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Licensed Unlicensed Requires Authentication Published by De Gruyter May 7, 2013

From exceptional collections to motivic decompositions via noncommutative motives

Matilde Marcolli and Gonçalo Tabuada

Abstract

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) of every smooth and proper Deligne–Mumford stack 𝒳, whose bounded derived category 𝒟b(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) decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover 𝒟b(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.

Funding source: NSF

Award Identifier / Grant number: DMS-0901221

Funding source: NSF

Award Identifier / Grant number: DMS-1007207

Funding source: NSF

Award Identifier / Grant number: DMS-1201512

Funding source: NSF

Award Identifier / Grant number: PHY-1205440

Funding source: NEC

Award Identifier / Grant number: 2742738

Funding source: Portuguese Foundation for Science and Technology

Award Identifier / Grant number: PEst-OE/MAT/UI0297/2011

The authors are very grateful to Roman Bezrukavnikov and Yuri Manin for stimulating discussions and precise references. They would like also to thank Marcello Bernardara, Alexander Kuznetsov, John Alexander Cruz Morales and Kirill Zaynullin for detailed comments on a previous draft. They are also very grateful to the anonymous referee for all his/her corrections, suggestions, and comments that greatly helped the improvement of the article.

Received: 2012-3-30
Revised: 2012-12-28
Published Online: 2013-5-7
Published in Print: 2015-4-1

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