Algebraic group actions on normal varieties
Résumé
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.