laurent polynomials in practice and will give experimental evidence supporting our conjectures.

07.01.2016 Andreas Gross (Universität Kaiserslautern)

Intersection Theory on Tropicalizations of Torodial Embeddings

A central goal of tropical geometry is to give combinatoric descriptions of algebro-geometric objects. In enumerative geometry, these description ideally give rise to so-called correspondence theorems, which state that some given algebraic enumerative problem can be translated into a tropical enumerative problem with the same solution. The tropical intersection theory of Allermann and Rau has become a useful tool in tropical enumerative geometry, its connection to algebraic geometry being based on the description of the intersection ring of complete toric varieties by Fulton and Sturmfels. Unfortunately, moduli spaces are rarely toric, yet in many cases they are toroidal.
In my talk I will outline how to extend the scope of tropical intersection theory to be able to describe certain intersections on toroidal varieties.

17.12.2015 Sara Filippini (Universität Zürich)

Refined curve counting and the tropical vertex group

The tropical vertex group of Kontsevich and Soibelman is generated by formal symplectomorphisms of the 2-dimensional algebraic torus. It plays a role in many problems in algebraic geometry and mathematical physics. Based on the tropical vertex group, Gross, Pandharipande and Siebert introduced an interesting Gromov-Witten theory on weighted projective planes which admits a very special expansion in terms of tropical counts.

I will describe a refinement or "q-deformation" of this expansion, motivated by wall-crossing ideas, using Block-Goettsche invariants. This leads naturally to the definition of a class of putative q-deformed curve counts. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined. This is joint work with Jacopo Stoppa.

29.10.2015 Aleksey Zinger (MPIM Bonn)

Normal Crossings Divisors for Symplectic Topology

I will describe purely symplectic notions of normal crossings divisor and configuration. They are compatible with the existence of the desired auxiliary almost Kahler structures, provided ``existence" is suitably interpreted. These notions lead to a multifold version of Gompf's symplectic sum construction. They also imply that Brett Parker's work on exploded manifolds concerns a multifold version of the usual symplectic sum (or degeneration) formula for Gromov-Witten invariants. We hope our approach can be extended to more general singularities and provide purely symplectic analogues of the singularities and their deformations appearing in the Gross-Siebert program. This is joint work with Mark McLean and Mohammad Tehrani.

25.06.2015 Peter Overholser (Leuven / Imperial College London)

Descendent tropical mirror symmetry for P2

The Gross-Siebert program can be seen as an attempt to understand mirror symmetry from a tropical perspective. Gross has realized this goal in a particular example, giving a tropical description of mirror symmetry for P2. I will show how his construction can be modified to to yield a novel mirror symmetric relationship.