Moduli spaces of holomorphic mappings into hyperbolically imbedding complex spaces and locally symmetric spaces (Q1106365)

Let X be a connected Zariski open subset of a compact reduced complex space \(\bar X\) such that X is complete hyperbolic and hyperbolically imbedded into \(\bar X,\) and let N be a Zariski open subset of a compact complex manifold \(\bar N\) such that \(\bar N-\)N is empty or a hypersurface with only normal crossings. The author proves the very general extension theorem for the holomorphic mappings f: \(N\to X\) and applies it to a study of the space Hol(N,X) of all holomorphic mappings. He proves that Hol(N,X) is a Zariski open subset of a compact complex space. Then the general theorem is applied to the special case when X is the quotient \(\Gamma\) \(\setminus D\) of a symmetric bounded domain D by a torsionfree discrete subgroup \(\Gamma\) of Aut(D) which is assumed to be uniform or arithmetic group. Denote by l(D) (resp. l(\(\Gamma)\)) the maximum dimension of proper boundary components (resp. \(\Gamma\)-rational boundary components) of D. Let also Hol(k,N,X) be the set of the maps f with \(rank f=k.\) Then Hol(k,N,X) is open and closed in Hol(N,X). It is proved that Hol(k,N,\(\Gamma\) \(\setminus D)\) is compact for \(k\geq 1(\Gamma)+1\) and finite for \(k\geq l(D)+1.\) Combining this with Schoen and Yau results on harmonic mappings the author proves that for any connected component Z of Hol(N,\(\Gamma\) \(\setminus D)\) and for any \(x\in N\) the evaluation map \(f\in Z\to f(x)\in \Gamma \setminus D\) is a proper holomorphic immersion of Z onto a totally geodesic submanifold in \(\Gamma\) \(\setminus D\). This gives the following rigidity theorem for the holomorphic mapping f:N\(\to \Gamma \setminus D\). f is a unique holomorphic mapping among the free homotopy class of f if f satisfies one of the following conditions: a) the image of f is not contained in a totally geodesic complex proper submanifold of \(\Gamma\) \(\setminus D\); \(b)\quad rank f\geq l(\Gamma)+1;\quad c)\quad \bar f^{-1}(\partial \Gamma \setminus D)\neq \emptyset\) and \(rank f\geq l(\Gamma)+1.\) When \(D=H\) n is a product of upper half planes in C a rigidity and finiteness theorem is proved for the non-constant holomorphic mappings into \(\Gamma\) \(\setminus H\) n.