On a deformation of gluing stability conditions

TL;DR

This paper investigates how gluing stability conditions on a triangulated category can be deformed continuously, constructing a family of stability conditions that maintain key properties, and applies this to the category of morphisms.

Contribution

It introduces a deformation of gluing stability conditions, proving continuousness and support property preservation, and explores the stability space of morphism categories.

Findings

Constructed a continuous family of stability conditions via deformation.

Proved each stability condition in the family satisfies the support property.

Analyzed the stability space of morphism categories in triangulated categories.

Abstract

On a triangulated category $\mathbf D$ equipped with a semiorthogonal decomposition $\mathbf D=\langle{\mathbf D_{1}},{\mathbf D_{2}}\rangle$, Collins and Polishchuk develop a gluing construction of stability condition on $\mathbf D$. The gluing construction gives a stability condition on $\mathbf D$ from these on $\mathbf D_{1}$ and $\mathbf D_{2}$. We study a deformation of gluing stability conditions on for a nice semiorthogonal decomposition. As a consequence, we construct a continuous family of stability conditions by showing a deformation property introduced by Bridgeland's original paper. Here the deformation property is weaker than the support property which is the standard solution for the continuousness. After proving the continuousness of the family, we show that each stability condition in the family satisfies the support property via specialization. More precisely we find a stability condition with support property at the boundary of the family. Finally applying these results, we study the space of stability conditions on the category of morphisms in a triangulated category.