Transversely product singularities of foliations in projective spaces

We prove that a transversely product component of the singular set of a holomorphic foliation on $\mathbb P^n$ is necessarily a Kupka component.


Introduction
Let U be an open set of a complex manifold M and let k ∈ N. Let η be a holomorphic k-form on U and let Sing η : = {p ∈ U : η(p) = 0} denote the singular set of η.We say that η is integrable if each point p ∈ U \ Sing η has a neighborhood V supporting holomorphic 1-forms η 1 , . . ., η k with η| V = η 1 ∧ • • • ∧ η k , such that dη j ∧ η = 0 for each j = 1, . . ., k.In this case the distribution defines a holomorphic foliation of codimension k on U \ Sing η.A singular holomorphic foliation F of codimension k on M can be defined by an open covering (U j ) j∈J of M and a collection of integrable k-forms η j ∈ Ω k (U j ) such that η i = g ij η j for some g ij ∈ O * (U i ∩ U j ) whenever U i ∩ U j = ∅.The singular set Sing F is the proper analytic subset of M given by the union of the sets Sing η j .From now on we only consider foliations F such that Sing F has no component of codimension one.
Given a singular holomorphic foliation F of codimension k on M as above, the Kupka singular set of F , denoted by K(F ), is the union of the sets This set does not depend on the collection (η j ) of k-forms used to define F .It is well known that, given p ∈ K(F ), the germ of F at p is holomorphically equivalent to the product of a one-dimensional foliation with an isolated singularity by a regular foliation of dimension (dim F − 1).More precisely, if dim M = k + m+ 1, there exist a holomorphic vector field with a unique singularity at the origin, a neighborhood V of p in M and a biholomorphism ψ : the foliation F X is also defined by the k-form ω = i X µ and the Kupka condition dω(0) = 0 is equivalent to the inequality div X(0) = 0.
Following [7], we say that F is a transversely product at p ∈ Sing F if as above there exist a holomorphic vector field X and a biholomorphism ψ : V → D k+1 × D m conjugating F with F X , except that it is not assumed that div X(0) = 0. We say that Γ is a local transversely product component of Sing F if Γ is a compact irreducible component of Sing F and F is a transversely product at each p ∈ Γ.In particular, if Γ ⊂ K(F ) we say that Γ is a Kupka component -for more information about Kupka singularities and Kupka components we refer the reader to [8,6,1,2,3,4,5].If Γ is a transversely product component of Sing F , we can cover Γ by finitely many normal coordinates like ψ, with the same vector field X: that is, there exist a holomorphic vector field X on D k+1 with a unique singularity at the origin and a covering of Γ by open sets (V α ) α∈A such that each V α supports a biholomorphism ψ α : V α → D k+1 × D m that maps Γ ∩ V α onto {0} × D m and conjugates F with the foliation F X .The sets (V α ) can be chosen arbitrarily close to Γ.
In [7], the author proves that a local transversely product component of a codimension one foliation on P n is necessarily a Kupka component.The goal of the present paper is to generalize this theorem to foliations of any codimension.
Theorem 1.Let F a holomorphic foliation of dimension ≥ 2 and codimension ≥ 1 on P n .Let Γ be a transversely product component of Sing F .Then Γ is a Kupka component.
This theorem is a corollary of the following result.
Theorem 2. Let F a holomorphic foliation of dimension ≥ 2 and codimension k ≥ 1 on a complex manifold M .Suppose that F is defined by an open covering (U j ) j∈J of M and a collection of k-forms η j ∈ Ω k (U j ).Let L be the line bundle defined by the cocycle (g ij ) such that

Proof of the results
Proof of Theorem 2. Let V be a tubular neighborhood of Γ.Then the map is an isomorphism and so it suffices to prove that c 1 (L| Γ ) = 0. Let dim M = k + m + 1.As explained in the introduction, there exist a holomorphic vector field X on D k+1 with a unique singularity at the origin and a covering of Γ by open sets (V α ) α∈A such that each V α is contained in V and supports a biholomorphism and conjugates F with the foliation F X generated by the commuting vector fields X, ∂ y1 , . . ., ∂ ym .Notice that div(X)(0 We can assume that the k-forms ψ * α (ω) belong to the family of k-forms (η j ) j∈J defining F .Therefore the cocycle (θ αβ ) define the line bundle L restricted to some neighborhood of Γ.Thus, in order to prove that c 1 (L| Γ ) = 0 it is enough to show that each we have that ψ * (ω) = θω, which means that ψ preserves the foliation F X .It suffices to prove that the derivatives θ y1 (p), . . ., θ ym (p) vanish if p ∈ {0} × D m .Since ∂ y1 is tangent to F X , then the vector field ψ * (∂ y1 ) is tangent to F X and so we can express where λ, λ 1 , . . ., λ m are holomorphic.Then where the last equality follows from the identity ω = i X µ.Thus, since In the same way we prove that θ y2 (p) = • • • = θ ym (p) = 0 if p ∈ {0} × D m , which finishes the proof.
Proof of Theorem 1. Suppose that Γ is not a Kupka component.Let L be the line bundle associated to F as in the statement of Theorem 2. We notice that c 1 (L) = 0, otherwise F will be defined by a global k-form on P n , which is impossible.Then, if we take an algebraic curve C ⊂ Γ, we have c 1 (L) • C = 0. Therefore, if Ω is a 2-form on P n in the class c 1 (L) and V is a tubular neighborhood of Γ, which contradicts Theorem 2.