Birational motives, I: pure birational motives

TL;DR

This paper introduces a new category of pure birational motives over a field, constructed by modifying effective motives and exploring their relations with cycle decompositions, Rost's cycle modules, and Albanese functors.

Contribution

It defines a novel category of pure birational motives based on algebraic cycles, expanding on previous frameworks and connecting with key concepts like Bloch's decomposition and Rost's cycle modules.

Findings

Defines a category of pure birational motives by 'killing' the Lefschetz motive.

Establishes relationships with Bloch's decomposition of the diagonal.

Analyzes Chow-K"unneth decompositions and links to Rost's cycle modules and Albanese functor.

Abstract

This is a considerably expanded version of the "pure" part of our 2002 preprint. We define a category of pure birational motives over a field, depending on the choice of an adequate equivalence relation on algebraic cycles. It is obtained by "killing" the Lefschetz motive in the corresponding category of effective motives. For rational equivalence, it encompasses Bloch's decomposition of the diagonal. We study the induced Chow-K\"unneth decompositions in this category, and establish relationships with Rost's cycle modules and the Albanese functor for smooth projective varieties.