From exceptional collections to motivic decompositions via noncommutative motives

Details From exceptional collections to motivic decompositions via noncommutative motives From exceptional collections to motivic decompositions via noncommutative motives

2012-02-28

arxiv.org/abs/1202.6297v2

Mathematics

Algebraic Geometry

13 pages; revised version

Scientific paper

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 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 to 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 tensor units. As an application we obtain a simplification of Dubrovin's conjecture.

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