Topics around CalabiYau 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 CalabiYau Manifolds and Related Geometries".
Organizers are: Simon Felten, Matej Filip, Andrea Petracci, Helge Ruddat
Regular times: Tuesday 56 pm & Wednesday 910 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 MaurerCartan Equation
19.05.2021 08:30 Taro Sano
Construction of nonKähler CalabiYau 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 3folds
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 CalabiYau 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 MaurerCartan Equation
The StromingerYauZaslow conjecture for understanding Mirror Symmetry geometrically, leads to the Fukaya's conjectural reconstruction of mirror manifolds which solves MaurerCartan equation near large limits using quantum corrections. In this talk, we will discuss progesses of the Fukaya's conjecture and the formulation of the MaurerCartan equation near large structure limits by constructing a dgBV algebra PV *(X), a generalized version of the KodairaSpencer dgLa, associated to possibly degenerate CalabiYau variety X equipped with local thickening data. This talk is based on joint works with Kwokwai Chan, Conan Leung and YatHin Suen.
19.05.2021 08:30
Taro Sano  Construction of nonKähler CalabiYau manifolds by log deformations
CalabiYau 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 nonKähler CalabiYau 3folds whose 2nd Betti numbers are zero. In this talk, I will present examples of nonKähler CalabiYau 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 CalabiYau 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 CalabiYau varieties in the GrossSiebert 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 PoincaréVerdier 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 3folds (notes/script)
The HochschildKostantRosenberg 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 nontrivial 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 3folds (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 CalabiYau subvarieties
The DoranHarderThompson conjecture is a duality on geometric structures under mirror symmetry. On one side we have smooth degenerations of CalabiYau'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 ChanLeungMa and FeltonFilipRuddat, together with the mirror construction of the GrossSiebert 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 (DoranKostiukYou) via more recent techniques in the GrossSiebert 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 GrossSiebert 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

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 DuisburgEssen)

Finite quotients of threedimensional 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)
Abstracts of former talks

 Abstracts
 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 DuisburgEssen)
 Finite quotients of threedimensional 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 MaurerCartan Equation
The StromingerYauZaslow conjecture for understanding Mirror Symmetry geometrically, leads to the Fukaya's conjectural reconstruction of mirror manifolds which solves MaurerCartan equation near large limits using quantum corrections. In this talk, we will discuss progesses of the Fukaya's conjecture and the formulation of the MaurerCartan equation near large structure limits by constructing a dgBV algebra PV *(X), a generalized version of the KodairaSpencer dgLa, associated to possibly degenerate CalabiYau variety X equipped with local thickening data. This talk is based on joint works with Kwokwai Chan, Conan Leung and YatHin Suen.
19.05.2021 Taro Sano
Construction of nonKähler CalabiYau manifolds by log deformations
CalabiYau 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 nonKähler CalabiYau 3folds whose 2nd Betti numbers are zero. In this talk, I will present examples of nonKähler CalabiYau 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 CalabiYau 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 CalabiYau varieties in the GrossSiebert 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 PoincaréVerdier 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
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
CalabiYau toric hypersurfaces using the GrossSiebert algorithm
We explain how to form a novel dataset of simply connected CalabiYau threefolds via the GrossSiebert algorithm. These are related to CalabiYau 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 4dimensional reflexive polytopes to obtain a polarised tropical manifolds. We also describe how to compute the Betti numbers of these CalabiYau threefolds. We also explain an extension of this project generalising work of BatyrevKreuzer 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 GromovWitten 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 CalabiYau varieties are realized as complete intersections in toric varieties.
In these examples the features of Mirror Symmetry are more or less direct consequences of convexcombinatorial 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 CalabiYau 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 noncomplete intersections based on embedded toric degenerations, following J. Böhm’s PhD thesis.
18.10.2017 Valentin Tonita
Ktheoretic mirror formulae
Permutation equivariant Ktheoretic 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 Ktheoretic Ifunctions for large classes of varieties
(e.g. toric, certain complete intersections), i.e. certain qhypergeometric 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 2categorical 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 KatoIllusie using the theory of Artin fans. Finally, given time, I will also report on an ongoing followup 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 CalabiYau3 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 DuisburgEssen)
Finite quotients of threedimensional 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 threedimensional 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 ShepherdBarron 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 wallcrossing formulae can be seen in terms of symplectic cohomology, which is a Floer theoretic invariant of noncompact symplectic manifolds. In the case of log CalabiYau manifolds, this invariant is supposed to be mirrordual to the polyvector fields. I will draw connections to the theory of cluster varieties as studied by GrossHackingKeelKontsevich. 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 longstanding problem in algebraic geometry. A new approach to this problem via mirror symmetry was recently proposed by CoatesCortiGalkinGolyshevKasprzyk. 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 algebrogeometric objects. In enumerative geometry, these description ideally give rise to socalled 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 2dimensional 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 GromovWitten theory on weighted projective planes which admits a very special expansion in terms of tropical counts.
I will describe a refinement or "qdeformation" of this expansion, motivated by wallcrossing ideas, using BlockGoettsche invariants. This leads naturally to the definition of a class of putative qdeformed curve counts. We prove that this coincides with another natural qdeformation, 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 GromovWitten 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 GrossSiebert 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 GrossSiebert 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.