A Poincaré-Hopf-type theorem for holomorphic one-forms
type: Item
instance of: scholarly article
title: A Poincaré-Hopf-type theorem for holomorphic one-forms
zbMATH Open document ID: 1076.32026
author: Toshikazu Ito
author: Bruno Scárdua
published in: Topology
publication date: 2005-02-23T00:00:00Z
review text: The authors study the singularities of codimension one (not necessarily integrable) holomorphic distributions which are transverse to the \(2n-1\) sphere in \(\mathbb C^{n}\), \(n>1\). The main result is an interesting generalization of the classical Poincaré-Hopf index theorem. More in detail, let \(\Omega\) be a holomorphic one form in a neighborhood of a closed ball \(B\subset\mathbb C^n\). The associated (possibly singular) codimension one distribution given by \(\operatorname {Ker}\Omega\subset T\mathbb C^n\) is transverse to the boundary sphere \(S^{2n-1}=\partial B\) provided that \(S^{2n-1}\) contains no singularities of \(\Omega\) and \(\operatorname {Ker}\Omega_p\not\subset T_pS^{2n-1}\) for all \(p\in S^{2n-1}\). The authors prove that if \(\operatorname {Ker}\Omega\) is transverse to \(S^{2n-1}\) then \(n\) is even and \(\Omega\) has exactly one simple singularity in \(B\). In particular, the theorem applies to codimension one (singular) foliations, providing a negative answer for odd dimensions \(n\) to the following question: is there any codimension one holomorphic foliation in a neighborhood of a ball \(B\subset\mathbb C^n\) transverse to \(\partial B\) for \(n>2\)? Among other consequences, the previous result is used to prove that for \(n\geq 3\) any holomorphic one-form which annihilates the radial vector field and has no singularities on the boundary of a ball is not integrable
reviewed by: Filippo Bracci
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 2137340
zbMATH Keywords: holomorphic foliations
zbMATH Keywords: transversality
zbMATH Keywords: index theorems
MaRDI profile type: Publication
full work available at URL: j.top.2004.03.005
OpenAlex ID: W2074368907
DOI: 10.1016/J.TOP.2004.03.005
description: scientific article; zbMATH DE number 2137340