A survey on hyperbolicity of projective hypersurfaces

TL;DR

This survey reviews recent advances in understanding the hyperbolicity of generic complex projective hypersurfaces, focusing on the behavior of entire curves and their geometric properties in high-degree cases.

Contribution

It compiles and explains recent results on Kobayashi hyperbolicity of projective hypersurfaces, providing a comprehensive overview of the current state of research.

Findings

Progress in establishing hyperbolicity for high-degree hypersurfaces

Insights into the behavior of entire curves in complex projective varieties

Compilation of recent techniques and results in complex hyperbolic geometry

Abstract

These are lecture notes of a course held at IMPA, Rio de Janiero, in september 2010: the purpose was to present recent results on Kobayashi hyperbolicity in complex geometry. Our ultimate goal is to describe the results obtained on questions related to the geometry of entire curves traced in generic complex projective hypersurfaces of high degree. For the convenience of the reader, this survey tries to be as self contained as possible.