Abstract Let 𝑋 be a smooth projective variety. We study admissible subcategories of the bounded derived category of coherent sheaves on 𝑋 whose support is a proper subvariety Z ⊂ X Z X. We show that any one-dimensional irreducible component of 𝑍 is a rational curve. When dim Z = 1 dimZ=1, we prove that at least one irreducible component in 𝑍 intersects the canonical class K X Kₗ 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 (Sat,) studied this question.