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# Motivic, logarithmic, and topological Milnor fibrations

Abstract : We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration. We give two equivalent constructions: the first one extending the classical Kato-Nakayama log-space and the second one, more geometric, based on the real oriented multigraph construction, a version of the real oriented deformation to the normal cone. In particular we recover A'Campo's model of the topological Milnor fibration, by quotienting the motivic Milnor fibration with suitable powers of $\mathbb R_{>0}$, and show that it determines the classical motivic Milnor fibre. We give precise formulae expressing how the introduced objects change under blowings-up thus showing, in particular, that the motivic Milnor fibre is well-defined as an element in a Grothendieck ring and not merely in its localisation.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-03460891
Contributor : Jean-Baptiste Campesato Connect in order to contact the contributor
Submitted on : Wednesday, December 1, 2021 - 10:45:19 AM
Last modification on : Monday, October 10, 2022 - 4:06:07 PM

### Identifiers

• HAL Id : hal-03460891, version 1
• ARXIV : 2111.14881

### Citation

Jean-Baptiste Campesato, Goulwen Fichou, Adam Parusiński. Motivic, logarithmic, and topological Milnor fibrations. {date}. ⟨hal-03460891⟩

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