Holomorphic curves of finite lower order with few inflection points

Abstract

We prove a conjecture proposed by the first-named author in 1998. Let $f:\mathbb{C}\to {\mathbb{P}}^n$ be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function $N_1(r,f)$ of its Wronskian zeros satisfies $N_1(r,f)=o(T(r,f))$, then its order and lower order coincide and belong to $\{1+k/q:k\in {\mathbb{Z}}_{\ge 0},2\le q\le n+1\}$, and its characteristic is regularly varying. Every order in this set occurs. We also prove the sharp inequality ${\mathrm{limsup}}_{r\to \infty }{N}_1(r,f)/T(r,f)\ge 1$ for transcendental linearly non-degenerate curves of order zero.

arXiv comment30 pages