Smoothness of stabilisers in generic characteristic

TL;DR

This paper proves that stabilisers of certain group actions are smooth in large positive characteristic, using algebraic and geometric techniques, and applies this to the structure of Lie algebras in positive characteristic.

Contribution

It establishes uniform smoothness of stabilisers and normalisers in large characteristic, extending classical results to positive characteristic settings.

Findings

Stabilisers are smooth for large enough prime characteristic.

Normalisers of subschemes are smooth in large characteristic.

Decomposition of Lie algebras into symplectic varieties in positive characteristic.

Abstract

Let $R$ be a commutative unital ring. Given a finitely presented affine $R$-group scheme $G$ acting on a separated scheme $X$ of finite type over $R$, we show that there is a prime $p_0$ such that for any $R$-algebra $k$ which is an algebraically closed field of characteristic $p\geq p_0$, the centraliser in $G_k$ of any closed subscheme of $X_k$ is smooth. When $X$ is not necessarily separated we show similarly that for any closed subscheme $Y \subseteq X$ there is a $p_1$ depending on $Y$ such that when $k$ has characteristic $p \geq p_1$ the normaliser of $Y$ in $G_k$ is smooth. We prove these results using the Lefschetz principle together with careful application of Gr\"obner basis techniques, and using a suitable notion of the complexity of an action. We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics. In particular, every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit.