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 t(z)=z+c, then f is periodic with period 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.
Normality criterion for a family of holomorphic curves that partially share wandering hyperplanes with their derivatives, and holomorphic functions lifted to curves in $P^2(\mathbb{C})$
We established a hyperplane restriction theorem for the local holomorphic mappings between projective spaces, which is inspired by the corresponding theorem of Green for homogeneous ideals in polynomial rings. Our theorem allows us to give the first proof for the existence of gaps (albeit smaller) at all level for the rational proper maps between complex unit balls, conjectured by Huang-Ji-Yin. In addition, our proof does not distinguish the unit balls from other generalized balls and thus it simultaneously demonstrates the same phenomenon for all generalized balls.
We consider a problem of whether a property of holomorphic curves on a subset $X$ of the complex plane can be extended to the whole complex plane. In this paper, the property we consider is uniqueness of holomorphic curves. We introduce the propagation set. Simply speaking, $X$ is a propagation set if linear relation of holomorphic curves on the part of preimage of hyperplanes contained in $X$ can be extended to the whole complex plane. If the holomorphic curves are of infinite order, we prove the existence of a propagation set which is the union of a sequence of disks (In fact, the method app…
We study holomorphic curves $f:\C\longrightarrow \C^3$ avoiding four complex hyperplanes and a real subspace of real dimension four or five in $\C^3$. We show that the projection of $f$ into the complex projective space $\C P^2$ is not necessarily constant.
We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map $g: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^n,0)$ is again singular. This provides a generalization of previous results of this nature by Ebenfelt-Rothschild [Comm. Anal. Geom. 15 (2007), no. 2, 491-507], Lebl [arXiv:0812.2498], and Denkowski [Manuscripta Math. 149 (2016), no. 1-2, 83-91]. The same statement is proved for pullbacks of singular codimension one holomorphic foliations.
Let $\mathcal{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^n$ ($n\geq 2$) having $0$ as a weakly hyperbolic singularity. Let $T$ be a positive harmonic current directed by $\mathcal{F}$ which does not give mass to any of the $n$ coordinate invariant hyperplanes $\{z_j=0\}$ for $1\leq j\leq n.$ Then we show that the Lelong number of $T$ at $0$ vanishes. Moreover, an application of this local result in the global context is given. We discuss also the relation between several basic notions such as directed positive harmonic currents, directed positive d…
On McQuillan's "tautological inequality" and the Weyl-Ahlfors theory of associated curves
1941年，L. Ahlfors给出了另一个证明1933年定理H。P^n中全纯曲线超平面的Cartan逼近Ahlfors的证明建立在H.和J. Weyl（1938），并通过研究全纯曲线的相关曲线证明了Cartan定理。这项工作后来由H. H. 1970年，M. Cowen和P.A. Griffiths在1976年进一步强调了曲率，Y. T. Siu在1987年和1990年提出的，强调亚纯联络。本文利用Schmidt子空间定理证明中的连续极小值，并利用McQuillan的重言式不等式，给出了证明的另一种变形。在这个证明中，基本上所有的分析都被封装在一个修改的McQuillan不等式中，因此大多数证明主要使用代数几何的方法，特别是标志种类。基于McQuillan不等式提出了一个丢番图猜想。
