Some remarks on phantom categories and motives

TL;DR

This paper investigates phantom categories in derived categories of smooth projective varieties, providing evidence that these categories are undetectable at the level of Chow motives, and explores their abstract properties.

Contribution

It offers new insights into the nature of phantom categories and their invisibility in Chow motives, advancing understanding in algebraic geometry.

Findings

Phantom categories have vanishing Grothendieck groups.

Evidence suggests phantom categories are invisible in Chow motives.

Results support the idea of phantom categories being abstract and elusive.

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.