Invariant tensor fields and orbit varieties for finite algebraic transformation groups

Abstract

Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $\pi_X:X\to X/G$. We characterize rational tensor fields $\tau$ on $X/G$ such that the {\it pull back} of $\tau $ is regular on $X$: these are precisely all $\tau$ such that $\operatorname{div}_{R_{X/G}}(\tau)\ge 0$ where $R_{X/G}$ is the {\it reflection divisor} of $X/G$ and $\operatorname{div}_{R_{X/G}}(\tau)$ is the {\it $R_{X/G}$-divisor} of $\tau$. We give some applications, in particular to the generalization of Solomon's theorem. In the last section we show that if $V$ is a finite dimensional vector space and $G$ a finite subgroup of $\operatorname{GL}(V)$, then each automorphism $\psi$ of $V/G$ admits a biregular lift $\phi: V\to V$ provided that $\psi$ maps the regular stratum to itself and $\psi_*(R_{X/G})=R_{X/G}$.