Stability condition on a singular surface and its resolution

Let $X$ be a surface with an ADE-singularity and let $\tilde{X}$ be its crepant resolution. In this paper, we show that there exists a Bridgeland stability condition ${\sigma }_X$ on ${\mathrm{D}}^b(X)$ and a weak stability condition ${\sigma }_{\tilde{X}}$ on the derived category of the desingularisation ${\mathrm{D}}^b(\tilde{X})$, such that pushforward of ${\sigma }_{\tilde{X}}$-semistable objects are ${\sigma }_X$-semistable We first construct Bridgeland stability conditions on ${\mathrm{D}}^b(\tilde{X})$ associated to the contraction $\tilde{X}⟶X$, generalizing the results of Tramel and Xia in \cite{TX22}, Then we deform it to a weak stability condition ${\sigma }_{\tilde{X}}$ and show that it descends to ${\mathrm{D}}^b(X)$, producing the stability condition ${\sigma }_X$. Finally, we study the moduli spaces of ${\sigma }_{{\pi }^{\ast }H,\beta ,z}$, of ${\sigma }_{\tilde{X}}$, 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 $\mathbb{C}$.