Equivariant Resolution of Singurlarities in Characteristic 0

0. Introduction We work over an algebraically closed field k of characteristic 0. 0.1. Statement. In this paper, we use techniques of toric geometry to reprove the following theorem: Theorem 0.1. Let X be a projective variety of finite type over k, and let Z ⊂ X be a proper closed subset. Let G ⊂ Aut k (Z ⊂ X) be a finite group. Then there is a G-equivariant modification r : X 1 → X such that X 1 is nonsingular projective variety, and r −1 (Z red) is a G-strict divisor of normal crossings. This theorem is a weak version of the equivariant case of Hironaka's well known theorem on resolution of singularities. It was announced by Hironaka, but a complete proof was not easily accessible for a long time. The situation was remedied by E. Bierstone and P. Milman [B-M2], who gave a construction of completely canonical resolution of singularities. Their construction builds on a thorough understanding of the effect of blowing up. They carefully build up an invariant pointing to the next blowup. The proof we give in this paper takes a completely different approach. It uses two ingredients: first, we assume that we know the existence of resolution of singularities without group actions. The method of resolution is not important: any of [H], [B-M1], [ℵ-dJ] or [B-P] would do. Second, we use equivariant toroidal