Geometric Invariant Theory, holomorphic vector bundles and the Harder-Narasimhan filtration

TL;DR

This survey explores the application of Geometric Invariant Theory to the construction of moduli spaces of holomorphic vector bundles, emphasizing stability, the Harder-Narasimhan filtration, and related stratifications.

Contribution

It provides a comprehensive overview of GIT's role in understanding stability and stratifications in the moduli space of holomorphic vector bundles.

Findings

Connection between GIT and stability notions clarified

Harder-Narasimhan filtration characterized within GIT framework

Stratification results on moduli spaces discussed

Abstract

This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed.