Nonexistence of semiorthogonal decompositions and sections of the canonical bundle

Abstract

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.

v2: Proof of Theorem 4.2 corrected. Many minor modifications. 23 pages

References in corpus (4)

• Log Crepant Birational Maps and Derived Categories
• Grothendieck Duality for Deligne-Mumford Stacks
• A simple counterexample to the Jordan-Hölder property for derived categories
• Some remarks on the group of derived autoequivalences

Cited by in corpus (7)

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• Stably semiorthogonally indecomposable varieties
• On nonexistence of semi-orthogonal decompositions in algebraic geometry
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• Semiorthogonal decomposition of ${\mathrm{D}}^b({\mathrm{Bun}}_2^L)$
• Frobenius and spherical codomains and neighbourhoods
• Indecomposability of the bounded derived categories of Brill-Noether varieties