Given a map of vector bundles on a smooth variety, consider the deepest
degeneracy locus where its rank is smallest. We show it carries a natural
perfect obstruction theory whose virtual cycle can be calculated by the
Thom-Porteous formula.
We show nested Hilbert schemes of points on surfaces can be expressed as
degeneracy loci. We show how to modify the resulting obstruction theories to
recover the virtual cycles of Vafa-Witten and reduced local DT theories.
The result computes some Vafa-Witten invariants in terms of Carlsson-Okounkov
operators. This proves and extends a conjecture of Gholampour-Sheshmani-Yau and
generalises a vanishing result of Carlsson-Okounkov.