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Stable bundles and differentiable structures on certain elliptic surfaces. (English) Zbl 0613.14018
The authors extend a result of S. K. Donaldson [C. R. Acad. Sci., Paris, Sér. I 301, 317-320 (1985; Zbl 0584.57010)] that shows that a rational elliptic surface obtained from the projective plane by blowing up 9 base points of a pencil of cubic curves is not diffeomorphic to its homogeneous space \(X_{2,q}\) with two multiple fibres of \(multiplicity^ 2\) and odd q. The fact that the both surfaces are simply-connected and homotopy equivalent was observed earlier by the reviewer. Using a similar technique based on computation of Donaldson’s invariant, the authors show that for two prime numbers p and q the surfaces \(X_{2,q}\) and \(X_{2,p}\) are diffeomorphic if and only if \(p=q\). A more general result about the diffeomorphy of surfaces \(X_{p,q}\) was independently obtained by R. Friedman and J. Morgan (to appear).
Reviewer: I.Dolgachev

14F45 Topological properties in algebraic geometry
57R55 Differentiable structures in differential topology
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J25 Special surfaces
57R50 Differential topological aspects of diffeomorphisms
Full Text: DOI EuDML
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