Inner rates of finite morphisms

Abstract Let (X, 0) be a complex analytic surface germ embedded in ({C}ⁿ, 0) (C n, 0) with an isolated singularity and = (g, f): (X, 0) ({C}², 0) 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 with the morphism Φ when fixing the topological data of the curve (gf) ^-1 (0) (g f) - 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) 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 of the curve (gf) ^-1 (0) (g f) - 1 (0).