# The space of stability conditions on the local projective plane

Arend Bayer, Emanuele Macri

Number | 47 |
---|---|

Author | Emanuele Macrì |

Project | C03 |

Year | 2009 |

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 Gamma1(3).

Finally, we show that there is a submanifold isomorphic to the universal
covering of a moduli space of elliptic curves with Gamma1(3)-level structure.
The morphism is Gamma1(3)-equivariant, and is given by solutions of
Picard-Fuchs equations. This result is motivated by the notion of Pi-stability
and by mirror symmetry.