Families of ICIS with constant total Milnor number

We give the definition of the Thom condition and we show that given any germ of complex analytic function $$f:(X,x)\rightarrow ({\mathbb {C}},0)$$ f : ( X , x ) → ( C , 0 ) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that $$f\in {\mathfrak {m}}_{X,x}^2$$ f ∈ m X , x 2 , where $${\mathfrak {m}}_{X,x}$$ m X , x is the maximal ideal of $${\mathcal {O}}_{X,x}$$ 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.

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Thom condition and monodromy

Conejero

Lê

Nuño–Ballesteros

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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The Bruce–Roberts Numbers of a Function on an ICIS

Lima-Pereira,

Nuno-Ballesteros,

Orefice-Okamoto

et al. 2023

The Quarterly Journal of Mathematics

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We relate a number of invariants of a function germ with isolated singularity over an isolated complete intersection singularity (ICIS), generalizing our previous results, [17, 18], as a consequence we present a new relatively elementary proof of Greuel’s theorem that for a weighted homogeneous ICIS the Milnor number is equal to Tjurina number. With this relation, we are able to prove that the relative logarithmic characteristic variety is Cohen–Macaulay at any point and the logarithmic characteristic variety is Cohen–Macaulay at any point not in $X\times \{0\}$.

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Families of ICIS with constant total Milnor number

Cited by 2 publications

References 9 publications

Thom condition and monodromy

Thom condition and monodromy

The Bruce–Roberts Numbers of a Function on an ICIS

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