Question

Recently I just learned the Kobayashi distance on complex manifolds and wants to get some feeling of how it looks like on exmaples of manifolds with positive Ricci curvature. I have a feeling that the Kobayashi distance on those manifolds should vanish since those are not very "hyperbolic".

A simple example is extended complex plane. then its Kobayashi distance vanishes since the automorphism group can contract one point very close to the origin while keeping the origin fixed. I believe it is similarly true for all complex projective spaces with usual complex structure since automorphism group is known. Can we have more examples? Or is there a counterexample (an example of Kaehler manifold with positive Ricci curvature but not identically vanishing Kobayashi distance)?

Answer 1

Positive Ricci curvature Kaehler manifolds are Fano, and therefore rationally connected. See Janos Kollar's book Rational Curves on Algebraic Varieties. Therefore any two points lie on a rational curve. Since the Kobayashi pseudodistance along a subvariety is never less than the pseudodistance in the ambient space, any pair of points lie at 0 pseudodistance.