Balanced hyperbolic and divisorially hyperbolic compact complex manifolds

Pub Date : 2024-07-17 DOI:10.4310/mrl.2023.v30.n6.a7
Samir Marouani, Dan Popovici
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Abstract

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.
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平衡双曲和分裂双曲紧凑复流形
我们为不一定是 Kähler $n$ 维紧凑复流形 $X$ 引入了两个双曲性概念。第一个概念称为平衡双曲性,它通过高杜洪的平衡度量对格罗莫夫的凯勒双曲性进行了概括。第二种双曲性被称为分裂双曲性,它通过排除从 $\mathbb{C}^{n-1}$ 到 $X$ 的非退化全态映射的存在来概括布罗迪双曲性,我们称之为亚指数增长。我们在论文第一部分的主要结果断言,每一个平衡双曲的 $X$ 也是分裂双曲的。我们提供了一些例子和反例,并讨论了这些流形的各种性质。在论文的第二部分,我们引入了阶数为 2$ 的可导 Kähler 和可导 nef 实 De Rham 同调类的概念,并研究了它们的性质。它们也适用于 $C^infty$,不一定是全形的复线束,并且有望在某些情况下被第一部分中引入的双曲性性质所隐含。虽然这四个新概念的动机是观察某些非凯勒流形的双曲性特性,但它们似乎在凯勒流形和投影流形中也能发挥作用。
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