Two Lemmas in Local Analytic Geometry

We prove two results about the local properties of generically one to one analytic mappings. Version: .76; Revised: 12–15–98; Run: March 23, 1999 §1: Introduction In this paper we consider two local properties of analytic maps which are generically one to one. First we consider holomorphic maps from open sets in C which behave locally like monoidal transformations: Let (z, w) denote coordinates for C and D,D′ ⊂ C be neighborhoods of (0, 0). We call a holomorphic map f : D → D′ a germ of a blowdown if (1) f(0, w) = (0, 0), (2) f is injective on D \ {z = 0}. We prove the following normal form result for such maps: Lemma 1. Suppose that f : D → D′ is a germ of a blowdown then there are local coordinates, (ζ, ξ) on a neighborhood of (0, 0) such that in these coordinates the map is either (1.1) f(ζ, ξ) = (ζ, ζξ), k ∈ N or f(ζ, ξ) = (ζ , ζ(α1 + ζ (α2 + . . . ζ (αp + ξ) . . . )), αi ∈ C, ki ∈ N, i = 1, . . . , p. (1.2) As a consequence of the lemma we obtain the following: