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

Gurjar, R. V.; Shastri, A. R.

Compos. Math. 54, 95-104 (1985).

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

MSC:

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

Keywords:

elliptic surfaces; holomorphic convexity; fundamental group; elliptic fibration; unramified covering

Citations:

Zbl 0527.14016

References:

[1]	I. Dolgacev : On the Severi hypothesis concerning simply-connected algebraic Surfaces , Soviet Math. Doklady 7 (1966). · Zbl 0173.22902
[2]	H. Grauert : Analytischen Faserungen uber Holomorph Vollstandigen Raumern , Math. Ann. 133. · Zbl 0081.07401
[3]	S. Iitaka : Deformations of compact complex Surfaces III , J. Math. Soc. Japan 23 (4) (1971). · Zbl 0219.32012
[4]	K. Kodaira : On compact analytic Surfaces II-III , Ann. of Math. 77 (1963); 78 (1963). · Zbl 0171.19601
[5]	K. Kodaira : On the structure of compact, complex analytic Surfaces I , Amer. J. Math. 86 (1964). · Zbl 0137.17501
[6]	K. Kodaira : On Homotopy K - 3 Surfaces. Essays on Topology and Related Topics Papers dedicated to George de Rham , Springer (1970). · Zbl 0212.28403
[7]	A. Morimoto : Non-compacted complex Lie groups . Proc. Conference on Complex Analysis Minneapolis (1964). · Zbl 0144.07902
[8]	M.V. Nori : Zariski’s conjecture and related problems . (to appear) École Normale Superieure. | · Zbl 0527.14016
[9]	I.R. Shafarevich : Basic Algebraic Geometry . Springer-Verlag (1974). · Zbl 0284.14001