Hyperbolically imbedded spaces and the big Picard theorem

Abstract

Recently several extension theorems have been obtained for holomorphic maps f : X A ~ M where M is an hyperbolically imbedded complex subspace of a complex space Y (see [12-15]). The purpose of this note is to clarify this concept by giving three equivalent characterizations of hyperbolically imbedded spaces. Using a result of Dufresnoy [3], it will also be shown that if H1, ..., H2, + 1 are 2n + t hyperplanes in general position in P.(C), then M = P,(C){Hi ..... H2.+x} is a complete hyperbolic manifold which is hyperbolically imbedded in P.(C). As a corollary, we obtain a refined version of a generalized big Picard theorem due to Fujimoto [6] and Green [10]. All complex spaces shall be reduced, second countable and connected. If X is a complex space with structure sheaf O, the Zariski tangent space is defined as the linear space over X whose fibre Tp(X) over p consists of all derivations v : ¢p--. C. By an hermitian metric h on X we shall mean: (1) h determines a positive definite hermitian form on each fibre Tp(X). If vs T,(X), then Ilvllh shall denote the length of v with respect to this form. (2) If ~b(t) is a continuously differentiable curve in X, then ][~b'(t)Hh is a continuous function of t. By (2), if q~:[a, b] ~ X is a piecewise continuously differentiable b curve in X, then the length L(4~)= j" H~b'(t)H,dt is well-defined. Thus, the