Holomorphic curves with shift-invariant hyperplane preimages
Holomorphic curves with shift-invariant hyperplane preimages
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DOI：
10.1090/s0002-9947-2014-05949-7
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发表时间：
2009-03
影响因子：
1.3
通讯作者：
R. Halburd;R. Korhonen;K. Tohge
中科院分区：
文献类型：
--
作者：
R. Halburd;R. Korhonen;K. Tohge
如果f：C！ P n 是小于 1 的超阶全纯曲线，其中一般位置的 2n + 1 个超平面相对于平移 �(z) = z +c 具有前向不变原像，则 f 是周期性的，周期为 c 2 C。这个结果可以描述为全纯曲线的 M. Green 皮卡德型定理的差分模拟，是从本文提出的更一般的结果得出的。该证明依赖于卡索拉蒂行列式的嘉当第二主定理的新版本和对数导数引理的差分类比的扩展版本，这两者都在这里得到证明。最后给出了亚纯函数唯一性理论的应用，并通过算例证明了所得结果的准确性。
If f : C ! P n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation �(z) = z +c, then f is periodic with period c 2 C. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples.