Friday, March 12, 2021 4:00 PM
Yuuji Tanaka (Kyoto University)

(warning:  notice unusual time)

I'll deliver an overview of studies on the virtual Euler 
characteristics of the moduli spaces of semistable sheaves on a complex 
projective surface. The virtual Euler characteristic is a refinement of 
the topological Euler characteristic for a proper scheme with a perfect 
obstruction theory,which was introduced by Fantechi and Goettsche, and 
by Ciocan-Fontanine and Kapranov. Motivated by the work of Vafa and 
Witten in the early 90's on the S-duality conjecture in N=4 super 
Yang-Mills theory in physics, Goettsche and Kool conjectured that the 
generating function of the virtual Euler characteristics, or other 
variants, of the moduli space of semistable sheaves on a complex 
projective surfaces could be written in terms of modular forms (and the 
Seiberg-Witten invariants), and they verified it in examples. I'll 
describe the recent progress around this topic, starting by mentioning 
more background materials such as the studies on the topological Euler 
characteristics of the moduli spaces.

The discussion for Yuuji Tanaka’s talk is taking place not in zoom-chat, but at (and will be deleted after ~3-7 days).