# CURRENT TRENDS IN THE CATEGORICAL APPROACH TO ALGEBRAIC AND SYMPLECTIC GEOMETRY

4-6 February 2023
Media Hall in Kashiwa Library
Asia/Tokyo timezone

### Contact

Dates: February 4 -6, 2023

Venue: Kavli IPMU

Speakers:

Agnieszka Bodzenta
Alexey Bondal
Will Donovan
Mikhail Kapranov
Tatsuki Kawagaki
Okawa Shinnosuke
Yukinobu Toda

SCHEDULE
Saturday 4 FEB 2023
Balcony room A, 4-th floor:
10:00 - 11:00 Free discussion (optional)
Seminar room A, ground floor:
14:00 - 15:00 Mikhail Kapranov (Kavli IPMU),
Perverse sheaves on symmetric products and Ran spaces
15:30 - 16:30 Tatsuki Kuwagaki (Kyoto University),
An introduction to sheaf quantization
Seminar room A, Balcony 4-th floor:
17:00 - 19:00 Free discussion

Sunday 5 Feb 2023
Balcony room A, 4-th floor:
10:00 - 11:00 Free discussion (optional)
Seminar room A, ground floor:
14:00 - 15:00 Will Donovan (Tsinghua University),
Homological comparison of resolution and smoothing
15:30 - 16:30 Shinnosuke Okawa (Osaka University),
Blowing down noncommutative cubic surfaces
Seminar room A, Balcony 4-th floor:
17:00 - 19:00 Free discussion

Monday 6 Feb 2023
Balcony room A, 4-th floor:
10:00 - 11:00 Free discussion (optional)
Conference Hall, ground floor:
14:00 - 15:00 Agnieszka Bodzenta (Warsaw University),
Reconstruction of a surface from reflexive sheaves
15:30 - 16:30 Yukinobu Toda (Kali IPMU),
Quasi BPS categories for C^3
Seminar room A, Balcony 4-th floor:
17:00 - 19:00 Free discussion

ABSTRACTS

Agnieszka Bodzenta (Warsaw University)

Reconstruction of a surface from reflexive sheavesConsider the category D of normal surfaces and open embeddings with complement of codimension two. I will show that any connected component of D contains a unique codim-2-saturated surface, i.e. such X that any morphism in D with domain X is an isomorphism. I will call such X the codim-2-saturated model of any surface X_0 in its connected component.The category Ref of reflexive sheaves is constant on a connected component of D. I will recover X from Ref(X_0). The main tool in the reconstruction will be the right abelian envelope of Ref(X_0) with its canonical exact structure. It will allow us to pass from Ref(X_0) to Ref(U) for any open U in X_0. A characterization of quasi-affine surfaces in terms of their categories of reflexive sheaves will finally allow us to define X as the colimit of spectra of centers of Ref(U) over all quasi-affine open U in X. This is based on a joint work with A. Bondal.

Will Donovan (Tsinghua University)

Homological comparison of resolution and smoothing

A singular space often comes equipped with (1) a resolution, given by a morphism from a smooth space, and (2) a smoothing, namely a deformation with smooth generic fibre. I will discuss existing results and work in progress on how these may be related homologically, starting with the threefold ordinary double point as a key example.

Mikhail Kapranov (Kali IPMU)

Perverse sheaves on symmetric products and Ran spaces

I will discuss the results, joint with V. Schechtman, describing perverse sheaveson two closely related types of spaces: $Sym^n(C)$, the symmetric products of the complex line $C$ and the Ran space $Ran(C)$. They are given in terms of the formal theory of Hopf algebras of certain type, i.e., of the PROBs of universal (co)operations in such algebras. The space $Ran(C)$ being infinite-dimensional, the next step of incorporating actual Hopf algebras (rather than the PROB of (co)operations) into our picture requires extending the concept of perverse sheaves to include objects not supported on any finite skeleton. Such an extension with good properties is not clear at the present.

Tatsuki Kuwagaki (Kyoto University)

An introduction to sheaf quantization

Abstract: Sheaf quantization of Lagrangian submanifold plays important role in symplectic topology and algebraic analysis. In symplectic topology, it is a model of A-branes. In algebraic analysis, it is an enrichment of constructible sheaves/perverse sheaves. In this talk, I’d like to give an introduction to the recent development of the theory of sheaf quantizations.

Shinnosuke Okawa (Osaka University)

Blowing down noncommutative cubic surfaces

In 2001 Van den Bergh established the notion of blowup of noncommutative surfaces and proved that the blowup of a noncommutative $\mathbb{P}^2$ in $6$ points is a cubic surface of a noncommutative $\mathbb{P}^3$. In this talk we identify the morphism from the configuration space of points to the moduli of noncommutative $\mathbb{P}^3$s by investigating the (global) monodromy of lines on noncommutative cubic surfaces. As a corollary we prove the converse to the aforementioned theorem under mild genericity conditions. If time permits, we also explain how it was helpful to consider the corresponding Poisson geometry. This is a joint work in progress with Ingalls, Sierra, and Van den Bergh.

Yukinobu Toda (Kavli IPMU)

Quasi BPS categories for $\mathbb{C}^3$Abstract:

The quasi BPS categories are some subcategories ofderived categories of derived commuting stacks, and appear as semiorthogonal summands of categories of matrix factorizations which categorify Donaldson-Thomas invariants on $\mathbb{C}^3$. In this talk, I will discuss several properties of quasi BPS categories in relation to BPS cohomology in CoDT theory. Among other things, I will discuss a conjecture on 3-dimensional analogue of Bridgeland-King-Reid-Haiman derived McKay correspondence for Hilbert schemes of points on surfaces via quasi BPS categories. This is a part of my series of joint works with Tudor Padurariu.

Organizers:Alexey Bondal (Kavli IPMU, Steklov Math Institute)

Kakenhi
Kavli IPMU

Starts
Ends
Asia/Tokyo
Media Hall in Kashiwa Library
Kashiwa Campus, the University of Tokyo