Residues of holomorphic vector fields relative to singular invariant subvarieties
Let \(V\) be a locally complete intersection subspace of a complex manifold \(W\). Assume that the normal bundle \(N\), originally defined only on \(V\), extends to a smooth vector bundle defined in a neighbourhood of \(V\) in \(W\). Let \(F\) be a holomorphic foliation (with possible singularities) which leaves \(V\) invariant. The main results of the paper deal with explicit computations of the residues of the characteristic forms of the triple \((F, V,N)\). The proofs use the Chern-Weil formalism and a detailed local analysis of the differential geometric data. Prior related results of Baum and Bott are thus extended to subspaces with singularities. [See \textit{P. Baum} and \textit{R. Bott}, `On the zeroes of holomorphic vector fields', Essays Topol. Relat. Top. 29-47 (1970; Zbl 0193.52201) and J. Differ. Geom. 7, 279-342 (1972; Zbl 0268.57011)].
scientific article; zbMATH DE number 844217