Holomorphic Mappings, the Schwarz-Pick Lemma, and Curvature

摘要

The main result of this paper bounds the dimension of the complex space of rank-k holomorphic mappings between two compact complex manifolds X and Y, which we denote by Hol_k(X, Y). The hypothesis of the main theorem involves curvature conditions on the image manifold Y. There are also two lemmas of independent interest. One shows that the evaluation mapping at any point x ∈ X, which we denote by eval(x): Hol_k(X, Y) →Y, does not reduce dimension. The other lemma is a variation on the Schwarz-Pick lemma. These results are part of the school of thought exemplified in [KSW], [K1], [NS], [No], [SY], and [La].