Motivic Cell Structures for Spherical Varieties

TL;DR

This paper develops a method to construct unstable motivic cell structures for spherical varieties, especially rank 1, using algebraic Morse theory, advancing the understanding of their geometric and motivic properties.

Contribution

It introduces a general approach for unstable motivic cell structures and applies it to spherical varieties, extending previous techniques with new results for rank 1 cases.

Findings

Constructs unstable motivic cell structures after finite suspensions.

Applies algebraic Morse theory to spherical varieties.

Provides new insights into the motivic decomposition of these varieties.

Abstract

In this note, we give a general method to obtain unstable motivic cell structures, following Wendt's application of the Bialynicki-Birula algebraic Morse theory. We then apply the method to spherical varieties, with special attention to the case of rank 1, to obtain unstable motivic cell structures after a finite number of $\mathbb{P}^1$-suspensions. This work is a partial derivative of the first two chapters of the author's 2016 PhD thesis.