Balanced hyperbolic and divisorially hyperbolic compact complex manifolds

We introduce two notions of hyperbolicity for not necessarily Kähler $n$-dimensional compact complex manifolds $X$. The first, called balanced hyperbolicity, generalises Gromov’s Kähler hyperbolicity by means of Gauduchon’s balanced metrics. The second, called divisorial hyperbolicity, generalises the Brody hyperbolicity by ruling out the existence of non-degenerate holomorphic maps from $\mathbb{C}^{n-1}$ to $X$ that have what we term a subexponential growth. Our main result in the first part of the paper asserts that every balanced hyperbolic $X$ is also divisorially hyperbolic. We provide a certain number of examples and counter-examples and discuss various properties of these manifolds. In the second part of the paper, we introduce the notions of divisorially Kähler and divisorially nef real De Rham cohomology classes of degree $2$ and study their properties. They also apply to $C^\infty$, not necessarily holomorphic, complex line bundles and are expected to be implied in certain cases by the hyperbolicity properties introduced in the first part of the work. While motivated by the observation of hyperbolicity properties of certain non-Kähler manifolds, all these four new notions seem to have a role to play even in the Kähler and the projective settings.