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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Z</m:mi> <m:mo>⊂</m:mo> <m:mi>X</m:mi> </m:mrow> </m:math> Z⊂ X . We show that any one-dimensional irreducible component of 𝑍 is a rational curve. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>dim</m:mi> <m:mo lspace="0.167em">⁡</m:mo> <m:mi>Z</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> dimZ=1 , we prove that at least one irreducible component in 𝑍 intersects the canonical class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mi>X</m:mi> </m:msub> </m:math> KX 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.