Some variants of the generalized Borel Theorem and applications

TL;DR

This paper extends generalized Borel's Theorem, constructs low-degree hyperbolically embedded surfaces in complex projective space, and establishes a Second Main Theorem estimate for certain hypersurfaces, advancing hyperbolicity theory.

Contribution

It provides new results on generalized Borel's Theorem, improves degree bounds for hyperbolic embeddings, and formulates a Second Main Theorem estimate for Fermat-Waring type hypersurfaces.

Findings

Constructed smooth surfaces of degree ≥19 with hyperbolic complements in CP^3.

Improved previous degree bound from 31 to 19 for hyperbolic embedding.

Established a Second Main Theorem estimate for specific Fermat-Waring hypersurfaces.

Abstract

In the first part of this paper, we establish some results around generalized Borel's Theorem. As an application, in the second part, we construct example of smooth surface of degree $d\geq 19$ in $\mathbb{CP}^3$ whose complements is hyperbolically embedded in $\mathbb{CP}^3$. This improves the previous construction of Shirosaki where the degree bound $d=31$ was gave. In the last part, for a Fermat-Waring type hypersurface $D$ in $\mathbb{CP}^n$ defined by the homogeneous polynomial \[ \sum_{i=1}^m h_i^d, \] where $m,n,d$ are positive integers with $m\geq 3n-1$ and $d\geq m^2-m+1$, where $h_i$ are homogeneous generic linear forms on $\mathbb{C}^{n+1}$, for a nonconstant holomorphic function $f\colon\mathbb{C}\rightarrow\mathbb{CP}^n$ whose image is not contained in the support of $D$, we establish a Second Main Theorem type estimate: \[ \big(d-m(m-1)\big)\,T_f(r)\leq N_f^{[m-1]}(r,D)+S_f(r). \] This quantifies the hyperbolicity result due to Shiffman-Zaidenberg and Siu-Yeung.