On the geometry of algebraic groups and homogeneous spaces

TL;DR

This paper explores the algebraic and geometric structures of algebraic groups and their homogeneous spaces, focusing on Chow rings, Picard groups, and fibrations to deepen understanding of their intrinsic properties.

Contribution

It provides explicit descriptions of Chow rings and Picard groups for algebraic groups and homogeneous spaces, and studies their Albanese and anti-affine fibrations.

Findings

Chow ring of G characterized explicitly

Rational Chow ring of X described in detail

Analysis of Albanese and anti-affine fibrations conducted

Abstract

Given a connected algebraic group G over an algebraically closed field and a G-homogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the "anti-affine" fibrations of G and X.