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Differentiable structures of elliptic surfaces with cyclic fundamental group. (English) Zbl 0663.14027

Let \(X_{p,q}\) be elliptic surfaces over \({\mathbb{P}}_ 1\) with at most 2 multiple fibres of multiplicities p and q. It is known that \(\pi_ 1(X_{p,q})={\mathbb{Z}}/k\) where \(k=g.c.d.(p,q)\). In the special case where \(k=1\) and \(p_ g=0\), one has the following theorem:
The surfaces \(X_{2,q}\) with \(q=2n+1\) are pairwise differentiably inequivalent [see e.g. C. Okonek and A. Van de Ven, Invent. Math. 86, 357-370 (1986; Zbl 0613.14018)].
This result is interesting for two reasons; first of all, the surfaces \(X_{p,q}\) are all homeomorphic to \({\mathbb{P}}_ 2\) with 9 points blown up [see M. H. Friedman, J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)]. Furthermore, this theorem is in sharp contrast with a recent result by M. Ue [Invent. Math. 84, 633-643 (1986; Zbl 0595.14028)] who shows that for elliptic surfaces with at least 3 multiple fibres or elliptic surfaces over base curves S with \(g(S)>1\), their diffeomorphism type is completely determined by their homeomorphism type.
In this paper, using their previous work and by computing Donaldson’s invariants, the authors generalize the previous theorem to the cases where \(k>1\). Also notice that, at least for odd k, the \(X_{p,q}\) are homeomorphic in view of recent results by Hambleton and Kreck.
Reviewer: Vo Van Tan

MSC:

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14E20 Coverings in algebraic geometry
57R50 Differential topological aspects of diffeomorphisms
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References:

[1] S.K. Donaldson : La topologie differentielle des surfaces complexes . C.R. Acad. Sc. Paris t.301, Série 1 No. 6, (1985) 317-320. · Zbl 0584.57010
[2] S.K. Donaldson : Irrationality and the h-cobordism conjecture . Preprint (1986). · Zbl 0631.57010 · doi:10.4310/jdg/1214441179
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[10] M. Ue : On the diffeomorphism types of elliptic surfaces with multiple fibres . Invent. Math. 84 (1986) 633-643. · Zbl 0595.14028 · doi:10.1007/BF01388750
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