A note on M. Soares' bounds
type: Item
instance of: scholarly article
title: A note on M. Soares' bounds
zbMATH Open document ID: 1089.32025
author: Eduardo Esteves
author: I. Vainsencher
publication date: 2006-03-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10142
review text: The Poincare-Hopf index \(i\) of a holomorphic map germ \(f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^n,0)\) is \[ \dim_{\mathbb{C}}\mathbb{C} [| x_1,\dots,x_n| ]/\langle f_1,\dots,f_n\rangle. \] In [Math. Nachr. 278, No. 6, 703--711 (2005; Zbl 1069.32014)] \textit{M. Soares} found some upper bounds for \(i\). He has constructed a foliation of \(\mathbb{C}\mathbb{P}^n\) with an isolated singularity at 0 of the same index as the original map germ. When all the singularities of the foliation are isolated, M. Soares deduced the bound using the Baum-Bott formula; in the general case he used an analytic deformation argument. In the present paper, the authors replace the deformation argument of M. Soares by a direct application of Fulton's intersection theory. They find a bound for the Poincaré-Hopf index of an isolated singularity of a foliation of \(\mathbb{CP}^n\). They also give special bounds for foliations that admit smooth invariant hypersurfaces not passing through the singularity.
reviewed by: Corina Baciu
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 37F75
zbMATH DE Number: 5015308
zbMATH Keywords: singularities
zbMATH Keywords: foliations
zbMATH Keywords: intersection theory
MaRDI profile type: Publication
OpenAlex ID: W2524030049
cites work: The Castelnuovo-Mumford regularity of an integral variety of a vector field on projective space.
cites work: Q3328651
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cites work: The Poincaré problem for hypersurfaces invariant by one-dimensional foliations
cites work: Bounding Poincaré-Hopf indices and Milnor numbers
cites work: Q3843136
DOI: 10.5802/AIF.2180
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5015308