From exceptional collections to motivic decompositions via noncommutative motives

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 semiorthogonal 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.