Complex versus differentiable classification of algebraic surfaces (Q908981)

This short note is an announcement of results of the authors on the holomorphic and differentiable structures on compact complex algebraic surfaces. One of several statements is that the natural map from \{algebraic surfaces mod deformation equivalence\} to \{smooth 4- manifolds mod orientation-preserving diffeomorphisms\} is finite-to-one. For the proof it is enough to consider elliptic surfaces. Main tools are Donaldson's polynomial invariants and the analysis of moduli spaces of stable 2-bundles.