On the local structure of holomorphic foliation singularities (Q5943950)

Let \(X\) be a connected complex \(n\)-dimensional manifold. A singular dimension \(k\) holomorphic foliation on \(X\) is given by assigning to a closed analytic subvariety \(S\) of \(X\) a dimension \(k\) holomorphic foliation of \(X\smallsetminus S\), i.e. a rank \(k\) holomorphic subbundle of the tangent bundle \(T(X\smallsetminus S)\) closed under the Lie bracket [\textit{P. Baum} and \textit{R. Bott}, J. Differ. Geom. 7, 279-342 (1972; Zbl 0268.57011)]. Fix integers \(n,k,s,a_i\), \(0\leqslant i\leqslant s\), \(b_i\), \(0\leqslant i\leqslant s\) such that \(s\geqslant 0\), \(n\geqslant a_0+2\), \(a_i>a_j\geqslant 0\) for \(s\geqslant j>i\geqslant 0\), \(a_i\geqslant b_i\) for every \(i\), \(b_i>b_{i+1}\geqslant 0\) for \(s>i\geqslant 0\). Under these assumptions, the author proves the existence of a dimension \(k\) singular holomorphic foliation \(F\) of a neighborhood of \(0\in \mathbb{C}^n\) with the following properties. If \(Z\) denotes the reduction of the singular set of \(F\), then \(Z\) is smooth at \(0\) and there is a chain of \(s+1\) closed smooth submanifolds \(0\in Z_s\subset Z_{s-1}\subset\cdots\subset Z_0=Z\) such that: (i) \(\dim(Z_i)=a_i\) for every \(i\); (ii) \(F\) has tangential rank \(b_i\) at each point of \(Z_i\smallsetminus Z_{i+1}\) (with the convention \(Z_{-1}:=\emptyset\)).