Some remarks on phantom categories and motives

Abstract

A phantom category is an admissible subcategory with vanishing Grothendieck group of the bounded derived category of coherent sheaves on a smooth projective variety. The goal of this paper is to study the abstract situation when such a category appears and establish some results which provide evidence for the idea that these categories are invisible on the level of Chow motives.

Derived categories and rationality of conic bundles

“…In particular, S satisfies Bloch's conjecture. In this case, Sosna [162,Cor. 4.8] (using results of [92]) remarked that any phantom category in D b (S) is a universal phantom, so that in any base change S K of S, the admissible subcategory A K is a phantom in the base change of the decomposition (6.1).…”

On strong exceptional collections of line bundles of maximal length on Fano toric Deligne-Mumford stacks

“…They also prove that these phantom categories have trivial K-motives and, hence, all their higher K-groups are trivial too. [21] Under certain assumptions on the semi-orthogonal decomposition, this result has implications for the structure of the Chow motive of a variety admitting a phantom category.…”