Desingularization of non-dicritical holomorphic foliations and existence of separatrices

Reduction of the singularities of codimension one singular foliations in dimension three

“…In [7], [9] we have proved a similar result, under the additional assumption of nondicriticalness. In this paper we treat in a unified way both dicritical and nondicritical foliations.…”

“…For n ≥ 3, the same statement is true (see [9], [11]) under the additional assumption of nondicriticalness, to be explained below. The proofs of these results depend strongly on statements of reduction of singularities of foliations: the two-dimensional case [26], the nondicritical three-dimensional case [7], [9] and a generic equireduction in any ambient dimension [11].…”

“…For instance, being dicritical does not mean to have infinitely many integral hypersurfaces: the foliation produces by transversality a codimension one foliation on a dicritical component, but the leaves are not necessarily closed and hence we do not have an integral hypersurface in the ambient space. In fact, there are dicritical singular foliations without integral hypersurfaces [9]. For the real case, in the paper [10] we have shown that the appearance of such dicritical components is important in order to understand transcendence properties of the leaves of the foliation.…”

Elimination of resonances in codimension one foliations

“…In ambient dimension n = 3, the problem has been first solved in the non-dicritical case ( [2], [4]) and the result has been used to prove the existence of germs of invariant surfaces in this case. In the general case, it has been solved in [3].…”

“…In [4] and [3] the authors have identified two kinds of points for codimension one singular foliations in higher dimensional ambient spaces. They are the pre-simple points and the simple ones.…”