A space of stability conditions that is not a length space

TL;DR

This paper demonstrates that the space of Bridgeland stability conditions, under the canonical metric, is not a length space, and proposes two alternative metrics that could have improved properties.

Contribution

It proves the non-length space property of Bridgeland stability conditions and introduces two modified metrics for better metric behavior.

Findings

The space of Bridgeland stability conditions is not a length space.

Two modified metrics are proposed that may have improved metric properties.

Addresses a question posed by Kikuta regarding the metric structure.

Abstract

We prove that the space of Bridgeland stability conditions, when equipped with the canonical metric, is not a length space in general. This resolves a question posed by Kikuta in the negative. Furthermore, we introduce two modified metrics on the stability spaces, which may exhibit better metric properties.