Hyperbolic and Diophantine analysis

In this survey we consider Kobayashi hyperbolicity, in which there is an interplay between five notions: analytic notions of distance and measure; complex analytic notions; differential geometric notions of curvature (Chern and Ricci form); algebraic notions of "general type" (pseudo ampleness); arithmetic notions of rational points (existence of sections). I am especially interested in the relations of the first four notions with diophantine geometry, which historically has intermingled with complex differential geometry. One of the main points of this survey is to arrive at a certain number of conjectures in an attempt to describe at least some of these relations coherently.