A survey on rational curves on complex surfaces

TL;DR

This survey reviews the existence and classification of rational curves on complex surfaces, exploring various approaches across different surface classes and highlighting open problems in the field.

Contribution

It systematically analyzes rational curves on complex surfaces within the Enriques–Kodaira classification, summarizing methods and open questions.

Findings

Rational curves exist in various classes of complex surfaces.

Different techniques are applied depending on the surface class.

Open problems remain in understanding rational curves on non-Kähler surfaces.

Abstract

In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the K\"ahler and non-K\"ahler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight the different approaches applied to the study of rational curves in each class. We also provide several examples and point out some open problems.