Virtual Segre and Verlinde numbers of projective surfaces

Recently, Marian–Oprea–Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending the work Johnson, they provided a conjectural correspondence between and Verlinde numbers. For surfaces with holomorphic 2-form, we propose generalizations their results moduli spaces stable sheaves any rank. Using Mochizuki's formula, derive universal function which expresses virtual 2-form in terms Seiberg–Witten invariants intersection products points. We prove that certain canonical general type are topological verify our conjectures examples. The power series algebraic functions, find expressions several cases permuted under Galois actions. Our imply an analog Mariño–Moore higher rank Donaldson invariants. ranks 3 4, obtain explicit