Seminar

Entire curves are non-constant holomorphic maps from the complex line to X, and so transcendental analogs of rational curves. We conjecture that X contains a dense entire curve if and only if it is `special', which means that none of its bundles of holomorphic p-forms, for p positive, contains a line bundle with top Iitaka dimension p. These are the manifolds `opposite' in a precise sense to those of `general' type. The main examples are manifolds either rationally connected or with zero Kodaira dimension, which are (in their `orbifold' version) the `building blocks' for all special manifolds. We prove (joint with J. Winkelmann) that rationally connected manifolds carry (a lot of) dense entire curves, and prove this in some cases for manifolds with zero first Chern class. Drawing (in S.Lang's spirit) a parallel with the arithmetic case (when X is defined over a number field), we deduce from this in the above cases a Nevanlinna version of the (arithmetic) Hilbert Property introduced by Corvaja-Zannier.