We discuss three applications of the results of the preceding chapter which in some way are all related to each other. We begin with defining a so-called period mapping for the miniversal deformation of an icis (X0, x). This mapping has the complement of the discriminant (or rather a covering thereof) as its source and the complex cohomology group of a Milnor fibre as its target. We prove that under certain conditions, this map is a local immersion. This implies that then τ(X0, x) ≤ μ(X0, x) and we thus recover a result of Greuel. The second section begins with discussing isolated singularities with good C*-action in general. In the complete intersection case, we define a certain pairing which we prove to be perfect. This enables us to conclude that τ(X0, x) = μ(X0, x) in that case (dim(X0, x) ≥ 1), a result which is also due to Greuel. In the final section we use the period mapping to investigate the miniversal deformation of a Kleinian singularity. In particular, we obtain an identification of the discriminant of its miniversal deformation with the discriminant of its associated Coxeter group.

Milnor number and Tjurina number

(9. 1) We begin with recalling some elementary linear algebra. Let V be a finite dimensional vector space over a field k. Any v ∈ V defines a contraction ιv : V* → k, φ ↦ φ(v).