On Complex Surfaces Diffeomorphic to Rational Surfaces

The first major step in proving that every complex surface diffeomorphic to a rational surface is rational was Yau’s theorem [40] that every complex surface of the same homotopy type as P is biholomorphic to P. After this case, however, the problem takes on a different character: there do exist nonrational complex surfaces with the same oriented homotopy type as rational surfaces, and the issue is to show that they are not in fact diffeomorphic to rational surfaces. The only known techniques for dealing with this question involve gauge theory and date back to Donaldson’s seminal paper [9] on the failure of the h-cobordism theorem in dimension 4. In this paper, Donaldson introduced analogues of polynomial invariants for 4-manifolds M with b2 (M) = 1 and special SU(2)-bundles. These invariants depend in an explicit way on a chamber structure in the positive cone in H(M ;R). Using these invariants, he showed that a certain elliptic surface (the Dolgachev surface with multiple fibers of multiplicities 2 and 3) was not diffeomorphic to a rational surface. In [13], this result was generalized to cover all Dolgachev surfaces and their blowups (the case of minimal Dolgachev surfaces was also treated in [28])