From exceptional collections to motivic decompositions via noncommutative motives

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12 Citations (Scopus)

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)(Q) of every smooth and proper Deligne-Mumford stack X, whose bounded derived category D-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)Q decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover D-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 circle times-units. As an application we obtain a simplification of Dubrovin's conjecture.
Original languageEnglish
Pages (from-to)153-167
Journal Journal für die reine und angewandte Mathematik
Volume2015
Issue number701
DOIs
Publication statusPublished - 2015

Keywords

  • QUANTUM COHOMOLOGY
  • PROJECTIVE SPACES
  • STOKES MATRICES
  • FANO THREEFOLDS
  • K-THEORY
  • CATEGORIES
  • VARIETIES

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