Motivic Decomposition of Projective Pseudo-homogeneous Varieties

TL;DR

This paper investigates the motives of projective pseudo-homogeneous varieties associated with semi-simple algebraic groups over perfect fields, establishing Rost nilpotence and decompositions, especially in positive characteristic.

Contribution

It introduces the study of motives of pseudo-homogeneous varieties, proving Rost nilpotence and deriving their motivic decompositions, extending previous work on homogeneous varieties.

Findings

Motives of pseudo-homogeneous varieties satisfy Rost nilpotence.

Motivic decompositions relate to those of homogeneous varieties.

Results apply to varieties over fields of positive characteristic.

Abstract

Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of $G$-homogeneous varieties appear. These are varieties over which $G$ acts transitively with possibly non-reduced isotropy subgroup. In this paper we study these varieties which we call ${\it \mbox{projective pseudo-homogeneous varieties}}$ for $G$ inner type over $k$ and prove that their motives satisfy Rost nilpotence. We also find their motivic decompositions and relate them to the motives of corresponding homogeneous varieties.