Thom condition and monodromy

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