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)].

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 |