The space of stability conditions on the local projective plane

Abstract

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 ${\Gamma }_1\left(3\right)$.

Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with ${\Gamma }_1\left(3\right)$-level structure. The morphism is ${\Gamma }_1\left(3\right)$-equivariant and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of $\Pi$-stability and by mirror symmetry.