Thom condition and Monodromy

TL;DR

This paper generalizes a theorem about monodromy in complex analytic spaces, showing existence of fixed-point-free monodromy under certain conditions and deriving implications for singularity theory.

Contribution

It extends known results on monodromy to singular spaces and proves a new theorem about the existence of fixed-point-free geometric monodromy.

Findings

Existence of fixed-point-free monodromy when $f otin rak{m}_{X,x}^2$

Generalization of Tibar's theorem to singular spaces

Application showing no coalescing of singularities in certain families

Abstract

We give the definition of the Thom condition and we show that given any germ of complex analytic function $f:(X,x)\to(\mathbb{C},0)$ on a complex analytic space $X$, there exists a geometric local monodromy without fixed points, provided that $f\in\mathfrak m_{X,x}^2$, where $\mathfrak m_{X,x}$ is the maximal ideal of $\mathcal O_{X,x}$. This result generalizes a well-known theorem of the second named author when $X$ is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that $X$ has maximal rectified homotopical depth at $x$ and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.