The space of stability conditions for $A_3$-quiver

TL;DR

This paper investigates the covering map property of the space of stability conditions for the $A_3$-quiver, extending previous work on the $n$-Kronecker quiver to a specific Dynkin quiver case.

Contribution

It demonstrates that the local homeomorphism for $A_3$-quiver stability conditions acts as a covering map outside certain codimension-one subspaces.

Findings

The local homeomorphism becomes a covering map on the complement of six codimension-one subspaces.

The study extends the understanding of stability condition spaces from $n$-Kronecker to $A_3$-quiver.

Provides geometric insight into the structure of stability conditions for Dynkin quivers.

Abstract

The author studied in \cite{shi} the covering map property of the local homeomorphism associating to the space of stability conditions over the $n$-Kronecker quiver. In this paper, we discuss the covering map property for stability conditions over the Dynkin quiver of type $A_3$. The local homeomorphism from a connected component of stability conditions over $A_3$ to 3-dimensional complex vector space becomes a covering map when we restrict it to the complement of six codimension one subspaces.