Lehmann-Suwa residues of codimension one holomorphic foliations and applications (Q2657837)

The authors essentially consider residues of singular holomorphic foliations. Firstly they use the transversal disc method to define the variational index of a foliation using integration currents along irreducible complex subvarieties. In this sense, they prove that the sum of the variational index over all irreducible components of the singular set of \(\mathcal{F}\) in \(V\) is equal to the first Chern class of the normal bundle of the foliation (Theorem 3.1). Secondly, the authors consider \(\mathcal{F}\) as the germ at \(0 \in \mathbb{C}^{n}\) of a simple almost Liouvillian foliation of codimension one and \(V\) as a divisor of poles. So it is shown that the residue of \(\mathcal{F}\) at \(Z\), an irreducible component of the singular set of \(\mathcal{F}\) of codimension 2, can be written as a sum (over the irreducible components of \(V\) that contain \(Z\)) in terms of Lehmann-Suwa residues [\textit{D. Lehmann} and \textit{T. Suwa}, Int. J. Math. 10, No. 3, 367--384 (1999; Zbl 1039.32041)]. This represents an effective way to compute residues: \[BB(\mathcal{F}, Z) = \sum_{j=1}^{k} \mathrm{Res}(\gamma_{0},V_{j})\mathrm{Var}(\mathcal{F},V_{j},Z).\] As an application of above results, sufficient conditions are given to \(\mathcal{F}\), a singular codimension one holomorphic foliation, to have dicritical singularities on a tangent irreducible singular Levi-Flat hypersurface \(M\).