Spectra of K–Unimodal Isolated Singularities of Complete Intersections

Summary

Abstract

We produce tables of spectra for all K–unimodal isolated complete intersection singularities. Their computation requires slight generalisations of the known methods for hypersurface singularities.

Introduction

The spectrum of an isolated complete intersection singularity (abbreviated: ICIS) has been denned by W. Ebeling and the author [2]. Given an ICIS (X, x), one chooses an ICIS (X′, x) with dim(X′) = dim(X) + 1 and a holomorphic function germ f : (X′, x) → (C, 0) such that X = f⁻¹(0) and the Milnor number μ(X′, x) is as small as possible. Using the mixed Hodge structure on the vanishing cohomology of a holomorphic function f, one may define its spectrum S_p(f) = Σ_α∈Qn_α (α) ∈ Z[Q]. If s_k(X′) is the k–th Hodge number of any one-parameter smoothing of X′ and S_p(f) denotes the spectrum of f, then S_p(X, x) := S_p(f) + Σ_ks_k(X′)(k). Two properties of the spectrum have been proved in [2]: the symmetry and its semicontinuity under deformation. The spectrum is constant under deformations with both μ(X, x) and μ(X′, x) constant.

In this paper we will give the spectra for all K–unimodal isolated singularities of complete intersections; here K is the group of contact equivalences.

The K–unimodal isolated complete intersection singularities have been classified by Wall [10]. Let ℛ stand for right equivalences, i.e. diffeomorphisms of the source of a map germ.