First order extensions of holomorphic foliations (Q1772577)

Let \(S\) be a subvariety of a complex manifold \(M\). Let \(\Theta_S\) be the holomorphic tangent sheaf of \(S\), and let \({\mathcal E}\subset\Theta_S\) be a coherent subsheaf. For a holomorphic foliation \(\mathcal F\) of \(S\), the concept of a first order of tangency extension with respect to \(\mathcal E\) is introduced as a family of local extensions of \(\mathcal F\) in \(M\) so that two different extensions of the same element coincide up to order two in the normal directions to \(\mathcal E\). Now the author assumes that \(\mathcal E\) and \(\mathcal F\) are locally free, and denotes by \(E\) and \(F\) the associated bundles. The main result asserts that, if \(E\) is compatible with \(F\) and \(\mathcal F\) has a first order tangency extension with respect to \(\mathcal E\), then there is a natural holomorphic action of \(F\) on \(TM| _S/E\). This yields localization of characteristic classes (residue theorems) by using the machinery of \textit{D. Lehmann} and \textit{T. Suwa} [J.~Differ.~Geom. 42, 165--192 (1995; Zbl 0844.32007)]. The compatibility of \(E\) with \(F\) is satisfied if \(E\) is involutive and \(F\subset E\). This result generalizes the classical Camacho-Sad index theorem [\textit{C. Camacho} and \textit{P. Sad}, Ann. Math. 115, 579--595 (1982; Zbl 0503.32007)], as well as new versions of that theorem.