We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply connected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to .
Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with -level structure. The morphism is -equivariant and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of -stability and by mirror symmetry.
Arend Bayer. Emanuele Macrì. "The space of stability conditions on the local projective plane." Duke Math. J. 160 (2) 263 - 322, 1 November 2011. https://doi.org/10.1215/00127094-1444249