全纯曲线族在导曲线与分担超平面情形下的正规定则

The Normality Criterion of the Family of Holomorphic Curves in the Case of Sharing Hyperplanes and Derivative Curves

摘要: 基于值分布理论、正规族理论及高等代数方法，本文研究了复射影空间中全纯曲线族与导曲线的关系，探讨了其在分担超平面情形下的正规性，并证明了当$N=5$ 时全纯曲线族的正规性准则。设$ℱ$ 是从$D\subset ℂ$ 到${\mathbb{P}}^5\left(ℂ\right)$ 的一族全纯映射，$H_{\ell }=\left\{x\in {\mathbb{P}}^5\left(ℂ\right):\langle x,a_{\ell }\rangle =0\right\}\ne H_0=\left\{x_0=0\right\}$ 是${\mathbb{P}}^5\left(ℂ\right)$ 上处于一般位置的超平面，其中${\alpha }_{\ell }={\left(a_{\ell 0},a_{\ell 1},a_{\ell 2},a_{\ell 3},a_{\ell 4},a_{l5}\right)}^{\text{T}},\ell =1,2,\cdots ,14$ 。假定对任意的$f\in ℱ$ 满足条件：$f\left(z\right)\in H_{\ell }$ 当且仅当$\nabla f\in H_{\ell },\ell =1,2,\cdots ,14$ ；若$f\left(z\right)\in H_{\ell }$ 的并集，则有$\left|f\left(z\right),H_0\right|/\Vert f\Vert \Vert H_0\Vert \ge \delta$ ，$\delta$ 是常数，则$ℱ$ 在$D$ 上正规。

Abstract: Based on the theory of value distribution, the theory of normal families and methods of advanced algebra, this paper studies the relationship between families of holomorphic curves and their derivative curves in complex projective space, explores their normality in the case of sharing hyperplanes, and proves the normality criterion for families of holomorphic curves when $N=5$ . Let $ℱ$ be a family of holomorphic maps of a domain $D\subset ℂ$ into ${\mathbb{P}}^5\left(ℂ\right)$ . Let $H_{\ell }=\left\{x\in {\mathbb{P}}^5\left(ℂ\right):\langle x,a_{\ell }\rangle =0\right\}\ne H_0=\left\{x_0=0\right\}$ be hyperplanes in ${\mathbb{P}}^5\left(ℂ\right)$ located in general position, where ${\alpha }_{\ell }={\left(a_{\ell 0},a_{\ell 1},a_{\ell 2},a_{\ell 3},a_{\ell 4},a_{l5}\right)}^{\text{T}},\ell =1,2,\cdots ,14$ Assume the following conditions hold for every $f\in ℱ$ : $f\left(z\right)$ belongs to $H_{\ell }$ , if and only if $\nabla f$ belongs to $H_{\ell },\ell =1,2,\cdots ,14$ ; if $f\left(z\right)$ belongs to the union set of $H_{\ell }$ , then $\left|f\left(z\right),H_0\right|/\Vert f\Vert \Vert H_0\Vert \ge \delta$ is equal or greater than $\delta$ , where $\delta$ is a constant. Then $ℱ$ is normal on $D$ .