Differential forms and smoothness of quotients by reductive groups

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 onX (resp. any regular differential k-form on Y ) is a Kähler 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. Introduction Let π : X −→ Y be a good quotient of a smooth variety X by a reductive algebraic group G. How one can bound the dimension of the singular locus of Y ? Since there exists no natural embedding of Y in some smooth variety, it seems difficult to describe the n-th Fitting ideal of the sheaf ΩY . J. Fogarty suggests a different approach to this problem by raising in [Fog88] the following questions (all schemes are assumed to be of finite type over a field of characteristic 0) : Question Let G be a finite group acting on a smooth variety X and π : X −→ Y the quotient. Is the natural morphism ΩY −→ (Ω 1 X) G surjective if and only if Y is smooth? In that article J. Fogarty verifies that the surjectivity condition is indeed necessary. He also proves that, when the group G is abelian, this condition is sufficient ([Fog88, Lemma 5]). Observe that the module (ΩX) G is naturally isomorphic to ΩY ∨∨ and, the variety Y being normal, also isomorphic to the module ω Y of regular 1-forms (cf. appendix A) and to the module i∗Ω 1 Ysmth (here i denotes the inclusion Ysmth ⊂ y). It is also