Residues of complex analytic foliation singularities
type: Item
instance of: scholarly article
title: Residues of complex analytic foliation singularities
zbMATH Open document ID: 0582.32017
author: Tatsuo Suwa
published in: Journal of the Mathematical Society of Japan
publication date: 1984-01-01T00:00:00Z
review text: Let M be an n-dimensional \({\mathbb{C}}\)-manifold and let \(\Omega_ M\) be the cotangent sheaf of M. Then a foliation is defined to be a full coherent subsheaf F of \(\Omega_ M\) satisfying the integrability condition. It is shown that this definition coincides with the ones introduced by \textit{P. Baum} and \textit{R. Bott} [J. Differ. Geom. 7, 279-342 (1972; Zbl 0268.57011)]. Let \(\Omega_ F:=\Omega_ M/F\) and let \(S:=\{z\in M| \Omega_{F,z}\quad is\quad not\quad a\quad free\quad O_ z-module\}\) be the singular set of F. Now let Z be a connected component of S and let assume that Z is compact, Baum and Bott defined the residues of Chern classes of F arising from Z and conjectured the rationality of these residue classes. In this paper the author considers the special case where F is a locally free O-module of rank q. Such an F is called a foliation of complete intersection type for which the rationality conjecture is settled down affirmatively. As was remarked by the author, in this case, one has a ''decomposition'' \[ 0\to \Theta_ M/F^ a\to \underline{Hom}_ O(F,O)\to \underline{Ext}^ 1_ O(\Omega_ F,O)\to 0 \] where \(\Theta_ M\) denotes the tangent sheaf of M and \(F^ a:=\{\theta \in \Theta_ M| \omega (\theta)=0\quad \forall \omega \in F\}.\)
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 32A27
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 32C35
zbMATH DE Number: 3931387
zbMATH Keywords: residues of complex analytic foliation singularities
zbMATH Keywords: residues of Chern classes
zbMATH Keywords: foliation of complete intersection type
zbMATH Keywords: rationality conjecture
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03610037
OpenAlex ID: W1970694161
DOI: 10.2969/JMSJ/03610037
description: scientific article; zbMATH DE number 3931387