From exceptional collections to motivic decompositions via noncommutative motives
Author(s)
Marcolli, Matilde; Trigo Neri Tabuada, Goncalo Jorge
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Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(𝒳)[subscript ℚ] of every smooth and proper Deligne–Mumford stack 𝒳, whose bounded derived category 𝒟[superscript b](𝒳) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(𝒳)[subscript ℚ] decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover 𝒟[superscript b](𝒳) admits a semi-orthogonal decomposition, then the noncommutative motive of each one of the pieces of the semi-orthogonal decomposition is a direct sum of ⊗-units. As an application we obtain a simplification of Dubrovin's conjecture.
Date issued
2013-05Department
Massachusetts Institute of Technology. Department of MathematicsJournal
Journal für die reine und angewandte Mathematik (Crelles Journal)
Publisher
Walter de Gruyter
Citation
Marcolli, Matilde, and Gonçalo Tabuada. “From Exceptional Collections to Motivic Decompositions via Noncommutative Motives.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 0, no. 0 (January 7, 2013). © De Gruyter 2015
Version: Final published version
ISSN
0075-4102
1435-5345