The space of stability conditions on the local projective plane

TL;DR

This paper investigates the structure of the space of stability conditions on the canonical bundle over the projective plane, revealing its connectedness, autoequivalence group, and links to elliptic curve moduli spaces through mirror symmetry.

Contribution

It explicitly describes a chamber of geometric stability conditions, shows the connected component is simply-connected, and relates the autoequivalence group to Gamma1(3).

Findings

The connected component of stability conditions is simply-connected.

Autoequivalence group preserving the component is related to Gamma1(3).

A submanifold corresponds to the universal cover of a moduli space of elliptic curves.

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 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.