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 a geometric quotient Π _X : X → X/G. We characterize rational tensor fields τ on X/G such that the pull back of τ is regular on X: these are precisely all τ such that $di{v_{{R_{X/G}}}}(\tau ) \geqslant 0$ where R_X/G is the reflection divisor of X/G and $di{v_{{R_{X/G}}}}(\tau )$ is the R_X/G-divisor of τ. We give some applications, in particular to a 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 GL(V), then each automorphism Ψ of V/G admits a biregular lift φ : V → V provided that Ψ maps the regular stratum to itself and Ψ_*(R_X/G) = R_X/G.

To C. S. Seshadri on the occasion of his 70th birthday