A counterexample to the Jordan-H\"older property for polarizable semiorthogonal decompositions

TL;DR

This paper demonstrates that the Jordan-H"older property does not hold for certain semiorthogonal decompositions in algebraic geometry and symplectic geometry, providing explicit counterexamples involving Fukaya categories and derived categories.

Contribution

It introduces counterexamples to the Jordan-H"older property for polarizable semiorthogonal decompositions, expanding understanding of stability conditions in derived and Fukaya categories.

Findings

Counterexamples in Fukaya categories of surfaces.

Counterexamples in derived categories of smooth projective varieties.

Existence of a dg category with positive rank Grothendieck group lacking a stability condition.

Abstract

We show that the Jordan-H\"older 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.