Algebraic degeneracy and Uniqueness Theorems for Holomorphic Curves with Infinite Growth Index from a Disc into ${\mathbb {P}}^n({\mathbb {C}})$ Sharing $2n+2$ Hyperplanes

Abstract

Let f and g be two holomorphic curves of a ball $\Delta (R)$ into ${\mathbb {P}}^n({\mathbb {C}})$ with finite growth index, and let $H_1,\ldots ,H_{2n+2}$ be $2n+2$ hyperplanes in general position. In this paper, our first purpose is to show that if f and g have the same inverse images for all $H_i\ (1\le i\le 2n+2)$ with multiplicities counted to level $l_i$ satisfying an explicitly estimate concerning $c_f$ and $c_g$, then the map $f\times g$ into ${\mathbb {P}}^n({\mathbb {C}})\times {\mathbb {P}}^n({\mathbb {C}})$ must be algebraically degenerated. Our second purpose is to prove that $f=g$ if they share $2n+2$ hyperplanes with some certain conditions (in particular, they share $2n+2$ hyperplanes with multiplicities counted to level $n+1$). Our results extend and improve the previous results for the case of holomorphic curve from ${\mathbb {C}}$ on these directions.