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Holomorphic maps of projective algebraic manifolds into compact $$C$$- hyperbolic manifolds. (English) Zbl 0835.32011
Let $$M$$ be a projective algebraic manifold and let $$N$$ be a compact $$C$$- hyperbolic (i.e. the Carathéodory pseudo-distance is actually a distance, on some normal covering of $$N)$$ complex manifold. Then the space $$\text{Hol} (M,N)$$ of all holomorphic mappings $$M \to N$$, in its compact open topology, is a compact complex space with finitely many irreducible components.
The first section of this paper is devoted to the following rigidity result: If holomorphic maps $$f_1, f_2 : M \to N$$ induce the same surjective monodromy $$\pi_1 (M) \to \pi_1 (N)$$ between the fundamental groups and if the images of $$f_1$$ and $$f_2$$ meet, then $$f_1 = f_2$$.
The second section is devoted to an extension of the results of J. Noguchi [Invent. Math. 93, No. 1, 15-34 (1988; Zbl 0651.32012)] who treated a more special class of $$N$$’s, where $$\text{Hol} (M,N)$$ is actually smooth. These results are complicated to state and concern $$c$$- biholomorphica equivalence between connected components of $$\text{Hol} (M,N)$$ and certain subsets of coverings of $$N$$.
In the last section a projective $$C$$-hyperbolic algebraic manifold $$N$$ of dimension 3 is constructed, such that for a certain Riemann surface $$C$$ the space $$\text{Hol} (C,N)$$ has a one-dimensional component with singular points.
Reviewer: P.Michor (Wien)
##### MSC:
 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 32G13 Complex-analytic moduli problems 58D15 Manifolds of mappings 14M99 Special varieties 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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