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| Original language | Undefined/Unknown |
|---|---|
| Publication status | E-pub ahead of print - 10 Feb 2020 |
Abstract
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2020.
Research output: Working paper/Preprint › Preprint
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TY - UNPB
T1 - A generalization of Milnor's formula
AU - Zach, Matthias
N1 - 30 pages, 4 figures
PY - 2020/2/10
Y1 - 2020/2/10
N2 - 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.
AB - 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.
KW - math.AG
M3 - Preprint
BT - A generalization of Milnor's formula
ER -