  For any admissible subcategory of the bounded derived category of coherent
sheaves on a smooth proper variety, we prove that sections of the canonical
bundle impose a strong constraint on the supports of the objects of the
subcategory or its semiorthogonal complement. We also show that admissible
subcategories are rigid under the actions of topologically trivial
autoequivalences. As applications of these results, we prove that the derived
category of various minimal varieties in the sense of the minimal model program
admit no non-trivial semiorthogonal decompositions, generalizing the result for
curves due to the second author to higher dimensions. The case of minimal
surfaces is further investigated in detail.
