Weekly Geometry Seminar (archive)

30) August 5, 2021: Makoto Miura (Research Institute for Mathematical Sciences, Kyoto University) 16:00–17:00 (KST) #Zoom

Geometric transitions for Calabi--Yau hypersurfaces

Geometric transition is a process of connecting two smooth Calabi--Yau 3-folds by a birational contraction followed by a complex smoothing. It has attracted the interest of both mathematicians and physicists, since it may give the right notion to connect moduli spaces of Calabi--Yau 3-folds. For Calabi--Yau hypersurfaces in toric varieties, a well-known idea of constructing geometric transitions is the use of nested reflexive polytopes. However, the method fails if we are going in a naive way. In this talk, I will report the current situation of this approach by looking at the counterexample by Fredrickson and discuss how to save this idea.

29) July 15, 2021: Younghan Bae (ETH Zürich) 16:00–17:00 (KST) #Zoom

A recent development in Abel-Jacobi theory

Abel-Jacobi theory studies when a divisor on a smooth curve is linearly equivalent to zero. When the curve degenerates over a family, the locus of the Abel-Jacobi question is related to the double ramification cycle. In this talk, we vary curves and line bundles simultaneously. The closure of the Abel-Jacobi section defines a locus on the universal Picard stack. We will see the closed formula of this locus in terms of tautological classes on the universal Picard stack. If time permits, we will discuss two interesting applications of the Abel-Jacobi problem on the universal Picard stack: (i) the multiple cover formula for GW invariants of K3 surfaces and (ii) the locus of meromorphic differentials.

28) July 1, 2021: Dhyan Aranha (Universität Duisburg-Essen) 16:00–17:00 (KST) #Zoom

The Intrinsic Normal for Artin Stacks

We will report on joint work with Piotr Pstragowski where we extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. In this work we are able to characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods to any relative perfect obstruction theory on Artin stacks we can associate a virtual fundamental class.

27) June 24, 2021: Emanuel Scheidegger (Beijing International Center for Mathematical Research) 16:00–17:00 (KST) #Zoom

On the quantum K-theory of the quintic

Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology. We will explain the main properties of quantum K-theory and we will discuss the examples of projective space and the quintic hypersurface in $\mathbb{P}^4$.

26) June 10, 2021: Zijun Zhou (Kavli IPMU) 16:00–17:00 (KST) #Zoom

$3d$ mirror symmetry, vertex function, and elliptic stable envelope

$3d$ mirror symmetry is a duality in physics, where Higgs and Coulomb branches of certain pairs of $3d$ $N=4$ SUSY gauge theories are exchanged with each other. Motivated from this, M. Aganagic and A. Okounkov introduced the enumerative geometric conjecture that the vertex functions of the mirror theories are related to each other. The two sets of $q$-difference equations satisfied by the vertex functions, in terms of the Kähler and equivariant parameters, are expected to exchange with each other. The conjecture therefore leads to a nontrivial relation between their monodromy matrices, the so-called elliptic stable envelopes. In this talk, I will discuss the proof in several cases of the conjecture for both vertex functions and elliptic stable envelopes. This is based on joint works with R. Rimányi, A. Smirnov, and A. Varchenko.

25) June 3, 2021: Jack Hall (University of Arizona & The University of Melbourne) 16:00–17:00 (KST) #Zoom

GAGA theorems

Given an algebraic variety $X$ over a topological field (e.g. $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{Q}_p$), one can often make some sort of analytic space $X_{\mathrm{an}}$ from $X$. The topology on $X_{\mathrm{an}}$ reflects the topology of the topological field and the functions on $X_{\mathrm{an}}$ should be appropriately holomorphic. The relationship between the vector bundles, subvarieties, cohomology, and coherent sheaves on $X$ and $X_{\mathrm{an}}$ is typically referred to as a “GAGA theorem”. Over the decades, corresponding to different types of analytifications of varieties and schemes, various GAGA theorems have been established (e.g., rigid, adic, and formal). I will discuss a new and unified GAGA theorem. This gives all existing results in the literature and also puts them into a broader context.

24) May 27, 2021: Pierrick Bousseau (ETH Zürich) 16:00–17:00 (KST) #Zoom

The flow tree formula for Donaldson-Thomas invariants of quivers with potentials

Very generally, Donaldson-Thomas invariants are counts of stable objects in Calabi-Yau triangulated categories of dimension 3. A natural source of examples is provided by the representation theory of quivers with potentials. I will present a proof of a formula, conjectured by Alexandrov-Pioline from string-theory arguments, which computes Donaldson-Thomas invariants of a quiver with potential in terms of a much smaller set of ``attractor invariants". The proof uses the framework of scattering diagrams to reorganize sequences of iterated applications of the Kontsevich-Soibelman wall-crossing formula. This is joint work with Hülya Argüz.

23) May 20, 2021: Fenglong You (Imperial College London) 16:00–17:00 (KST) #Zoom

Relative Gromov-Witten theory and mirror symmetry

Absolute Gromov-Witten theory is known to have many nice structural properties, such as quantum cohomology, WDVV equation, Givental's formalism, mirror theorem, CohFT etc.. In this talk, I will explain how to obtain parallel structures for relative Gromov-Witten theory via the relationship between relative and orbifold Gromov-Witten invariants. If time permits, I will also talk about the generalization to simple normal crossings pairs and some applications in mirror symmetry.

22) May 13, 2021: Daniel Pomerleano (University of Massachusetts) 10:00–11:00 (KST) #Zoom

Intrinsic Mirror Symmetry and Categorical Crepant Resolutions

A general expectation in mirror symmetry is that the mirror partner to an affine log Calabi-Yau variety is "semi-affine" (meaning it is proper over its affinization). We will discuss how the semi-affineness of the mirror can be seen directly as certain finiteness properties of Floer theoretic invariants of $X$ (the symplectic cohomology and wrapped Fukaya category). One interesting consequence of these finiteness results is that, under fairly general circumstances, the wrapped Fukaya of $X$ gives an ("intrinsic") categorical crepant resolution of the affine variety $\mathrm{Spec}(SH^0(X))$. This is based on arxiv:2103.01200 .

21) May 6, 2021: Navid Nabijou (University of Cambridge) 16:00–17:00 (KST) #Zoom

$(2+1)$ ways of counting tangent curves

Logarithmic Gromov-Witten theory is a framework for counting curves in a fixed variety $\mathrm{X}$, with specified tangency orders to a fixed normal crossings divisor $\mathrm{D}$. The associated moduli spaces of logarithmic stable maps have been extensively studied over the past decade. Despite this, calculating invariants remains a hard problem, and there are relatively few targets for which the theory has been "solved".

In this talk I will explain how tropical combinatorics can be leveraged to control the geometry of these moduli spaces and, ultimately, compute numbers. This point of view leads us to construct a natural iterated blowup of the moduli space of (ordinary) stable maps, whose intersection theory can then be exploited to relate the logarithmic Gromov-Witten invariants to other, better-understood curve counts.

This is joint work with Dhruv Ranganathan. No prior knowledge of logarithmic geometry will be assumed.

20) April 29, 2021: Tony Yue YU (CNRS, Université Paris-Saclay (Paris-Sud)) 16:00–17:00 (KST) #Zoom

Frobenius structure conjecture and application to cluster algebras

I will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry, and an application towards cluster algebras. Let U be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in U uniquely determine a commutative associative algebra equipped with a compatible multilinear form. Although the statement of the theorem involves only elementary algebraic geometry, the proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. I will explain various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, our algebra generalizes, and gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK, as well as the positivity in the Laurent phenomenon, follow readily from the geometric description. This is joint work with S. Keel, arXiv:1908.09861. If time permits, I will mention another application towards the moduli space of KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs, joint with P. Hacking and S. Keel, arXiv: 2008.02299.

19) April 22, 2021: Genki Ouchi (Nagoya University) 16:00–17:00 (KST) #Zoom

Hochschild entropy and categorial entropy

In this talk, I will talk about categorical entropy, which is introduced by Dimitrov-Haiden-Katzarkov-Kontsevich, via homological mirror symmetry of K3 surfaces. We show the existence of a symplectic Torelli mapping class of positive categorical entropy, which is a counterexample of Gromov-Yomdin type conjecture. Furthermore, I will introduce the notion of Hochschild entropy and related topics.

18) April 15, 2021: Matthew Habermann (University College London) 16:00–17:00 (KST) #Zoom

Homological mirror symmetry for nodal stacky curves

In this talk, I will explain the proof of homological mirror symmetry where the B-model is a ring or chain of stacky projective lines joined nodally, and where each irreducible component is allowed to have a non-trivial generic stabiliser. This generalises the work of Lekili and Polishchuk, and is motivated by the study of invertible polynomials. In particular, this work establishes homological mirror symmetry where the B--model is taken to be an invertible polynomial in two variables with not-necessarily-maximal grading group. In the maximally graded case, one can show that the mirror is graded symplectomorphic to the Milnor fibre of the transpose invertible polynomial, thus establishing the Lekili-Ueda conjecture in complex dimension one.

17) April 8, 2021: JunWu Tu (ShanghaiTech University) 16:00–17:00 (KST) #Zoom

On the categorical enumerative invariants of a point

We briefly recall the definition of categorical enumerative invariants (CEI) first introduced by Costello around 2005. Costello's construction relies fundamentally on Sen-Zwiebach’s notion of string vertices $V_{g,n}$’s which are chains on moduli space of smooth curves $M_{g,n}$’s. In this talk, we explain the relationship between string vertices and the fundamental classes of the Deligne-Mumford compactification of $M_{g,n}$. More precisely, we obtain a Feynman sum formula expressing the fundamental classes in terms of string vertices. As an immediate application, we prove a comparison result that the CEI of the field $\mathbb{Q}$ is the same as the Gromov-Witten invariants of a point.

16) April 1, 2021: Adeel A. Khan (IHES) 16:00–17:00 (KST) #Zoom

Microlocal analysis and Donaldson-Thomas theory

I will describe a way to generalize microlocal sheaf theory to singular spaces using derived geometry. I will then explain how this formalism can be used to construct new cohomological invariants conjectured by Joyce, which recover among other things the Donaldson-Thomas invariants of Calabi-Yau fourfolds recently constructed by Borisov-Joyce and Oh-Thomas. This is joint work in progress with Tasuki Kinjo.

15) March 25, 2021: Andrea Ricolfi (SISSA) 16:00–17:00 (KST) #Zoom

Refinements of higher rank DT invariants

Let $n > 0$ be an integer. The Quot scheme of length $n$ quotients of the free sheaf $\mathcal{O}^r$ on affine space $\mathbb{A}^3$ is the main character in “rank $r$ (local) Donaldson-Thomas theory”. We will explain how to attach several types of invariants (enumerative, motivic, cohomological, $\mathrm{K}$-theoretic) to this Quot scheme, and show that the resulting generating functions (varying $n$) have nice plethystic expressions. In particular, the $\mathrm{K}$-theoretic formula completely solves the higher rank DT theory of $\mathbb{A}^3$, confirming the Awata-Kanno Conjecture in String Theory. This part is joint work with Nadir Fasola and Sergej Monavari.

14) March 18, 2021: Charanya Ravi (Universität Regensburg) 16:00–17:00 (KST) #Zoom

Negative algebraic $\mathrm{K}$-groups of stacks

In this talk, we start by giving an introduction to the algebraic $\mathrm{K}$-theory of stacks. After recalling some basic properties, we will explain Weibel's conjecture for stacks. This concerns the vanishing of certain negative $\mathrm{K}$-groups and was proven in joint work with Tom Bachmann, Adeel Khan and Vladimir Sosnilo.

13) March 4, 2021: David Kern (Université d'Angers) 16:00–17:00 (KST) #Zoom

Categorification of the quantum Lefschetz principle

The quantum Lefschetz principle allows one to know the Gromov-Witten invariants of the zero locus of a section of a vector bundle from those of the ambient space. More generally, it expresses a relation between the virtual fundamental classes of the moduli stacks of stable maps to these schemes, algebraic cycles obtained from a construction which can appear somewhat mysterious. In this talk, I will explain how, through derived geometry, one can interpret this result as a purely geometric phenomenon, and deduce a categorification of the formula for virtual classes.

12) February 25, 2021: Hyeonjun Park (SNU) 16:00–17:00 (KST) #Zoom

Localizing virtual cycles for Donaldson-Thomas invariants of Calabi-Yau $4$-folds

Recently, Oh-Thomas constructed a virtual cycle for a moduli space of stable sheaves on a Calabi-Yau $4$-fold. In this talk, I will discuss how to localize the virtual cycle by an isotropic cosection and define a reduced virtual cycle when the cosection is surjective. Also, a vanishing result for a non-isotropic cosection will be discussed. I will explain how these results can be applied to various examples of cosections. This is joint work with Young-Hoon Kiem ( arXiv:2012.13167).

11) February 18, 2021: Yaoxiong Wen (Peking University) 16:00–17:00 (KST) #Zoom

$3d$ mirror symmetry

One of the recent remarkable discoveries is the connection between quantum K-theory and $3d$ TQFT. For a long time, quantum K-theory has been viewed as a variant to quantum cohomology, which comes from a $2d$ TQFT. $3d$ physics has its own mirror symmetry phenomenon, which falls into two versions, for $\mathcal{N}=4$ theories versus $\mathcal{N}=2$ theories. There are many mathematical results on the enumerative-geometric aspect of $3d$ $\mathcal{N}=4$ mirror symmetry by Okounkov's group. Our main interest is in $3d$ $\mathcal{N}=2$ theories, which apply to a general Kähler manifold. Its mirror symmetry was poorly understood even in physics. In this talk, I will introduce a new version of $3d$ mirror symmetry for toric stacks, inspired by a $3d$ $\mathcal{N}=2$ abelian mirror symmetry construction in physics introduced by Dorey-Tong. More precisely, for a short exact sequence $$ 0 \rightarrow \mathbb{Z}^k \rightarrow \mathbb{Z}^n \rightarrow \mathbb{Z}^{n-k} \rightarrow 0,$$ we consider the toric Artin stack $[C^n/(C^*)^k]$, and its mirror is given by the Gale dual of the above exact sequence, i.e., $[C^n/(C^*)^{n-k}]$. We introduce the modified equivariant $\mathrm{K}$-theoretic $\mathrm{I}$-functions for the mirror pair; they are defined by the contribution of fixed points. Under the mirror map, which switches the Kälher parameters and equivariant parameters and maps $q$ to $q^{-1}$, we see that modified $\mathrm{I}$-functions with the effective level structure of mirror pair coincide. This talk is based on the joint work with Yongbin Ruan and Zijun Zhou.

10) February 4, 2021: Jeongseok Oh (KIAS) 16:00–17:00 (KST) #Zoom

Multiplicative property of localised Chern characters

We would like to discuss the multiplicative property of localised Chern characters for complexes constructed by Baum-Fulton-MacPherson and Polishchuk-Vaintrob. Theirs use MacPherson's graph construction for a complex. It allows us to obtain tautological bundles which are, in a certain sense, extensions of terms of the complex. The key idea of our discussion is to show this is not only extensions of terms but also an extension of the complex.

9) January 28, 2021: Pranav Pandit (ICTS) 16:00–17:00 (KST) #Zoom

Calabi-Yau structures and Landau-Ginzburg models

Sheaves of categories provide a framework for the study of the symplectic geometry of Landau-Ginzburg models. After introducing this framework, I will explain how it leads to the construction of Calabi-Yau structures on generalized Fukaya categories and derived symplectic structures on various moduli spaces. This is based on joint work with L. Katzarkov and T. Spaide.

8) January 7, 2021: Bumsig Kim (KIAS) 16:00–17:00 (KST) #Zoom

Categorical Chern Characters

I will review categorical Chern characters for dg categories, especially following papers:
Shklarov, Hirzebruch-Riemann-Roch-type formula for DG algebras,
Brown and Walker, A Chern-Weil formula for the Chern character of a perfect curved module.

7) December 10, 2020: Bhamidi Sreedhar (KIAS) 16:00–17:00 (KST) #Zoom

Learning seminar on Balmer spectrum of matrix factorization categories (Lecture 5)

We shall discuss the classification of thick subcategories and Balmer spectrum of the derived category of matrix factorizations. Reference: Section 5 and Section 6 of the paper 'Relative singular locus and Balmer spectrum of matrix factorizations' by Yuki Hirano.