Inner rates of finite morphisms

TL;DR

This paper introduces inner rates as new invariants for finite morphisms of complex surface germs, enabling detailed analysis of polar curves and Milnor fibers, and provides a formula linking these invariants with topological data.

Contribution

It defines inner rates for finite morphisms and applies them to study polar curves and Milnor fibers, offering a novel formula connecting these invariants with topological features.

Findings

Inner rates are effective invariants for analyzing finite morphisms.

A new formula relates inner rates, polar curves, and topological invariants.

Application to Milnor fibers reveals geometric insights.

Abstract

Let $(X, 0)$ be a complex analytic surface germ embedded in $(\mathbb{C}^n,0)$ with an isolated singularity and $\Phi=(g,f):(X,0) \longrightarrow (\mathbb{C}^2,0)$ be a finite morphism. We define a family of analytic invariants of the morphism $\Phi$, called inner rates of $\Phi$. By means of the inner rates we study the polar curve associated to the morphism $\Phi$ 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 $f:(X,0) \longrightarrow (\mathbb{C},0)$. 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 $(gf)^{-1}(0)$.