Generalized sheaf counting (Young-Hoon Kiem, Korea Institute for Advanced Study)

02.04.2025 14:15 – 16:00

Abstract:
It is well known that many problems in algebraic geometry are reduced to finding vector bundles or sheaves with desired properties. To enumerate them, we construct their moduli spaces and apply intersection theory. To get a moduli space with an intersection theory, we have to pick a stability condition and delete unstable objects. In good circumstances where there are no strictly semistable sheaves, integrating cohomology classes against the (virtual) fundamental class gives us enumerative invariants like Donaldson invariant, Seiberg-Witten invariant and Donaldson-Thomas invariant. However when there are strictly semistable sheaves, the moduli space is an Artin stack on which integration doesn't make sense under current technology. Generalized sheaf counting is about finding a way to modify the moduli space of semistable sheaves to get a Deligne-Mumford stack by which an enumerative invariant can be defined. The first method is due to Frances Kirwan who constructed a partial desingularization of the moduli space of vector bundles over a smooth projective curve of fixed degree and rank. The second method is due to Takuro Mochizuki who applied the idea of Michael Thaddeus to construct a generalized Donaldson invariant. In this talk, I will report recent progresses on generalized sheaf counting from curves to 3-folds.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Seminaire "Fables géométriques"

entrée libre