Stability Manifold of P^1

TL;DR

This paper computes the stability manifold of the derived category of coherent sheaves on P^1, revealing it to be isomorphic to C^2, thus providing the first complete example for a non-Calabi-Yau manifold.

Contribution

It explicitly determines the stability manifold for P^1, extending the understanding of stability conditions beyond Calabi-Yau cases.

Findings

Stability manifold of D^b(Coh P^1) is isomorphic to C^2

First complete description for a non-Calabi-Yau manifold

Advances the understanding of stability conditions in derived categories

Abstract

T. Bridgeland defined the notion of a stability manifold for a triangulated category, motivated by Douglas's work on \Pi-stability for D-branes. We show that the stability manifold of the bounded derived category of the coherent sheaves on P^1 is C^2. This is the first complete picture of a stability manifold for a non-Calabi-Yau manifold.