Perversity and Exponential Sums II: Estimates for and Inequalities among A-Numbers

Abstract

This chapter describes the estimates for and inequalities among A-numbers. A priori inequalities between the A-numbers of different varieties are presented. The ideas here are entirely finite field, monodromy, and weight theoretic in nature, based on a sheaf-theoretic method of computing in principle. It is supposed that X is a closed n-dimensional subscheme, which is equidimensional, and that the dense open set is affine. It would be interesting to understand the A-number in some case intermediate between ordinary double points and cone singularities. The chapter highlights an algebraically closed field K and a prime. It discusses an irreducible projective hypersurface that has as its only singularities a finite set of ordinary double points. The Jordan decomposition of a tensor product of two unipotent Jordan blocks is well-known, by viewing a Jordan block of size d as the action of the upper nilpotent element in the Lie algebra. The canonical extension of M to a lisse sheaf which is tame at zero is, therefore, a multiplicative translate of a Kloosterman sheaf of rank d.