Beijing Algebraic Geometry Day    

September 24th, 2025  MCM110  

Invited Speakers

Organizers

Dr. Chenjing Bu (Oxford) 

Intrinsic cohomological integrality     

Cohomological integrality, as originally conjectured by Kontsevich–Soibelman, states that the cohomological Hall algebra (CoHA) of a symmetric quiver is a free supercommutative algebra, generated by a graded vector space called BPS cohomology, whose dimension is an integral version of Donaldson–Thomas invariants. This conjecture was later proved by Efimov, and better understood and generalized in the works of Davison, Meinhardt, Reineke, and others. In this talk, I will explain how to generalize this phenomenon from moduli stacks of quiver representations to a much more general class of stacks, including moduli stacks of G-bundles and G-Higgs bundles on a curve for a reductive group G, and moduli stacks of coherent sheaves on Calabi–Yau threefolds. The statement is that the cohomology of the stack can be decomposed as a direct sum of simpler pieces, called BPS cohomology, whose dimension can be understood as integral Donaldson–Thomas invariants. This talk is based on a joint work with Ben Davison, Andrés Ibáñez Núñez, Tasuki Kinjo, and Tudor Pădurariu.  

Prof. Paolo Cascini (ICL)

On the birational geometry of algebraically integrable foliations

I will review recent progress on extending the Minimal Model Program to foliations. Building on techniques from both birational geometry and the theory of foliations, we obtain new results on the structure of foliated pairs. I will focus on applications such as the boundedness of algebraically integrable Fano foliations. Joint work with Han, Liu, Meng, Spicer, Svaldi, and Xie.  

Prof. Shuai Guo (Peking Univ)

Graph sums in higher-genus enumerative geometry and their applications   

In the computation of higher-genus Gromov-Witten invariants, dual graphs provide a natural combinatorial framework for describing the stratification of the moduli space of stable curves. Furthermore, decorated graphs offer an effective way of encoding the contributions from various localization graphs in the virtual localization formula.

In this talk, we will explore two significant applications of these stable graph sum formulas. This work is based on joint work with Chang-Li-Li-Zhou and with Tian-Xu.

Prof. Yujiro Kawamata (Tokyo Univ & MCM)

On NC deformations of sheaves and the moduli space

The NC deformations of sheaves have formal versal families. But calculating the versal deformations is difficult. I will take Grassmann variety as an easy example, and show that even a global moduli space exists as an NC scheme.

Prof. 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 and Donaldson-Thomas 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 have any intersection theory for 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 hour, I will focus on the curve case (singular moduli spaces of vector bundles over curves) with a prospect towards higher dimensional cases.

Note:

There is no registration fee for this conference.

Please click here for registration for the lunch.

If you have any other questions, please send email to us (mcmoffice@math.ac.cn).