Stability condition on a singular surface and its resolution

TL;DR

This paper constructs and relates Bridgeland stability conditions on a singular surface with ADE singularities and its crepant resolution, analyzing their moduli spaces and stability properties.

Contribution

It introduces new stability conditions on the derived categories of singular surfaces and their resolutions, extending previous results and establishing moduli space properties.

Findings

Existence of Bridgeland stability condition on the singular surface.

Construction of a weak stability condition on the resolution.

Moduli spaces are Artin stacks of finite type.

Abstract

Let $X$ be a surface with an ADE-singularity and let $\widetilde{X}$ be its crepant resolution. In this paper, we show that there exists a Bridgeland stability condition $\sigma_X$ on ${\rm D}^b(X)$ and a weak stability condition $\sigma_{\widetilde{X}}$ on the derived category of the desingularisation ${\rm D}^b(\widetilde{X})$, such that pushforward of $\sigma_{\widetilde{X}}$-semistable objects are $\sigma_X$-semistable We first construct Bridgeland stability conditions on ${\rm D}^b(\widetilde{X})$ associated to the contraction $\widetilde{X} \longrightarrow X$, generalizing the results of Tramel and Xia in \cite{TX22}, Then we deform it to a weak stability condition $\sigma_{\widetilde{X}}$ and show that it descends to ${\rm D}^b(X)$, producing the stability condition $\sigma_X$. Finally, we study the moduli spaces of $\sigma_{\pi^\ast H,\beta,z}$, of $\sigma_{\widetilde{X}}$, and of $\sigma_X$-semistable objects, and we show that the moduli spaces satisfy boundedness and openness, and hence are all Artin stacks of finite type over $\mathbb{C}$.