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 proper
Deligne-Mumford stack X, whose bounded derived category D(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 D(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 the tensor
unit. As an application we obtain a simplification of Dubrovin's conjecture.Comment: 14 pages; revised versio
