Thom condition and monodromy

Gimenez Conejero, Roberto and Lê, D.T. and Nuño-Ballesteros, J.J. (2023) Thom condition and monodromy. REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, 117 (1). ISSN 1578-7303

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Abstract

We give the definition of the Thom condition and we show that given any germ of complex analytic function f : (X , x) → (C, 0) on a complex analytic space X , there exists a geometric local monodromy without fixed points, provided that f ∈ m2 X,x , where mX,x is the maximal ideal of OX,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.

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