Negative Curvature on Line Bundles

Abstract

This chapter gives sufficient conditions for a complex manifold to be hyperbolic in terms of differential forms. The key word here is curvature, which I find very misleading since what is involved are invariants from linear algebra and how they are related to distance or measure decreasing properties of holomorphic maps. The historical terminology, as it evolved from the real case, constitutes a serious psychological impediment for a beginner to learn the complex theory, or at least for me when I was a beginner. Not too much can be done about this, since the terminology of “curvature” is too well established to be discarded. But I have tried to speak systematically of Chern or Ricci forms, and to avoid “curvature” terminology, except in the title of the section as a code word, to obviate linguistic problems.