Big Diffeomorphism Groups and Minimal Models

Abstract

The last chapters contained much general information about the definition and basic properties of the Donaldson polynomial invariants. Here we shall study these invariants in more detail for certain classes of algebraic surfaces. In general, it seems to be quite difficult to say much about the Donaldson polynomials without having some a priori information on what they must look like. Such information is provided by the property that they are natural under orientation-preserving diffeomorphisms of spinor norm 1. More precisely, let A(M) be the arithmetic group of of all automorphisms of (H ₂ (M),q _M ) and let D(M)be the subgroup of A(M)consisting of those automorphisms which are of the form ψ _*, whereψ is an orientation-preserving diffeomorphism of M. Let D* (M) be the subgroup of index at most two of D(M) consisting of elements of D(M)which have real spinor norm one. Then the Donaldson polynomials, which are polynomials in _p with coefficients in the symmetric algebra on H ₂ (M),have coefficients invariant under D* (M). Thus, if a simply connected 4-manifold M has a relatively large diffeomorphism group, or more precisely, if D* (M)is a large subgroup of A(M),then the subring of polynomials invariant under D* (M) will be correspondingly smaller and the nature of the Donaldson invariants will be restricted significantly.