Open Access Publications from the University of California

## Virtual invariants of Quot schemes of surfaces

• Author(s): Lim, Woonam
The main result expresses the virtual Quot scheme invariants universally in terms of Seiberg-Witten invariants and certain cohomological data of a surface. When a curve class is of Seiberg-Witten length N, we prove the multiplicative structural formula for the generating series of equivariant virtual Quot scheme invariants in terms of the universal series and Seiberg-Witten invariants. Furthermore, the universal series are completely determined up to the change of variables. As an application, we prove the rationality of the homological and K-theoretic descendent series for any curve classes of Seiberg-Witten length N. Explicit formulas are available in several cases for specializations to the generating series of virtual Euler characteristics and virtual $\chi_{-y}$-genera.
For K3 surfaces, the usual virtual Quot scheme invariants vanish. We thus define and study the reduced invariants of Quot schemes. Rather surprisingly, we show that the reduced $\chi_{-y}$-genera of Quot schemes and Pair spaces are equal when N=1. This implies that the reduced $\chi_{-y}$-genera of Quot schemes are given by the Kawai-Yoshioka formula.