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.

arXiv commentv2: Proof of Theorem 4.2 corrected. Many minor modifications. 23 pages

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• Cycles, derived categories, and rationality
• Stably semiorthogonally indecomposable varieties
• On nonexistence of semi-orthogonal decompositions in algebraic geometry
• Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories
• 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