Onb-function, spectrum and rational singularity

Abstract

by duality [18], because R~Tr,CCy, = 0 for i > 0 by [6, 31] (this follows also from [13, 21, 23]) where rc is assumed projective. Here COy denotes the dualizing sheaf (i.e., the dualizing complex [18] shifted by the dimension to the right). The trace morphism (0.2) is injective, and its image is independent of the choice of resolution, because (0.2) is an isomorphism if Y is smooth. We will denote by CSy the image of (0.2). See (2.4.8) below. Now assume Y is a reduced divisor D on a complex manifold X of dimension n. Let f be a reduced defining equation of D on a neighborhood of a: C D. The b-function (i.e., Bernstein polynomial) of f is a monic polynomial bf(s) in s with rational coefficients, and is a generator of the ideal whose element b(s) satisfies the relation