We express nested Hilbert schemes of points and curves ona smooth projective surface as “virtual resolutions” of degeneracy lociof maps of vector bundles on smooth ambient spaces.We show how to modify the resulting obstruction theories to producethe virtual cycles of Vafa-Witten theory and other sheaf-counting prob-lems. The result is an effective way of calculating invariants (VW, SW,local PT and local DT) via Thom-Porteous-like Chern class formulae.