Non-Kähler Hyperbolicity: Surfaces and Codimension One

Abstract

We study hyperbolicity notions for compact complex manifolds defined by special Hermitian metrics and by holomorphic maps from $\mathbb C^{n-1}$ (divisorial hyperbolicity). For surfaces we show that divisorial and Brody hyperbolicity coincide and that Hermitian-symplectic and Kähler hyperbolicity coincide when $h^{2,0}=0$. In any dimension we prove obstructions to Gauduchon hyperbolicity, show that projective spaces are not divisorially hyperbolic, and give divisorially hyperbolic manifolds that are not balanced hyperbolic.