Note on Milnor numbers of irreducible germs

TL;DR

This paper investigates the behavior of Milnor numbers under finite holomorphic mappings for irreducible hypersurface germs with isolated singularities, establishing an inequality relating the Milnor numbers of the original and mapped germs.

Contribution

It proves that the Milnor number of the preimage under a finite holomorphic map is at least as large as that of the original hypersurface germ, extending understanding of singularity invariants.

Findings

Milnor number of preimage ≥ Milnor number of original germ

Inequality holds for irreducible germs with isolated singularities

Results contribute to singularity theory and complex hypersurface mappings

Abstract

Let $(\bf {V,0})\subset (\mathbb{C}^n,0)$ be a germ of a complex hypersurface and let $f: (\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ be a germ of a finite holomorphic mapping. If germs $(\bf {V,0})$ and ${\bf W}:=(F^{-1}(\bf{ V})),0)$ are irreducible and with isolated singularities, then $$\mu(F^{-1}(\bf{ V}))\ge \mu(\bf {V}),$$ where $\mu$ denotes the Milnor number.