Thurston compactifications of spaces of stability conditions on curves

TL;DR

This paper constructs a Thurston-like compactification of the space of Bridgeland stability conditions on curves, compares it with classical cases, and classifies autoequivalences, revealing new geometric insights.

Contribution

It introduces a novel compactification of stability condition spaces on curves and applies it to classify autoequivalences, connecting to Teichmüller theory and mirror symmetry.

Findings

Compactification analogous to Thurston's in Teichmüller theory.

Comparison with classical torus case via homological mirror symmetry.

Nielsen-Thurston classification of autoequivalences.

Abstract

In this paper, we construct a compactification of the space of Bridgeland stability conditions on a smooth projective curve, as an analogue of Thurston compactifications in Teichm\"uller theory. In the case of elliptic curves, we compare our results with the classical one of the torus via homological mirror symmetry and give the Nielsen-Thurston classification of autoequivalences using the compactification. Furthermore, we observe an interesting phenomenon in the case of the projective line.