Generalising Bogomolov's Inequality to Unstable Sheaves

TL;DR

This paper extends Bogomolov's inequality, originally valid for stable sheaves, to all coherent torsion-free sheaves on smooth projective surfaces, broadening its applicability in algebraic geometry.

Contribution

It provides a generalized form of Bogomolov's inequality applicable to unstable sheaves, which was previously limited to stable sheaves.

Findings

Inequality holds for all torsion-free sheaves on smooth projective surfaces.

The generalization includes unstable sheaves, expanding the scope of the original inequality.

Potential applications in the study of moduli spaces of sheaves.

Abstract

We generalise Bogomolov's inequality to all coherent torsion-free sheaves on a smooth projective surface.