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

Degeneracy loci, virtual cycles and nested Hilbert schemes I