Partially Hyperbolic Compact Complex Manifolds

TL;DR

Kasuya and Popovici develop two complementary notions of partial hyperbolicity for compact complex manifolds in directions given by a subbundle $E\subset T^{1,0}X$: a metric notion, partially $p$-Kähler hyperbolic in $E$, and a growth/entire-map notion, partially $p$-hyperbolic in $E$, with two variants. They show how special Hermitian metrics induce growth controls and Ahlfors currents, relate the notions through a chain of implications, and provide a curvature-based criterion via $f_\omega$ and $\star\rho_\omega$ that guarantees strong partial hyperbolicity in the $E$-directions. The paper furnishes explicit examples across OT and MO manifolds, complex parallelisable solvmanifolds, and Vaisman manifolds, illustrating both metric and map-based hyperbolicity in non-Kähler settings. A central theme is the interplay between metric curvature data, growth of higher-dimensional entire maps, and the existence or nonexistence of Ahlfors currents, offering tools for classifying hyperbolicity-like behavior beyond the classical Kähler/Brody framework. Overall, the work broadens hyperbolicity notions to higher-dimensional and non-Kähler contexts, linking curvature, growth conditions, and current theory in compact complex geometry.

Abstract

We propose and investigate two types, the latter with two variants, of notions of partial hyperbolicity accounting for several classes of compact complex manifolds behaving hyperbolically in certain directions, defined by a vector subbundle of the holomorphic tangent bundle, but not necessarily in the other directions. A key role is played by certain entire holomorphic maps, possibly from a higher-dimensional space, into the given manifold $X$. The dimension of the origin $\C^p$ of these maps is allowed to be arbitrary, unlike both the classical $1$-dimensional case of entire curves and the $1$-codimensional case introduced in previous work of the second-named author with S. Marouani. The higher-dimensional generality necessitates the imposition of certain growth conditions, very different from those in Nevanlinna theory and those in works by de Thélin, Burns and Sibony on Ahlfors currents, on the entire holomorphic maps $f:\C^p\longrightarrow X$. The way to finding these growth conditions is revealed by certain special, possibly non-Kähler, Hermitian metrics in the spirit of Gromov's Kähler hyperbolicity theory but in a higher-dimensional context. We then study several classes of examples, prove implications among our partial hyperbolicity notions, give a sufficient criterion for the existence of an Ahlfors current and a sufficient criterion for partial hyperbolicity in terms of the signs of two curvature-like objects introduced recently by the second-named author.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.7
• Lemma 2.8
• Proposition 3.1