Generalized Tjurina numbers of an isolated complete intersection singularity

Abstract

We define a series of non-negative integer valued invariants Open image in new window the generalized Tjurina numbers, of an analytic n-dimensional isolated complete intersection singularity. The Tjurina number τ equals τ(1). We show that all these numbers are bounded by the Milnor number μ by establishing explicit formulas for the differences μ−τ(p) involving the mixed Hodge numbers of the cohomology of the link and a new range of analytic invariants for (X,x). This generalizes earlier results of Looijenga and Steenbrink on the relation between the Milnor number and the Tjurina number. We show that singularities satisfying μ=τ(1)=⋯=τ(n−1) behave cohomologically like weighted homogeneous singularities, and we expect this to be a criterion for weighted homogeneity.