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Holomorphic maps of compact Riemann surfaces into 2-dimensional compact C-hyperbolic manifolds. (English) Zbl 0556.32014
Let Hol(R,M) be the Douady space of a compact Riemann surface R and a 2- dimensional compact C-hyperbolic manifold M, that is, Hol(R,M) is the set of all holomorphic maps of R into M. A complex manifold M is said to be C-hyperbolic if there exists a covering of M whose Carathéodory pseudo- distance is a distance.
The purpose of this paper is to study concretely the structure of Hol(R,M) by function-theoretic methods. Let F(R,M) be the set of all non- constant holomorphic maps of R into M. The main result is following: F(R,M) has a positive dimensional irreducible component X for some R if and only if M has an unramified finite covering \(S_ 0\times R_ 0\), where \(S_ 0\) and \(R_ 0\) are compact Riemann surfaces of genus \(>1\). - Moreover, X is bimeromorphic to \(S_ 0.\)
The following results are also proved. \((1)\quad \#F(R,M)=\infty\) for some R if and only if M has an unramified covering space \(S_ 0\times R_ 0\), where \(S_ 0\) and \(R_ 0\) are compact Riemann surfaces of genus \(>1\). - \((2)\quad \#F(R,M)=\infty\) for some R if and only if the universal covering space of M is biholomorphic to the 2-dimensional unit polydisc and its covering transformation group \(\Gamma\) contains a subgroup \(H_ 0\times G_ 0\) such that \(H_ 0\) and \(G_ 0\) are discrete subgroups of the analytic automorphism group of the unit disc and such that the index \([\Gamma,H_ 0\times G_ 0]<\infty.\)- (3) Assume that the universal covering space of M is not biholomorphic to the 2-dimensional unit polydisc, or it is biholomorphic to the 2-dimensional unit polydisc and its covering transformation group \(\Gamma\) does not contain any subgroups \(H_ 0\times G_ 0\) such that \(H_ 0\) and \(G_ 0\) are discrete subgroups of the analytic automorphism group of the unit disc and \([\Gamma,H_ 0\times G_ 0]<\infty.\) Then \(\#F(R,M)<\infty\) for any R. In particular, for any fixed Riemann surface R, there exist only finitely many analytic curves on M which are bimeromorphic to R. \((4)\quad \dim F(R,M)\leq 1\) for any R and M.
MSC:
32H99 Holomorphic mappings and correspondences
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
30F10 Compact Riemann surfaces and uniformization
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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