A counterexample to the Jordan-Hölder property for polarizable semiorthogonal decompositions

2025/02/17 by Fabian Haiden, Haiden, Fabian, Dongjian Wu +1 · 2 citations
Medicine · #Algebraic Geometry (math.AG) #FOS: Mathematics #Head and Neck Surgical Oncology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2502.12075

openalex publication_date 2025/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the Jordan-Hölder property fails for polarizable semiorthogonal decompositions -- those where every factor admits a Bridgeland stability condition. Counterexamples exist among Fukaya categories of surfaces and bounded derived categories of smooth projective varieties. Furthermore, we give an example of a smooth and proper pre-triangulated dg category with positive rank Grothendieck group which does not admit a stability condition.

Cited by

• Frobenius manifold for the nodal quiver
• Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops

Related

• Stability conditions and semiorthogonal decompositions I: quasi-convergence
• A simple counterexample to the Jordan-Hölder property for derived categories
• Openness results for uniform K-stability
• Stability conditions in geometric invariant theory
• Relative stability conditions on Fukaya categories of surfaces