Contractibility and total semi-stability conditions of Euclidean quivers

TL;DR

This paper explores the structure of stability conditions in the derived category of Euclidean quivers, showing that the moduli space of total semi-stability conditions can contract to any non-concentrated stability condition, with special results for type A quivers.

Contribution

It provides a detailed description of the moduli space of total semi-stability conditions and demonstrates its contractibility, offering new insights into the geometry of stability conditions for Euclidean quivers.

Findings

The moduli space of total semi-stability conditions can contract to any non-concentrated stability condition.

For type A_{p,q}, the entire space of stability conditions is contractible.

The results give an alternative proof of the contractibility of stability spaces for certain quivers.

Abstract

We study the bounded derived category $\mathcal{D}$ of an Euclidean quiver, or equivalently, that of coherent sheaves on a tame weighted projective line. We give a description of the moduli space $\mathrm{ToSS}$ of the total semi-stability conditions on $\mathcal{D}$, which implies that $\mathrm{ToSS}$ can linearly contract to any chosen non-concentrated stability condition in it. For type $\widetilde{A_{p,q}}$, this gives an alternative proof of the contractibility of the whole space of stability conditions.