Abstract Let ๐ be a smooth projective variety. We study admissible subcategories of the bounded derived category of coherent sheaves on ๐ whose support is a proper subvariety Z โ X Z X. We show that any one-dimensional irreducible component of ๐ is a rational curve. When dim โก Z = 1 dimZ=1, we prove that at least one irreducible component in ๐ intersects the canonical class K X Kโ negatively. In particular, this implies that a surface with a nef and effective canonical bundle has indecomposable derived category, confirming the conjecture by Okawa. We also prove that a configuration of curves with non-negative self-intersections on a surface cannot support an admissible subcategory.
Dmitrii Pirozhkov (2026) studied this question.