Stability condition on a singular surface and its resolution

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 $\mathbb{C}$.