ICTP/IGAP seminar on Algebraic Geometry: Moduli spaces of sheaves, Vafa-Witten formula
Channel: ICTP Mathematics
Date: 2022-03-03

Speaker: Lothar GOETTSCHE (ICTP)

ICTP/IGAP seminar on Algebraic Geometry: Moduli spaces of sheaves, Vafa-Witten formula and virtual invariants

2022_02_16-MTH-SEM.mp4

Abstract:
After the overview lecture of December, we start going through the material in more detail:
 
We introduce the Hilbert scheme of points on a surface, define moduli spaces of stable sheaves, and state the Vafa-Witten formula for the Euler numberts of moduli spaces of rank 2 sheaves. Â  The moduli spaces of sheaves are in general singular, of a dimension different from the expected dimension. We introduce perfect obstruction theories and the virtual fundamental class. Integrating cohomology classes against this virtual fundamental class allows to define virtual versions of enumerative invariants, which behave similar to the invariants of smooth varieties, among them a virtual version of the Euler number.
We conjecture that the Vafa-Witten formula computes this virtual Euler number.