How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:25:41Z http://mathoverflow.net/feeds/question/79182 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79182/how-to-compute-kobayashi-distance-of-compact-kaehler-manifolds-with-postive-ricci How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature? Bo 2011-10-26T18:52:56Z 2011-10-26T19:27:26Z <p>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".</p> <p>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)?</p> http://mathoverflow.net/questions/79182/how-to-compute-kobayashi-distance-of-compact-kaehler-manifolds-with-postive-ricci/79185#79185 Answer by Ben McKay for How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature? Ben McKay 2011-10-26T19:27:26Z 2011-10-26T19:27:26Z <p>Positive Ricci curvature Kaehler manifolds are Fano, and therefore rationally connected. See Janos Kollar's book <a href="http://books.google.com/books?hl=en&amp;lr=&amp;id=oqW3GabJLjgC&amp;oi=fnd&amp;pg=PA1&amp;dq=janos+kollar+rational+curves&amp;ots=_a9ah5mhdV&amp;sig=-xKWuKioqlb6J_U_L_bxP8Kd-Wg#v=onepage&amp;q=janos%2520kollar%2520rational%2520curves&amp;f=false" rel="nofollow">Rational Curves on Algebraic Varieties</a>. 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.</p>