Relations between the Chow motive and the noncommutative motive of a smooth projective variety

TL;DR

This paper explores the connections between classical Chow motives and noncommutative motives of smooth projective varieties, highlighting their similarities, differences, and implications through examples and applications.

Contribution

It establishes a relationship between Chow motives and noncommutative motives, clarifying how properties like decomposability and isomorphism translate between these frameworks.

Findings

Relations between Lefschetz type and unit type motives

Counter-examples illustrating differences between motives

Applications in understanding motive decompositions

Abstract

In this note we relate the notions of Lefschetz type, decomposability, and isomorphism, on Chow motives with the notions of unit type, decomposability, and isomorphism, on noncommutative motives. Examples, counter-examples, and applications are also described.