Artin shapes

TL;DR

This paper introduces the concept of Artin shapes for motives related to projective homogeneous varieties, providing new insights into their motivic decompositions through examples involving involution varieties and Weil transfers.

Contribution

It presents a novel notion of Artin shape for motives and applies it to explicitly determine motivic decompositions of certain projective homogeneous varieties.

Findings

Discovered unexpected motivic decompositions for specific varieties.

Provided examples that challenge prior assumptions on motivic structures.

Demonstrated the utility of Artin shapes in motivic analysis.

Abstract

We introduce and study on examples a notion of the Artin shape for a motive related to a projective homogenous variety. We apply it to the problem of finding the complete motivic decomposition of the variety. Our examples cover unitary involution varieties as well as some varieties given by a quadratic Weil transfer. Some of the decompositions obtained dispel prior expectations on how motivic decompositions of projective homogeneous varieties can look like.