Holomorphic maps of projective algebraic manifolds into compact \(C\)- hyperbolic manifolds (Q1342813)

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 \textit{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.