A generalization of Milnor's formula

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  • Matthias Zach

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Original languageEnglish
Pages (from-to)901–942
Number of pages42
JournalMathematische Annalen
Volume382
Issue number1-2
Early online date23 Jul 2021
Publication statusPublished - Feb 2022

Abstract

We describe a generalization of Milnor's formula for the Milnor number of an isolated hypersurface singularity to the case of a function \(f\) whose restriction \(f|(X,0)\) to an arbitrarily singular reduced complex analytic space \((X,0) \subset (\mathbb C^n,0)\) has an isolated singularity in the stratified sense. The corresponding analogue of the Milnor number, \(\mu_f(\alpha;X,0)\), is the number of Morse critical points in a stratum \(\mathscr S_\alpha\) of \((X,0)\) in a morsification of \(f|(X,0)\). Our formula expresses \(\mu_f(\alpha;X,0)\) as a homological index based on the derived geometry of the Nash modification of the closure of the stratum \(\mathscr S_\alpha\). While most of the topological aspects in this setup were already understood, our considerations provide the corresponding analytic counterpart. We also describe how to compute the numbers \(\mu_f(\alpha;X,0)\) by means of our formula in the case where the closure \(\overline{ \mathscr S_\alpha} \subset X\) of the stratum in question is a hypersurface.

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    math.AG

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A generalization of Milnor's formula. / Zach, Matthias.
In: Mathematische Annalen, Vol. 382, No. 1-2, 02.2022, p. 901–942.

Research output: Contribution to journal › Article › Research › peer review

Zach M. A generalization of Milnor's formula. Mathematische Annalen. 2022 Feb;382(1-2):901–942. Epub 2021 Jul 23. doi: 10.1007/s00208-021-02223-5
Zach, Matthias. / A generalization of Milnor's formula. In: Mathematische Annalen. 2022 ; Vol. 382, No. 1-2. pp. 901–942.
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