On the local structure of holomorphic foliation singularities

Here we prove the following result. Fix integers n,k,s, ai, 0⩽i⩽s, bi, 0⩽i⩽s, such that s⩾0, n⩾a0+2, ai>aj⩾0 for s⩾j>i⩾0, ai⩾bi for every i, bi>bi+1⩾0 for s>i⩾0. Then there exists a dimension k singular holomorphic foliation F of a neighborhood of 0∈Cn with the following properties. Let Z be 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∈Zs⊂Zs−1⊂⋯⊂Z0=Z such that: (i) )=ai for every i; tangential rank bi at each point of Zi⧹Zi+1 (with the convention Z−1:=∅).