Geometric model for module categories of Dynkin quivers via hearts of total stability conditions

TL;DR

This paper introduces a geometric framework for understanding module categories of Dynkin quivers through hearts of total stability conditions, and proves Reineke's conjecture on stability functions.

Contribution

It provides a novel geometric model linking module categories and stability conditions, and confirms Reineke's conjecture about stability functions for indecomposables.

Findings

Established a geometric model for module categories of Dynkin quivers.

Proved Reineke's conjecture on the existence of stability functions.

Demonstrated that any indecomposable can be made stable under a suitable stability function.

Abstract

We derive a geometric model for the module category $\operatorname{mod} \mathbf{k} Q$ of a Dynkin quiver $Q$ via the heart of a total stability condition on the bounded derived category of $\operatorname{mod} \mathbf{k} Q$. As an application, we prove Reineke's conjecture that there is a stability function on $\operatorname{mod} \mathbf{k} Q$ making any indecomposables stable.