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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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