Equivariant $G$-structure on versal deformations

Abstract:

Let ${X_0}$ be an algebraic variety, and $(\chi , \Sigma )$ its versal deformation. Now let G be an affine algebraic group acting algebraically on ${X_0}$. It gives rise to a definite linear G-action on the tangent space of $\Sigma$. In this paper we establish that if G is linearly reductive then there is an equivariant G-action on $(\chi ,\Sigma )$ which induces given G-action on the special fibre ${X_0}$ and its linear G-action on the tangent space of the formal moduli $\Sigma$. Furthermore, such equivariant G-structure is shown to be unique up to noncanonical isomorphism.