Basic Properties

Abstract

Kobayashi defined a natural semi distance on any complex space. Instead of linking two points by a chain of real curves and taking their lengths and inf via a hermitian or Riemannian metric, he joins points by a chain of discs and takes the inf over the hyperbolic metric. He calls a complex space hyperbolic when the semi distance is a distance. We shall describe the basic properties of such spaces. The most fundamental one is that a holomorphic map is distance decreasing, and hence that a family of holomorphic maps locally is equicontinuous. This gives the possibility of applying Ascoli’s theorem for families of such maps. Such an application has wide ramifications, including possible applications to problems associated with Mordell’s conjecture (Faltings’ theorem) and possible generalizations.