Virtual cycles of 3-term complexes and the Hilbert schemes of surfaces

Abstract

Given a 3-term perfect complex E over a quasi-projective variety X and a nonnegative integer r, we define two virtual cycles and their refinements supported over the r-th degeneracy loci of E. This is done by modifying the complex E after pulling it back to certain blow ups of X. We establish several Thom-Porteous, comparison, duality and wall-crossing formulas for these virtual cycles. We apply this construction to perfect complexes arising from the universal objects over the Picard variety and the Hilbert schemes of non-singular complex projective surfaces. We recover, reprove and strengthen some of the known results involving the reduced cycles and the virtual cycles of the Hilbert schemes related to the curve counting theory and Vafa-Witten theory, respectively. In the case of elliptic surfaces, we provide an explicit calculation generalizing that of Seiberg-Witten invariants.