论文标题
$ \ partial $ - 操作员和真实的全态矢量字段
The $\partial$-Operator and Real Holomorphic Vector Fields
论文作者
论文摘要
令$（m，h）$为hermitian歧管，$ψ$在$ m $上的平稳重量功能。 $ \ partial $ -complex在加权伯格曼空间上的$ a^2 _ {（p，0）}（m，h，e，e^{ - ψ}）$ holomorphic $（p，0）$ - 表格最近在[[10]和[9]中研究了。结果表明，如果$ h $是kähler且合适的密度状况，则$ \ partial $ - complex在$（\ bar \partialψ）^{\partialψ）^{\ part sharp} $是holomorphic（即，当真实的渐变$ \ mathrm {granm {granm {grarm {grarm {gronm {gronm fient）中表现出有趣的holomorthicity/duality属性。对于一般的Hermitian指标，如果没有扭转张量$ t_p {}^{rs} $的全体形态性，则该属性无法持有。 在本文中，我们研究了Kähler上的实际全态梯度领域的实价重量函数的存在，并保留了Kähler歧管及其与加权伯格曼空间上的$ \ partial $ complex的关系。对于具有$ \ mathbb c^n $的多主势函数的Kähler指标，我们确定所有具有实际全体形态梯度字段的多主重量功能。对于复杂空间形式的同轴kähler指标，我们首先确定具有全体形态扭转的指标，导致几个有趣的例子，例如Hopf歧管$ \ Mathbb {s}^{2n-1} \ times \ times \ times \ Mathbb {s} s}^1 $，以及“半个”的“半元”。对于其中一些指标，我们将进一步确定具有实际全态梯度字段的权重函数$ψ$。他们提供了赫尔米利亚非kähler歧管的三倍（m，h，e^{ - ψ}）$，带有$ \ partial $ complex的重量，表现出上述holomorphicity/diality属性。在这些示例中，我们详细研究了单位球上的$ \ partial $ complex，其中一半双曲线度量标准，并获得了$ \ partial $ - equation的新估计。
Let $(M,h)$ be a Hermitian manifold and $ψ$ a smooth weight function on $M$. The $\partial$-complex on weighted Bergman spaces $A^2_{(p,0)}(M,h, e^{-ψ})$ of holomorphic $(p,0)$-forms was recently studied in [[10] and [9]. It was shown that if $h$ is Kähler and a suitable density condition holds, the $\partial$-complex exhibits an interesting holomorphicity/duality property when $(\bar\partialψ)^{\sharp}$ is holomorphic (i.e., when the real gradient field $\mathrm{grad}_hψ$ is a real holomorphic vector field). For general Hermitian metrics this property does not hold without the holomorphicity of the torsion tensor $T_p{}^{rs}$. In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds and their relationship to the $\partial$-complex on weighted Bergman spaces. For Kähler metrics with multi-radial potential functions on $\mathbb C^n$ we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally Kähler metrics on complex space forms we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold $\mathbb{S}^{2n-1} \times \mathbb{S}^1$, and the "half" hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions $ψ$ with real holomorphic gradient fields. They provide a wealth of triples $(M,h,e^{-ψ})$ of Hermitian non-Kähler manifolds with weights for which the $\partial$-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the $\partial$-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the $\partial$-equation.
