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

Fabian Haiden, Dongjian Wu

TL;DR

The paper presents a counterexample to the Jordan–Hölder property for polarizable semiorthogonal decompositions by linking algebraic and symplectic perspectives. It identifies a genus-one Fukaya-category realization $P^ot$ that does not admit any Bridgeland stability condition under the standard grading, and shows that a nonstandard grading on a related derived category yields a contrasting polarizable decomposition, thereby violating the polarizable Jordan–Hölder property. The authors use gluing of stability conditions, gap/finite-heart criteria, and graded gentle algebras to establish these obstructions, and then invoke Orlov’s geometric results to transfer the counterexample to a smooth projective variety $X$ with $D^b(X)$ lacking polarizable J–H. This work deepens the interaction between stability conditions, SODs, and Fukaya-category models, illustrating fundamental limits to Jordan–Hölder-type rigidity in noncommutative and geometric settings.

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.

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

TL;DR

The paper presents a counterexample to the Jordan–Hölder property for polarizable semiorthogonal decompositions by linking algebraic and symplectic perspectives. It identifies a genus-one Fukaya-category realization that does not admit any Bridgeland stability condition under the standard grading, and shows that a nonstandard grading on a related derived category yields a contrasting polarizable decomposition, thereby violating the polarizable Jordan–Hölder property. The authors use gluing of stability conditions, gap/finite-heart criteria, and graded gentle algebras to establish these obstructions, and then invoke Orlov’s geometric results to transfer the counterexample to a smooth projective variety with lacking polarizable J–H. This work deepens the interaction between stability conditions, SODs, and Fukaya-category models, illustrating fundamental limits to Jordan–Hölder-type rigidity in noncommutative and geometric settings.

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.

Paper Structure

This paper contains 10 sections, 9 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contents
  3. Preliminaries
  4. Bridgeland stability conditions
  5. Polarizable Jordan--Hölder property
  6. Partially wrapped Fukaya categories and graded gentle algebras
  7. Genus one examples
  8. Proofs of the main results
  9. The non-existence of stability conditions
  10. The failure of the polarizable Jordan-Hölder property

Key Result

Theorem 1.1

The triangulated category $P^{\perp}$, equipped with the standard grading, does not admit a stability condition.

Figures (3)

  • Figure 1: Left: The graded surface $(S,\{p\},\eta)$ associated with $\mathbf{k} Q'/I'$ consists of a torus with one boundary and one marked point (in red) on that boundary, and the arc system $A'$ (in blue) consists of two arcs. Right: The graded surface $(S,M,\eta)$ associated with $\mathbf{k} Q/I$ consists of a torus with one boundary and two marked points (in red) on that boundary, and the arc system $A$ (in blue) consists of three arcs.
  • Figure 2: A sequence of mutations of the S-graphs, with each mutation being relative to a specific blue line.
  • Figure 3: The graded surface $(S,M,\eta)$ consists of a surface of genus $g\ge1$ with one boundary and two marked points in red. The object $P\in\mathcal{W}(S,M,\eta)$ is determined by the arc in blue.

Theorems & Definitions (18)

  • Theorem 1.1: \ref{['thm:Existense of stab']}
  • Corollary 1.3
  • Theorem 1.4: \ref{['thm:Polarizable J-H']}
  • Corollary 1.5
  • Definition 2.1
  • Theorem 2.2: HKK17Lekili2018DerivedEO
  • Definition 4.1: Collins2009GluingSC
  • Lemma 4.2: Collins2009GluingSC
  • Definition 4.3
  • Lemma 4.4: broomhead2024simpletiltslengthhearts
  • ...and 8 more