Zero Rank Asymptotic Bridgeland Stability

TL;DR

This paper characterizes the conditions under which objects with zero rank become asymptotically stable in Bridgeland stability, linking it to classical sheaf stability notions.

Contribution

It establishes an equivalence between asymptotic Bridgeland stability for zero-rank objects and classical sheaf Gieseker-Simpson stability.

Findings

Objects with zero rank are asymptotically stable iff they satisfy sheaf Gieseker-Simpson stability.

The stability condition is equivalent to a dual of sheaf stability depending on the curve.

Provides criteria for asymptotic stability in the context of Bridgeland stability.

Abstract

In this paper we examine the conditions that an object $E$ with $\text{ch}_0(E)=0$ has to satisfy in order for it to be asymptotically (semi)stable with regard to Weak or Bridgeland stability conditions. This notion turned out to be equivalent to sheaf Gieseker-Simpson (semi)stability or a dual of it, depending on the curve considered.