Virtual $\chi_{-y}$-genera of Quot schemes on surfaces

This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics in [OP]. We first prove a structural result expressing the equivariant $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants of [DKO]. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: arbitrary surfaces with the zero curve class, relatively minimal elliptic surfaces with curve classes supported on fibers, and minimal surfaces of general type with $p_g>0$ with multiples of the canonical class. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, the generating series of the virtual $\chi_{-y}$-genera are shown to be given by rational functions, for all surfaces with $p_g>0$, addressing a conjecture of [OP]. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula.