Algebraic geometric interpretation of Donaldson's polynomial invariants (Q688332)

Let \(S\) be a smooth projective surface over the complex numbers, and \(H\) be an ample divisor on \(S\). For any integer \(d\) let \(M_ d\) be the moduli space of Gieseker-Maruyama \(H\)-semistable rank-two torsion-free sheaves on \(S\) with trivial determinant and 2nd Chern class equal to \(d\). The author shows that, associated to \(H\), there exists a natural determinant line bundle \(L_ d\) on \(M_ d\). This line bundle is not ample but it is shown that it is base-point-free. Let \(\gamma\) be the morphism to a suitable projective space defined by sections of a high multiple of \(L_ d\). The main result of the paper asserts that \(\gamma (M_ d)\) is homeomorphic to the Uhlenbeck compactification of the moduli space of \(ASD\) connections on the \(SU(2)\)-bundle on \(S\) with 2nd Chern class equal to \(d\). The author also discusses the possibility of defininig Donaldson polynomials via algebraic geometry.