From exceptional collections to motivic decompositions via noncommutative motives

TL;DR

This paper establishes a connection between exceptional collections in derived categories and motivic decompositions using noncommutative motives, providing new insights into the structure of motives for certain algebraic stacks.

Contribution

It proves that smooth proper Deligne-Mumford stacks with full exceptional collections have Chow motives decomposing into Lefschetz motives, and relates semi-orthogonal decompositions to noncommutative motives.

Findings

Chow motive of stacks with full exceptional collections decomposes into Lefschetz motives.

Semi-orthogonal decompositions imply noncommutative motives are sums of the tensor unit.

Application to simplifying Dubrovin's conjecture.

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 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.