Polynomial Bridgeland Stability Conditions on the Category of Coherent Sheaves

TL;DR

This paper introduces a broad class of stability conditions on coherent sheaves that generalize Gieseker stability, explores their properties, and examines their relation to polynomial Bridgeland stability and gauge theory.

Contribution

It provides new examples of stability conditions adapted to coherent sheaves and analyzes their connection to polynomial Bridgeland stability and gauge theoretical frameworks.

Findings

Existence of a broad class of stability conditions generalizing Gieseker stability

These stability conditions admit Harder--Narasimhan and Jordan--H"older filtrations

Relation established between polynomial Bridgeland stability and gauge theoretical stability

Abstract

In this short note, we provide a broad class of examples of stability conditions on the category of coherent sheaves which generalise Gieseker stability. We refer to them as "adapted to coherent sheaves" and they admit Harder--Narasimhan and Jordan--H\"older filtrations. We also study the particular case of Bayer's polynomial Bridgeland stability conditions, and their relation to the gauge theoretical counterpart introduced by Dervan--McCarthy--Sektnan.