全纯曲线正规族及其应用

Since H.Cartan proved the second fundamental theorem about holomorphic curves concerning hyperplanes in general position, the value distribution theory for holomorphic curves has been developed for several decades. Especially, E.I.Nochka, by using the methods of linear algebra, introduced Nochka weight and proved the Cartan's conjecture about the number of hyperplanes which were intersected by linearly degenerate holomorphic curves. Prof. Min Ru extended the above results to holomorphic curves f intersecting hypersurfaces by establishing the inequalities about Chow's weight and Hilbert weight in algebraic geometry. The applicant and collaborators have given the definition for the derived curves of holomorphic curves, and we hope to study the normal family theory and its application by using it. The main contents are as follows: 1. The exact form of Pang-Zalcman lemma of holomorphic curves and its application; 2. To establish the Milloux and Hayman inequality concerning derived curves, and give some Picard type theorems and normal criterion; 3. To obtain some uniqueness theorems for holomorphic curves and their derived curves sharing hyperplanes or hypersurfaces in general position; 4. The application of normal family theory of meromorphic functions to value distribution theory.

自从H.Cartan证明了涉及处于一般位置的超平面的第二基本定理以后，全纯曲线的值分布理论已历经数十年的发展，尤其是E.I.Nochka利用线性代数的方法，通过引入Nochka weight 解决了线性退化时全纯曲线取超平面个数的问题，证明了Cartan猜想。汝敏教授通过建立代数几何中的Chow weight和Hilbert weight间的不等式，将Cartan第二基本定理中涉及超平面的密指量推广到超曲面的情形。申请人及合作者已初步给出了全纯曲线的某种导曲线的定义，本项目将继续研究涉及导曲线的正规族理论及其应用：1.全纯曲线中的Pang-Zalcman引理的精确形式及其应用；2.涉及导曲线的Milloux和Hayman不等式，及相关的Picard型定理与正规定则；3涉及导曲线分担处于一般位置的超平面或超曲面的唯一性；4.亚纯函数正规族理论在值分布中的应用。

全纯曲线的正规族理论对进一步研究全纯曲线的性质起着至关重要的作用，我们首先研究了涉及导曲线分担超平面的全纯曲线的正规定则，得到了一系列的结果；其次.Pang-Zalcman引理在经典的亚纯函数正规族理论中起着至关重要的作用，其在多复变函数中的推广是一个重要的研究方向。我们通过给出多复变全纯函数的球面导数这一概念，证明了关于多复变全纯函数的Pang-Zalcman引理，并利用其得到了相关的涉及多复变全纯函数方向导数的正规定则；最后，我们还得到了一系列涉及例外函数的亚纯函数的Picard型定理和相对应的正规定则。