# Are there any interesting examples of geometric triangulated categories with the Jordan-Holder property?

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?

• 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. – pbelmans Jul 29 '16 at 7:51