Logarithmic vector fields and hyperbolicity

TL;DR

This paper proves that the complement of a very generic high-degree curve in the complex projective plane is hyperbolic, using a new method that improves previous bounds, especially for curves with two components.

Contribution

Introduces a novel method to establish hyperbolicity of complements of generic curves, improving the degree bound from prior results.

Findings

Complement of a very generic degree ≥14 curve is hyperbolic.

Improved bounds for hyperbolicity in the case of curves with two components.

New method surpasses previous techniques by El Goul.

Abstract

In this article we prove that the complement of a very generic curve of degree at least equal to 14 in the complex projective plane is hyperbolic in the sense of Kobayashi. Thus, using a new method, we improve the former known bound obtained by El Goul. We also improve the case of a very generic curve with two components.