A generalization of Puiseux's theorem and lifting curves over invariants

TL;DR

This paper extends Puiseux's theorem to the setting of reductive group actions, showing that generic curves in the quotient space can be locally lifted to the representation space with controlled regularity, including smooth and Denjoy--Carleman classes.

Contribution

It generalizes classical lifting results to broader classes of curves and regularity conditions, providing new insights into the structure of lifts in invariant theory.

Findings

Existence of smooth and $C^M$ lifts for generic curves in the quotient.

Characterization of curves with differentiable lifts when $G$ is finite.

Any germ of a $C^ ext{infty}$ lift of a quasianalytic $C^M$ curve is $C^M$.

Abstract

Let $\rho: G \to \operatorname{GL}(V)$ be a rational representation of a reductive linear algebraic group $G$ defined over $\mathbb C$ on a finite dimensional complex vector space $V$. We show that, for any generic smooth (resp. $C^M$) curve $c : \mathbb R \to V // G$ in the categorical quotient $V // G$ (viewed as affine variety in some $\mathbb C^n$) and for any $t_0 \in \mathbb R$, there exists a positive integer $N$ such that $t \mapsto c(t_0 \pm (t-t_0)^N)$ allows a smooth (resp. $mathbb C^M$) lift to the representation space near $t_0$. ($C^M$ denotes the Denjoy--Carleman class associated with $M=(M_k)$, which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in $V // G$ admits locally absolutely continuous (not better!) lifts. Assume that $G$ is finite. We characterize curves admitting differentiable lifts. We show that any germ of a $C^\infty$ curve which represents a lift of a germ of a quasianalytic $C^M$ curve in $V // G$ is actually $C^M$. There are applications to polar representations.