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On the diffeomorphism types of certain algebraic surfaces. I. (English) Zbl 0669.57016
[Part II is reviewed below (see Zbl 0669.57017).]
S. K. Donaldson [C. R. Acad. Sci., Paris, Sér. I, Math. 301, 317- 320 (1985; Zbl 0584.57010)] provided the first example of simply connected h-cobordant 4-manifolds which are not diffeomorphic, by showing that two well-known simply connected algebraic surfaces of type (1,9) are not diffeomorphic. Using the same invariant (introduced by Donaldson [loc. cit.]) the authors strengthen this result considerably, by providing the existence of infinitely many diffeomorphism types in the homotopy class of a rational elliptic surface. In particular, they study certain minimal elliptic surfaces S(p,q) which arise from rational elliptic surfaces via logarithmic transformations of coprime indices p and q. The paper contains several other important results, mostly concerning the surfaces S(p,q). An exposition of this work can be found in a survey by the authors [Algebraic surfaces and 4-manifolds: some conjectures and speculations, Bull. Am. Math. Soc., New Ser. 18, No.1, 1- 19 (1988; Zbl 0662.57016)].
Reviewer: D.Repovš

##### MSC:
 57R55 Differentiable structures in differential topology 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 14J15 Moduli, classification: analytic theory; relations with modular forms 32G05 Deformations of complex structures 32J15 Compact complex surfaces
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