Two Theorems on Extensions of Holomorphic Mappings

In this paper we shall prove two theorems about extending holo-morphic mappings between complex manifolds. Both results involve extending such mappings across pseudo-concave boundaries. The first is a removable singularities statement for meromorphic mappings into compact K~ihler manifolds. The precise result and several illustrative examples are given in Section 1. The second theorem is a Hartogs'-type result for holomorphic mappings into a complex manifold which has a complete Hermitian metric with non-positive holomorphic sectional curvatures. This theorem answers one of Chern's problems posed at the Nice Congress [3]. The precise statement and further discussion is given in Section 4. The proofs of both theorems use the class of pluri-sub-harmonic (p. s.h.) functions, which is intrinsically defined on any complex mani-fold [9]. The second proof is rather elementary and essentially relates the p.s.h, functions on the domain off to the curvature assumption on the image manifold. The first theorem is technically a little more delicate and makes use of the removable singularity theorems for analytic sets due to Bishop-Stoll [14] together with the strong estimates available for the amount of singularity which the Levi form of a p. s. h. function may have at an isolated singularity of such a function. At the end of this paper there are two appendices. The first contains a brief survey of some removable singularity theorems for holomorphic mappings between complex manifolds. In the second appendix we give an informal discussion of the general problem of defining the "order of growth" of a holomorphic mapping and using this notion to study such maps. The basic open question here is what might be termed "Bezout's theorem for holomorphic functions of several variables," and this problem is discussed and precisely formulated there. It is my pleasure to acknowledge many helpful discussions with H. Wu concerning the material presented below. In particular, several of the ideas and results in Appendix 2 were communicated to me by him.