# Algebraic group actions on normal varieties

## TL;DR

This paper generalizes Sumihiro's classical result by showing that a normal variety with a connected algebraic group action can be covered by open, quasi-projective, G-stable subvarieties, each embeddable into projectivized G-linearized bundles on abelian quotients.

## Contribution

It extends Sumihiro's theorem to actions of connected algebraic groups on normal varieties, providing new equivariant embeddings into projectivized bundles.

## Key findings

- Normal varieties are covered by G-stable quasi-projective subvarieties.
- Each subvariety admits an equivariant embedding into a projectivized G-linearized bundle.
- Generalizes classical results for smooth connected affine algebraic group actions.

## Abstract

Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a $G$-linearized vector bundle on an abelian variety, quotient of $G$. This generalizes a classical result of Sumihiro for actions of smooth connected affine algebraic groups.