部分双曲紧复流形
Partially Hyperbolic Compact Complex Manifolds
AI总结：
该研究提出两类含变体的部分双曲性概念，通过高维整全纯映射及适配增长条件刻画特定方向双曲的紧复流形，证明概念间蕴含关系并给出相关充分判据。
摘要：
我们提出并研究两类部分双曲性概念（第二类包含两种变体），它们用于描述若干类紧复流形：这类流形在由全纯切丛的一个向量子丛定义的特定方向上呈现双曲性，但在其他方向上未必如此。某些从高维空间到给定流形$X$的整全纯映射在研究中起到关键作用。与经典的1维整曲线情形，以及第二作者与S. Marouani此前工作中提出的1余维情形不同，这些映射的定义域$\boldsymbol{C}^p$的维数可以是任意的。由于涉及高维一般性，我们需要对整全纯映射$f:\boldsymbol{C}^p\to X$施加特定的增长条件，这些条件与Nevanlinna理论中的条件，以及de Thélin、Burns和Sibony关于Ahlfors current（阿尔福斯流）的工作中的条件差异很大。这些增长条件的构造思路源自一类特殊的（可能是非Kähler的）埃尔米特度量，这类度量秉承Gromov的Kähler双曲性理论的思想，但适用于高维情形。我们随后研究了若干类例子，证明了我们提出的各部分双曲性概念之间的蕴含关系，给出了Ahlfors current存在性的一个充分判据，还基于第二作者近期引入的两类类曲率对象的符号，给出了部分双曲性的一个充分判据。
英文摘要：
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.
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