A blowup formula for Mochizuki's virtual Donaldson invariants on projective surfaces

2021/11/12 Fri 11:00 - 12:00
Yuuji Tanaka
Kyoto University

This is joint work with Nikolas Kuhn. We prove a blowup formula for the Donaldson-Mochizuki invariant, a virtual analogue of the Donaldson invariant, on projective surfaces.

Takuro Mochizuki introduced his virtual Donaldson invariants in the early 2000s, and they did play prominent roles in the progress of Gauge Theory in Mathematics, such as the determination of the wall-crossing terms of Donaldson invariants and a proof of the equivalence of Donaldson and Seiberg-Witten ones on a projective surface both by Goettsche-Nakajima-Yoshioka. More recently, the work of Mochizuki revealed anew, and confirmed in several examples, a fascinating
conjecture by Vafa and Witten on the modular properties of the generating series of virtual Euler characteristics of the moduli spaces of semistable sheaves on a surface in the course of works by
Goettsche-Kool and other people.

A remaining problem in the programme of Mochizuki by himself was to prove a blowup formula for his invariants. We establish it in the style of Nakajima-Yoshika using perverse coherent sheaves on a blowup, but also with perfect obstruction theories on the moduli spaces of them. An application of our formula is a direct proof of the equivalence of Mochizuki's invariants and the classical Donaldson invariants in our setting, which was once pointed out by Goettsche-Nakajima-Yoshioka in the above mentioned work D=SW on projective surfaces. We utilise Mochizuki's enhanced moduli space technique to handle automorphisms on the moduli stacks in a coherent way and apply Kiem-Li's version of master space construction to get around an issue of the absence of GIT descriptions of the moduli spaces in our setting.