A going down theorem for Grothendieck Chow motives

TL;DR

This paper introduces a motivic tool that provides conditions under which certain motives of varieties over field extensions can be lifted back to the base field, aiding in the classification of motivic decompositions.

Contribution

It presents a new going down theorem for Chow motives of geometrically split, irreducible varieties satisfying Rost nilpotence, with applications to classifying motivic decompositions.

Findings

Provides sufficient conditions for lifting outer motives to the base field

Enables classification of motivic decompositions of certain projective homogeneous varieties

Supports answering conjectures related to motivic structures

Abstract

Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost nilpotence principle. Consider a field extension E/F and a finite field K. We provide in this note a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the Chow motive of X_E with coefficients in K to be lifted to the base field. This going down result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type E_6 and to answer a conjecture of Rost and Springer.