On the C°° Invariance of the Canonical Classes of Certain Algebraic Surfaces

1. The results announced in this article concern certain aspects of the diffeomorphism classification of algebraic surfaces, and in particular, the role of the canonical class. We establish our results by developing a general criterion under which the possibilities for Donaldson's polynomial invariants for smooth 4-manifolds [2] are severely limited. We then use these limitations to conclude that in many cases the canonical class of an algebraic surface is a diffeomorphism invariant up to a multiple. Two classes of surfaces satisfying our general criterion are complete intersections and simply connected elliptic surfaces with pg = 0 (mod 2) (see Corollary 8). A third class of such surfaces are certain abelian branched coverings of CP x CP which are surfaces of general type (see §4). These latter surfaces provide infinitely many examples of pairs of homeomorphic, nondiffeomorphic, simply connected surfaces of general type. §2 gives a brief review of the part of Donaldson's theory needed for what we do here. The material described in §3 represents the work of the first and last authors and is some evidence for a general conjecture described in [5]. The material described in §4 represents the work of the second author and will be explained in detail in [6]. The authors owe much to Simon Donaldson. Not only were we inspired by his general theory, but also the earliest versions of Theorem 5 were worked out jointly with him. It is a pleasure to express our gratitude to him.