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 $\mathbb{ C}$.
Categories: math.AG