Residues of complex analytic foliation singularities
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 \(F^ a:=\{\theta \in \Theta_ M| \omega (\theta)=0\quad \forall \omega \in F\}.\)
residues of complex analytic foliation singularities
scientific article; zbMATH DE number 3931387