Inner rates of finite morphisms

Abstract

Let (X, 0) be a complex analytic surface germ embedded in (C n , 0) with an isolated singularity and Φ = (g, f ) : (X, 0) −→ (C 2 , 0) be a finite morphism. We define a family of analytic invariants of the morphism Φ, called inner rates of Φ. By means of the inner rates we study the polar curve associated to the morphism Φ when fixing the topological data of the curve (gf ) −1 (0) and the surface germ (X, 0), allowing to address a problem called polar exploration. We also use the inner rates to study the geometry of the Milnor fibers of a non constant holomorphic function f : (X, 0) −→ (C, 0). The main result is a formula which involves the inner rates and the polar curve alongside topological invariants of the surface germ (X, 0) and the curve (gf ) −1 (0).