In this paper, Kuznetsov mentions that the following triangulated categories have the Jordan-Holder property

  • $\mathbf{D}(\Bbb P^1)$ and $\mathbf{D}(\Bbb P^1/\Gamma)$
  • connected Calabi-Yau categories
  • derived categories of curves of positive genus

These are the only examples of geometric triangulated categories with the Jordan-Holder property that I know of. Are there any others?

  • $\begingroup$ One paper that comes to my mind containing related results is Hügel--Koenig--Liu, Jordan Hölder theorems for derived module categories of piecewise hereditary algebras. These examples are probably not geometric in your sense, but they are in Orlov's when considered as dg categories. $\endgroup$ – pbelmans Jul 29 '16 at 7:51

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