Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree

TL;DR

This paper proves a second main theorem estimate for entire holomorphic curves into projective spaces intersecting high-degree generic hypersurfaces, advancing understanding of the Green–Griffiths conjecture in complex geometry.

Contribution

It establishes a new second main theorem estimate for non-degenerate holomorphic curves intersecting generic hypersurfaces of high degree.

Findings

Quantifies the Green–Griffiths conjecture for high-degree hypersurfaces.

Provides explicit bounds relating the order and counting functions in Nevanlinna theory.

Extends previous results to a broader class of entire curves in projective spaces.

Abstract

In this note, we establish the following Second Main Theorem type estimate for every entire non-algebraically degenerate holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})$, in present of a {\sl generic} hypersuface $D\subset\mathbb{P}^n(\mathbb{C})$ of sufficiently high degree $d\geq 15(5n+1)n^n$: \[ T_f(r) \leq \,N_f^{[1]}(r,D) + O\big(\log T_f(r) + \log r \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,D)$ stand for the order function and the $1$-truncated counting function in Nevanlinna theory. This inequality quantifies recent results on the logarithmic Green--Griffiths conjecture.