A virtual fundamental class construction for the moduli space of (C⁎)n-equivariant morphisms

Let X be a smooth projective variety with the action of (C⁎)n. The article describes the moduli space of (C⁎)n equivariant morphisms from stable toric varieties into X as the inverse limit of the GIT quotients of X and their flips when these spaces are enhanced with a naturally associated Deligne-Mumford stack structure. This description is used for constructing a class in the Chow group of the moduli space of dimension dim(X)−n which is invariant under equivariant deformations of X.