Some metric properties of spaces of stability conditions

Understand

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.