Some properties of balanced hyperbolic compact complex manifolds

Abstract

We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the [Formula: see text] harmonic spaces on the universal cover of these manifolds. Other results include a Hard Lefschetz-type theorem for certain compact complex balanced manifolds and the non-existence of certain [Formula: see text] currents on the universal covering space of a balanced hyperbolic manifold.

Some Properties of Balanced Hyperbolic Compact Complex Manifolds

“…In this paper, we continue the study of compact complex balanced hyperbolic manifolds that we introduced very recently in [MP21] as generalisations in the possibly non-projective and even non-Kähler context of the classical notions of Kähler hyperbolic (in the sense of Gromov) and Kobayashi/Brody hyperbolic manifolds. Let X be a compact complex manifold.…”

“…What we mean by f having a subexponential growth is spelt out in Definition 2.3 of [MP21]. It refers to the growth of the volume of the ball B r ⊂ C n−1 of radius r > 0 centred at 0 ∈ C n−1 , with respect to the degenerate metric f ω on C n−1 that is the pullback under f of an arbitrary Hermitian metric ω on X, as r tends to +∞.…”

“…It refers to the growth of the volume of the ball B r ⊂ C n−1 of radius r > 0 centred at 0 ∈ C n−1 , with respect to the degenerate metric f ω on C n−1 that is the pullback under f of an arbitrary Hermitian metric ω on X, as r tends to +∞. Since we do not deal with divisorially hyperbolic manifolds in this paper, we do not repeat here the details of Definition 2.3 of [MP21].…”

“…In [MP21], we generalised the notion of degenerate balanced compact complex manifolds starting from the observation that this is a kind of hyperbolicity property.…”

“…(b) Our study of the cohomology of degree 2 in this setting centres on seeking out possible positivity properties of balanced hyperbolic manifolds. As an alternative to Question 1.4. in [MP21], wondering about possible positivity properties, in the senses defined therein, of the canonical bundle K X of any balanced hyperbolic manifold X, we concentrate this time on whether there are "many" (in a sense to be determined) closed positive currents T of bidegree (1, 1) on such a manifold.…”