- Adeel Khan (AS): title
- Young-Hoon Kiem (KIAS): Generalized sheaf counting
- Counting bundles or sheaves with desired properties is a
fundamental problem in algebraic geometry. There are three major
issues concerning sheaf counting, namely compactification,
singularity and automorphism. Firstly, the issue of compactifying
moduli spaces has many solutions such as Hilbert schemes and moduli
spaces of stable sheaves, and we have to deal with the comparison
problem of the moduli spaces and invariants. Geometric invariant
theory and various wall crossing techniques are of key importance in
this direction. Secondly, it is well known that moduli spaces may
acquire arbitrarily bad singularities and ordinary intersection
theory does not make sense. However, we can often utilize a deeper
structure (sometimes called the hidden smoothness) to construct a
nice homology class, called the virtual fundamental class, which
enables us to define sheaf counting invariants like Donaldson,
Seiberg-Witten, Donaldson-Thomas and DT4 invariants in algebraic geometry. In this talk, our main focus lies in the third issue.
Difficulties arise in sheaf counting due to automorphism groups even when the moduli stacks are smooth as in the case of semistable vector bundles over smooth projective curves because the moduli stacks are not Deligne-Mumford and we don’t know how to integrate cohomology classes on Artin stacks in general. To remove infinite automorphism groups, we should modify the moduli stack and there are several ways such as Kirwan’s partial desingularization (with intrinsic blowups for quasi-smooth stacks or canonical stabilizer reduction in derived algebraic geometry) and constructing smooth morphisms from Deligne-Mumford stacks as in the works of T. Mochizuki and Joyce-Song. After these modifications, we obtain virtual fundamental classes which provide us with generalized sheaf counting invariants, and it is an open problem to compare these generalized invariants. For instance, we want to compare the generalized Donaldson-Thomas invariants of Kiem-Li-Savvas (by partial desingularization) and Joyce-Song (by stable pairs) for Calabi-Yau 3-folds.
In this talk, I will focus on the curve case (singular moduli spaces of vector bundles over curves) with a prospect towards higher dimensional cases. I will show how we can compare all the generalized intersection pairings by partial desingularization, parabolic bundles, pairs and Joyce's wall crossing mechanism. Based on a joint work with Chenjing Bu.
- Emile Bouaziz (AS) : Elliptic objects and non-commutative geometry:
- I'll discuss some recent joint work with Adeel Khan which constructs a very general theory of elliptic loop spaces and uses this to give a clean construction of equivariant elliptic cohomology. There will be a lot of motivation coming from more classical objects, and I will include a variety of classical computations. I will also sketch a cochain model coming from the theory of meromorphic loop spaces. This latter is part of an ongoing project with Scherotzke, Sibilla and Tomasini constructing non-commutative versions of (equivariant) elliptic cohomology. I will attempt to pepper the talk with some vague speculations about the rôle of vertex algebras and conformal blocks to this theory, which I do not yet understand to my satisfaction.
- Hyeonjun Park (KIAS) : Lagrangian classes, Donaldson-Thomas theory, and gauged linear sigma models
- In this talk, I will explain the construction of Lagrangian
classes for perverse sheaves in cohomological Donaldson-Thomas
theory, whose existence was conjectured by Joyce. The two key
ingredients are a relative version of the DT perverse sheaves and a
hyperbolic version of the dimensional reduction theorem. As a
special case, we recover Borisov-Joyce/Oh-Thomas virtual classes in
DT4 theory.
As applications, I will explain how to construct the following
structures from the Lagrangian classes: (1) cohomological field
theories for gauged linear sigma models, (2) cohomological Hall
algebras for 3-Calabi-Yau categories, (3) relative
Donaldson-Thomas invariants for Fano 4-folds with anti-canonical
divisors, (4) refined surface counting invariants for Calabi-Yau
4-folds.
This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.
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- Borislav Mladenov (AS) : Algebraically coisotropic subvarieties in holomorphic symplectic varieties
- If (Χ,σ) is a holomorphic symplectic variety, a subvariety Z
is coisotropic if the restriction of the holomorphic sympelctic form
σ to Z has constant minimal rank. In particular hypersurfaces are
coisotropic. It turns out that being coisotropic is a cohomological
property, hence the notion of coisotropic classes. This leads
naturally to the definition of coisotropic Hodge classes. We say Z
is algebraically coisotropic if the foliation defined by the kernel
of σ is algebraically integrable.
A conjecture of Voisin states that the space of coisotropic Hodge classes over \Q is generated by classes of (special) algebraically coisotropic subvarieties. In this talk, I'm going to introduce the main characters, discuss constructions of examples of these algebraically coisotropic subvarieties, and present old and new cases of this conjecture, focusing on moduli of sheaves on K3 surfaces.
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- Woonam Lim (Yonsei U): On the \chi-independence phenomenon
- The moduli space of one-dimensional sheaves on surfaces and Calabi–Yau 3-folds has been extensively studied due to its connections with the Hitchin integrable system and curve counting theories. These moduli spaces depend in a subtle way on the holomorphic Euler characteristic, often denoted by \chi. Although the geometry and topology of the moduli spaces vary significantly with \chi, certain cohomological phenomena are known or conjectured to be \chi-independent. In this talk, I will review the \chi-independence phenomenon and explain some recent results. If time permits, I will also discuss some speculations about how a (-1)-shifted Lagrangian correspondence via Hecke modifications may provide a useful framework for studying \chi-independence in the Calabi–Yau 3-fold case. This talk is based on joint works (some in progress) with Y. Kononov, M. Moreira and W. Pi.
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