“Rational curves on open manifolds”
Iitaka’s philosphy claims that whenever there is a theorem for compact projective complex manifolds, there should be a counter-theorem for smooth quasiprojective manifolds. In this talk, I will first introduce this philosophy with explicit examples. Then I will report the recent progress on the theory of ”log rational curves” on smooth quasiprojective manifolds, which has various applications in mathematical physics and in arithmetic geometry. As another piece of evidence of Iitaka’s philosophy, this theory generalizes the classical theory of rational curves on compact projective manifolds developed by Campana and Kollar-Miyaoka-Mori.