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

15.06.2021 17:00 Pieter Belmans

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

 

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 - 

15.06.2021 17:00
Pieter Belmans - 

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

 

12.05. 2021 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 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 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 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 

 

 

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.