Some metric properties of spaces of stability conditions

TL;DR

This paper proves the completeness of the space of numerical Bridgeland stability conditions on certain triangulated categories, computes the metric for curves, and explores how hearts change within the stability space.

Contribution

It establishes the completeness of the stability condition space under mild conditions and provides explicit metric computations for complex curves.

Findings

The space of stability conditions is complete under mild conditions.

The metric on the space of stability conditions for a complex curve is explicitly computed.

Hearts of stability conditions are related by finite tilts, and limits preserve hearts via right tilts.

Abstract

We show that, under mild conditions, the space of numerical Bridgeland stability conditions Stab(T) on a triangulated category T is complete. In particular the metric on a full component of Stab(T) for which the central charges factor through a finite rank quotient of the Grothendieck group K(T) is complete. As an example, we compute the metric on the space of numerical stability conditions on a smooth complex projective curve of genus greater than one, and show that in this case the quotient Stab(T)/C by the natural action of the complex numbers is isometric to the upper half plane equipped with half the hyperbolic metric. We also make two observations about the way in which the heart changes as we move through the space of stability conditions. Firstly, hearts of stability conditions in the same component of the space of stability conditions are related by finite sequences of tilts. Secondly, if each of a convergent sequence of stability conditions has the same heart then the heart of the limiting stability condition is obtained from this by a right tilt.