Codimension two singularities of a vector field (Q5953594)

The paper deals with one-dimensional holomorphic foliations with singularities of complex codimension two on complex manifolds. It is known that an isolated singularity of a foliation on a complex surface of dimension two always has a complex separatrix, i.e., a holomorphic curve (that may be singular) passing through it and tangent to the foliation (by results of the paper [\textit{C. Camacho} and \textit{P. Sad}, Ann. Math. (2) 115, No. 2, 579-595 (1982; Zbl 0503.32007)]). The paper under review extends this result to higher dimensions. A brief statement of the main result (Theorem 2.5) is the following. Let \(D\) be a foliation as above (in arbitrary dimension), \(Z\) be its singular set (then \(\operatorname {Codim} Z=2\)). Then for almost each point \(z\in Z\) there exists a germ of a \(D\)-invariant complex analytic hypersurface passing through \(z\) and containing a germ of the singular set \(Z\). For the proof of this Theorem the author states and proves a generalization (Theorem 1.1) of Seidenberg's theorem on reduction of singularities of two-dimensional vector fields [\textit{A. Seidenberg}, Am. J. Math. 90, 248-269 (1968; Zbl 0159.33303)]. Afterwards the author states and proves a global analogue (Theorem 2.6) of the previous Theorem 2.5, which gives sufficient conditions for the existence of a ``global'' \(D\)-invariant analytic hypersurface and for existence (for almost every point \(z\in Z\)) of a germ of analytic solution of \(D\) passing through \(z\) that is not contained in the singular set \(Z\). These sufficient conditions are expressed in terms of the blowing up tree of \(D\) along \(Z\). The author also gives sufficient conditions for the existence of a formal \(D\)-invariant hypersurface containing \(Z\) (Theorem 2.2). At the end of the paper the author presents some ``counterexamples'' to Theorems 2.2, 2.6, when the previously mentioned sufficient conditions are not satisfied.