Weak Stability Conditions as Limits of Bridgeland Stability Conditions

TL;DR

This paper introduces a new type of weak stability condition on triangulated categories, allowing non-maximal phases in the kernel, and constructs four such conditions on Weierstrass elliptic surfaces as limits of Bridgeland stability conditions.

Contribution

It defines weak stability conditions with a relaxed kernel phase condition and constructs four natural examples on elliptic surfaces as limits of Bridgeland stability.

Findings

Defined weak stability conditions with non-maximal phases in the kernel.

Constructed four types of weak stability conditions on elliptic surfaces.

Showed these conditions arise as limits of Bridgeland stability.

Abstract

In this paper, we give a definition of weak stability condition on a triangulated category. The difference between our definition and existing definitions is that we allow objects in the kernel to have non-maximal phases. We then construct four types of weak stability conditions that naturally occur on Weierstrass ellitpic surfaces as limites of Bridgeland stability conditions.