Two theorems on higher dimensional singularities

Abstract

Definition 1.t. Let p be a normal isolated singularity in the n-dimensional analytic space Vand let zr : M ~ Vbe a resolution of V. p is a rational singularity if R~zr.(d~), the i-th direct image sheaf of the structure sheaf ~ on M, vanishes near p for i>0. Equivalently, since W~t,(O)=0 at the regular points of V, p is rational if dirlimHi(Tr-l(U),~)=0 for i>0, where U runs over a fundamental set of neighborhoods ofp in V. Several examples were found of such singularities by Burns [5], especially the Arnold singularities of [1], and all quotient singularities. The condition for p to be rational is in fact independent of the choice of the resolution by [9], Corollary 2, p. 153. In this paper the following two theorems are proved.