Differential Forms and Smoothness of Quotients by Reductive Groups

Abstract

Let π : X → Y be a good quotient of a smooth variety X by a reductive algebraic group G and 1≤k≤ dim (Y) an integer. We prove that if, locally, any invariant horizontal differential k-form on X (resp. any regular differential k-form on Y) is a Kahler differential form on Y then codim(Ysing)>k+1. We also prove that the dualizing sheaf on Y is the sheaf of invariant horizontal dim(Y)-forms.