Some computations of Donaldsonʹs invariants via flat connections

The Donaldson invariants for the elliptic surfaces X with pg(X) > 0 were first studied by Donaldson (1990), and Friedman and Morgan (1994) using stable bundles. Kronheimer (1991) gave a topological calculation of the degree d = 0 Donaldson invariant of the K3 surface. Here we generalize Kronheimerʹs approach to calculate a degree d = g + 1 invariant for each elliptic surface X with pg(X) = g > 0. We shall construct a moduli space of flat connections M with virtual dimension 2g + 2, derive that the degree g + 1 Donaldson invariant satisfies qd = aQkg − 1 + bkg + 1, and compute the leading coefficient a and how it changes under logarithmic transformations. The result agrees with Friedman and Morganʹs, but our proofs do not use algebraic geometry. l also prove a relation over the anti-self-dual moduli space between the Pontryagin class of the base-point fibration and Donaldsonʹs μ-class for certain smoothly embedded two-spheres with self-intersection −2.