Deformations of locally constant stability conditions and good moduli spaces

TL;DR

This paper demonstrates that the set of locally constant stability conditions forms a complex manifold and explores how properties like mass-hom bounds relate to its connected components, linking to moduli spaces.

Contribution

It establishes the complex manifold structure of locally constant stability conditions and connects their properties to moduli space existence and flat families.

Findings

The set of locally constant stability conditions is a complex manifold.

Property of having mass-hom bounds depends only on connected components.

Locally constant stability conditions are equivalent to flat families.

Abstract

We give a structure result on the set of locally constant stability conditions, $\operatorname{Stab}(\mathcal{D}/R)$, defined by Halpern-Leistner-Robotis showing that it has the structure of a complex manifold, in total analogy with Bridgeland's work. As a consequence, we show that the property of having relative mass-hom bounds and the existence of good moduli spaces depends only on the connected components of $\operatorname{Stab}(\mathcal{D}/R)$. Lastly, we observe that the datum of a locally constant stability condition is equivalent to that of a flat family of stability conditions, as described by Bayer et al. in the context of noncommutative algebraic geometry.