Exceptional singularities of codimension one holomorphic foliations (Q555261)

The paper considers co-dimension one holomorphic foliations \({\mathcal F}\) of complex manifolds of dimension larger than \(2\). The main result of the paper states that if \(Z\) is a connected component of the co-dimension \(2\) part of the singular set of \({\mathcal F}\), \(\text{Sing\,}({\mathcal F})\), and \(N_{\mathcal F}\) is nef, \(N_{\mathcal F}\) being the normal sheaf of the foliation, then the Baum-Bott residue of each irreducible component of \(Z\) vanishes and \(N_{\mathcal F|Z}\) is flat. This result is proved in Section 2. The paper contains another two sections where two consequences of that main result are shown. The first one asserts that if Pic\((X)={\mathbb Z}\), then every leaf of \({\mathcal F}\) accumulates to Sing\(({\mathcal F})\). The second one considers the analytic space \(X_0\) of dimension three defined by the equation \(z_1 z_2 = z_3 z_4\) in \({\mathbb C}^4\) and states that if \({\mathcal F}\) is a nonsingular foliation as above on \(V \setminus \{0\}\), where \(V \subset X_0\) is a neighborhood of \(0\), then, up to restricting \(V\), either \({\mathcal F}\) has a holomorphic first integral or each leaf of \({\mathcal F}\) is a proper analytic subset of \(V\) passing through the origin.