On the diffeomorphism types of elliptic surfaces (Q1071839)

An elliptic surface is a complex surface M with a holomorphic map \(\pi\) : \(M\to S\) onto a Riemann surface S such that the fibers of \(\pi\) at general points of S are elliptic curves. We can consider S as an orbifold such that images by \(\pi\) of multiple fibers are cone points with appropriate angles. Let \(\pi\) : \(M\to S\) and \(\pi\) ': M'\(\to S'\) be the relatively minimal elliptic surfaces. Suppose that S and S' are either euclidean or hyperbolic. The author announces that then M is diffeomorphic to M' iff the Euler numbers of M, M' are equal and \(\pi_ 1(M)=\pi_ 1(M').\) This contrasts with examples of exotic smooth structures among elliptic Dolgachev surfaces discovered recently by \textit{S. K. Donaldson} [cf. C. R. Acad. Sci., Paris, Sér. I 301, 317-320 (1985)].