Dense entire curves in Rationally Connected manifolds

TL;DR

This paper proves the existence of dense entire curves in rationally connected complex projective manifolds, confirming a conjecture linking such curves to the 'special' property, and explores related questions in Nevanlinna theory and singular surfaces.

Contribution

It establishes the existence of dense entire curves in rationally connected manifolds and shows they can avoid lifting to ramified covers, addressing conjectures in complex geometry and Nevanlinna theory.

Findings

Dense entire curves exist in rationally connected manifolds.

Such curves can be chosen not to lift to ramified covers.

The paper discusses entire curves in manifolds with trivial first Chern class.

Abstract

We show the existence of metrically dense entire curves in rationally connected complex projective manifolds confirming for this case a conjecture according to which such entire curves on projective manifolds exist if and only if these are "special". We also show that such a dense entire curve may be chosen in such a way that it does not lift to any of its ramified covers, answering in this case a question of Corvaja and Zannier about the Nevanlinna analog of the `weak Hilbert property' of arithmetic geometry. We consider briefly the other test case of the conjecture, namely manifolds with $c_1=0$. Furthermore we discuss entire curves in normal rational surfaces avoiding the singular locus.