2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119488In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms.72 pages, 2 figuresAlgebraic GeometryHigh Energy Physics - TheoryDifferential Geometry14D20; 14D21; 57R57; 81T13; 81T60text