We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of $A_n$-singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in A. Ishii and H. Uehara, "Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces", J. Differential Geom., 71(3):385–435, 2005, and the homological mirror symmetry for $A_n$-singularities.
"Stability Conditions on An-Singularities." J. Differential Geom. 84 (1) 87 - 126, January 2010. https://doi.org/10.4310/jdg/1271271794