Admissible subcategories supported on curves

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 $Z\subset X$. We show that any one-dimensional irreducible component of $Z$ is a rational curve. When $\dim Z=1$, we prove that at least one irreducible component in $Z$ intersects the canonical class $K_X$ 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.