Holomorphic maps of projective algebraic manifolds into compact \(C\)- hyperbolic manifolds.

*(English)*Zbl 0835.32011Let \(M\) 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.

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 |