A remark on universal coverings of holomorphic families of Riemann surfaces. (English) Zbl 1082.30032

Imayoshi, Yoichi; Nishimura, Minori

Kodai Math. J. 28, No. 2, 230-247 (2005).

The authors study the universal covering space \(\widehat{M}\) of a holomorphic family \((M, \pi, R)\) of Riemann surfaces over a Riemann surface \(R.\) It is well-known as Koebe’s uniformization theorem for a Riemann surface that the universal covering space \(\widetilde{R}\) of a complex manifold \(R\) of dimension one is given as follows: \(\overline{{\mathbb C}}, {\mathbb C}\) and the unit disk. However, universal coverings and fundamental groups of complex manifolds of higher dimension are very complicated.
Theorem 1. The universal covering space \(\widehat{M}\) of a holomorphic family of Riemann surfaces \((M, \pi, R)\) of type \((g, n)\) is not biholomorphically equivalent to the two-dimensional unit ball \({\mathbb B}_{2}\) provided that \((M, \pi, R)\) is locally trivial, \(n > 0,\) or \(R\) is not compact.
Theorem 2. The universal covering space \(\widehat{M}\) of a holomorphic family of Riemann surfaces \((M, \pi, R)\) is biholomorphically equivalent to the two-dimensional polydisc \(\Delta^{2}\) if and only if all the fibers \(S_{t} = \pi^{-1}(t)\) are biholomorphically equivalent.

Reviewer: V. V. Chueshev (Kemerovo)

MSC:

30F10	Compact Riemann surfaces and uniformization
30F60	Teichmüller theory for Riemann surfaces

Keywords:

universal covering space; holomorphic family of Riemann surfaces; biholomorphically equivalent