Balanced Hyperbolic and Divisorially Hyperbolic Compact Complex Manifolds

TL;DR

This paper introduces two new notions of hyperbolicity for complex manifolds, generalizing existing concepts, and explores their properties, examples, and implications for both Kähler and non-Kähler manifolds.

Contribution

It defines balanced and divisorial hyperbolicity, proves their relation, and introduces divisorially Kähler and nef classes, extending hyperbolicity concepts beyond Kähler manifolds.

Findings

Balanced hyperbolic manifolds are divisorially hyperbolic.

Examples and counter-examples illustrate the new notions.

Hyperbolicity properties influence cohomology classes and line bundles.

Abstract

We introduce two notions of hyperbolicity for not necessarily K\"ahler $n$-dimensional compact complex manifolds $X$. The first, called {\it balanced hyperbolicity}, generalises Gromov's K\"ahler hyperbolicity by means of Gauduchon's balanced metrics. The second, called {\it divisorial hyperbolicity}, generalises the Brody hyperbolicity by ruling out the existence of non-degenerate holomorphic maps from $\C^{n-1}$ 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 {\it divisorially K\"ahler} and {\it 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-K\"ahler manifolds, all these four new notions seem to have a role to play even in the K\"ahler and the projective settings.