Covering spaces of an elliptic surface. (English) Zbl 0583.14013

Let S be an irreducible complex projective nonsingular algebraic surface endowed with an elliptic fibration \(p: S\to \Delta\). The authors prove the following theorem: (i) the universal covering space of S is holomorphically convex; (ii) any unramified covering of S is holomorphically convex provided that at least one singular fibre of p is not of type \(mI_ 0\) (i.e. not all singular fibres of p are smooth elliptic curves with some multiplicity). They also give an example of an abelian surface, which is the product of two elliptic curves, having an infinite cyclic covering with no nonconstant holomorphic functions. - As a corollary of (ii) the authors prove the following. Assume that p has at least one singular fibre not of type \(mI_ 0\); if \(C\subset S\) is an irreducible curve with \(C^ 2>0\), then the image of \(\pi_ 1(\bar C)\), (\(\bar C\) a nonsingular model of C) has finite index in \(\pi_ 1(S)\). In particular, if C is rational, it turns out that \(\pi_ 1(S)\) is finite. This is related to a question discussed by M. V. Nori [Ann. Sci. Éc. Norm. Super., IV. Sér. 16, 305-344 (1983; Zbl 0527.14016)].
Reviewer: A.Lanteri


14J25 Special surfaces
32J15 Compact complex surfaces
14E20 Coverings in algebraic geometry


Zbl 0527.14016
Full Text: Numdam EuDML


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