Algebro-geometric analogues of Donaldson's polynomials

Abstract

Donaldson [5] has introduced polynomials on the second homology of any simply connected closed four-manifold, M, with b~odd (at least 3) which, up to + , are invariant under orientation preserving diffeomorphisms. For every c > Ibm+ there is a polynomial 7c(M)~ Syma(~ H2(S), where d(c) = 4c Z2(b~ + 1); to define it one considers J'/c, the set (which is in a natural way a manifold) of Gauge equivalence classes of connections on the SU(2)-bundle on M with c2 = c, antiself-dual with respect to a generic metric on M. There is a natural map /~: H2(M) ~ H2(j/c), namely slant product with 88 where P is the universal SO(3) principal bundle on M x J/c. One way of defining 7c(M) [10] is to construct a compactification of .//r Xc, carrying a fundamental class [Xc]. One proves that # extends to a map #: H2(M) ~ H2(Xc), then one defines