Local transversely product singularities

In the main result of this paper we prove that a codimension one foliation of $\mathbb{P}^n$, which is locally a product near every point of some codimension two component of the singular set, has a Kupka component. In particular, we obtain a generalization of a known result of Calvo Andrade and Brunella about foliations with a Kupka component.

In particular, the germ of the singular set of F at p is smooth of dimension k − 1: Sing(F p ) = ϕ −1 (0). Definition 1. We say that Γ is a local transversely product component (briefly l.t.p component) of Sing(F ) if Γ is an irreducible component of Sing(F ) and F is a transversely product at all points of Γ. b. There exists a singular one dimensional foliation G, on a polydisc V of C n−m , with an isolated singularity at 0 ∈ V , such that for any p ∈ Γ there exists a local chart (U, z) around p ∈ U satisfying the following conditions: b.1. z = (x, y) : U → C n−m × C m with x(U ) = V . b.2. F | U = x * (G). In the chart z = (x, y) the submersion of the definition is ϕ = x : U → V and the leaves of the non-singular foliation are the levels x −1 (a), a ∈ V . Moreover, The germ of G at 0 ∈ Q is called the normal type of F along Γ. Remark that, if T is a germ at p ∈ Γ of n − m manifold transverse to Γ then the restricted foliation F | T is holomorphically equivalent to the normal type of F along Γ.
Moreover, since G is one dimensional we can assume that it is defined by a holomorphic vector field X = n−m j=1 A j (x) ∂ ∂xj , or by the (n − m − 1)-form where in (1) dx j means omission of dx j in the product. The form η, considered as a form on U in the coordinates (x, y), defines F | U . Let us see some examples: Recall that a point p ∈ M is a Kupka singularity of the foliation F if p ∈ Sing(F ) and F is represented in a neighborhood of p by an integrable (n − k)form η such that dη(p) = 0. The form dη defines a k + 1 distribution D = ker(dη) in a neighborhood of p, where D(q) = ker(dη(q)) := {v ∈ T q M | i v (dη(q)) = 0} .
Definition 2. We say that K is a Kupka component of a foliation F (of dimension k ≥ 2) if K is a l.t.p. component of Sing(F ) and the normal type of F along K is of Kupka type.
Example 2. Let P and Q be homogeneous polynomials on C n+1 , n ≥ 3, where def (P ) = p and deg(Q) = q. The levels of the rational function f = P q Q p define a singular foliation of P n , that will be denoted by F (P, Q). We say that P and Q are transverse if the set {z ∈ C n+1 | P (z) = Q(z) = 0 and dP (z) ∧ dQ(z) = 0} is either {0}, or empty (if p = q = 1). If P and Q are transverse then the subset Γ of P n defined in homogeneous coordinates by (P = Q = 0) is a Kupka component on F (P, Q). The normal type of F (P, Q) at the points of Γ is given by the linear vector field X = p. x ∂ ∂x + q. y ∂ ∂y . In fact, the following result is known (cf. [1], [3], [4] and [9]): Theorem 1.1. Let F be a holomorphic foliation of codimension one on P n , n ≥ 3. If F has a Kupka component then F = F (P, Q), where P and Q are transverse polynomials.
In this paper we generalize theorem 1.1: Theorem 1. Let F be a holomorphic foliation of codimension one on P n , n ≥ 3.
Assume that F has a l.t.p. component Γ. Then Γ is a Kupka component of F . In particular, F is like in example 2.
Let us state some consequences of theorem 1. Corollary 1. Let F be a codimension one holomorphic foliation on P n , n ≥ 4. Assume that there is a linear embedding i : P 3 → P n such that i * (F ) has a l.t.p component. Then F has a rational first integral that can be written in homogeneous coordinates as P q /Q p , where P and Q are homogeneous polynomials on C n+1 with deg(P ) = p and deg(Q) = q.
The proof of corollary 1 is based in the fact that if there exists a linear embedding i : P 3 → P n such that i * (F ) has a first integral then F has also a first integral (see [10]).
Corollary 2. Let F be a codimension one foliation on P n , n ≥ 3. Assume that all components of its singular set are l.t.p. Then F has degree zero: the first integral of corollary 1 is of the form L 2 /L 1 , where L 1 and L 2 are linear.
Remark 1.2. Corollary 3 was proved in [11] in the case n = 4. We would like to observe that the assertion is not true in the case of distributions of C 4 . The following example, due to Krishanu and Nagaraj [8]: define a 2-form θ on C 4 by The proof of theorem 1 will be done in section 2. Since this proof is technical, in section 2.1 we give an idea of the proof by stating the main objects and results that will be used. In sections 2.2 and 2.4 we will prove the main auxiliary results used in the proof and stated in section 2.1. Section 3 is dedicated to the proof of corollaries 2 and 3.
Aknowledgement. I would like to thank J. Vitório Pereira for helpfull conversations that suggested me a simplification of the proof of lemma 2.3.

2.
Proof of theorem 1 2.1. Preliminaries and idea of the proof. Let F be a codimension one foliation on P n , n ≥ 3, with a l.t.p. component Γ ⊂ Sing(F ). The definition implies that cod C (Γ) = 2, so that the transversal type of F at the poins of Γ is a germ of singular foliation at (C 2 , 0) with an isolated singularity at 0 ∈ C 2 (see remark 1.1 and example 1). We can assume that this transversal type is given by germ at 0 ∈ C 2 of vector field X = X 1 (x, y) ∂ ∂x + X 2 (x, y) ∂ ∂y , where X 1 , X 2 ∈ O 2 and X 1 (0, 0) = X 2 (0, 0) = 0. Recall that Γ is a Kupka component if, and only if, we have T r(DX(0)) = 0, where is the trace of the linear part DX(0) of X at 0 ∈ C 2 . In this case, as we have pointed out before, F is like in example 2 (see theorem 1.1). Another useful ingredient is the normal Baum-Bott index of the component Γ, that we will denote as BB(F , Γ). Since Γ is a l.t.p. component of Sing(F ) then BB(F , Γ) coincides with the Baum-Bott of X at the singularity 0 of X, denoted by BB(X, 0) (see [12]) (for the definition of BB(X, 0) see [2]). In lemma 3.4 of §3.2 of [12] it is proven that if BB(F , Γ) = 0 and DX(0) ≡ 0 then Γ is a Kupka component and we are done.
One of the tools used in the proof of lemma 3.4 of [12] is the existence of a smooth analytic separatrix along Γ. Below we define the concept of separatrix in a way that will be used in the proof of theorem 1.
Definition 3. Let F be a holomorphic foliation of dimension k on a n dimensional compact complex manifold, 2 ≤ k < n, and Γ be l.
Remark 2.1. Let Σ be a separatrix of F of dimension ℓ along Γ, as in definition 3. Fix p ∈ Γ and (x, y) : U → C n−k+1 × C k−1 , a local coordinate system around p as in remark 1.1. It follows from the definition that x −1 (x(Σ ∩ U )) = Σ ∩ U . Let T is a germ at p of a n − k + 1 dimensional manifold transverse to Γ. As we have observed before, F | T is equivalent to the normal type of F along Γ. In particular, the intersection Σ ∩ T is invariant by F | T .
In the case of theorem 1, where F has codimension one, then dim(T ) = 2 and the normal type is a germ G of one dimensional foliation on (C 2 , 0). In this case Σ ∩ T is a finite number of analytic separatrixes of G as considered in [6]. The next result will be used in proof of theorem 1. If the normal type of F along Γ is not equivalent to the radial foliation of (C 2 , 0) then F admits an irreducible separatrix Σ along Γ with dim(Σ) = n − 1.
Recall that the radial foliation of C 2 is defined by the form x dy − y dx and their leaves are the straight lines through 0. Lemma 2.1 will be proved in section 2.2.
From now on, in this section, we will assume that F is a codimension one holomorphic foliation on the compact manifold M , dim C (M ) ≥ 3, with a l.t.p. component Γ and with a separatrix Σ along Γ, dim(Σ) = n − 1. Next we will introduce the normal bundle of Σ along Γ.
Of course g = (g αβ ) Uα∩U β =∅ is a multiplicative cocycle. We define the normal bundle of Σ along Γ as the line bundle on P icc(Γ) induced on a tubular neighborhood U ⊂ α U α by the cocycle g = (g αβ ) Uα∩U β =∅ .
It will be denoted by N Σ . Let C 1 (N Σ ) be the first Chern class of N Σ , considered as an element of As we have seen in remark 1.1, the normal type of F along Γ can be represented by a germ at 0 ∈ C 2 of holomorphic vector field X = A 1 (x, y) ∂ ∂x + A 2 (x, y) ∂ ∂y with an isolated at 0. When we intersect Σ with a germ of transversal section T ≃ (C 2 , 0) we obtain a separatrix of X, say γ : On the other hand, we have the following: As a consequence of lemma 2.3 we get the following: In particular, theorem 1.1 will imply theorem 1. Lemma 2.1 will be proved in the next section.

2.2.
Proof of lemma 2.1. Let F be a holomorphic codimension one foliation on a compact complex manifold M with dim(M ) ≥ 3. Assume that F has a l.t.p. component Γ with normal type G, where G is a germ of foliation on (C 2 , 0) with an isolated singularity at 0 ∈ C 2 . As before, we will assume that G is the foliation defined by a germ at (C 2 , 0) of vector field X = X 1 ∂ ∂x + X 2 ∂ ∂y with an isolated singularity at the origin of C 2 . The germ of foliation G can be defined also by the 1-form ω = i X (dx ∧ dy) = X 1 dy − X 2 dx , so that, dω(0) = T r(DX(0)) dx ∧ dy. We can assume that ω has a representative, denoted byω, defined in the polydisc Q = D 2 with an isolated singularity at 0 ∈ D 2 .
By the definition of l.t.p. component, we can find a covering U = (U α ) α∈A of Λ by open sets biholomorphic to polydiscs, a collection of local charts ((z α , U α )) α∈A and a multiplicative cocicle (k αβ ) Uα∩U β =∅ with the following properties: We will assume that U satisfies the following: Remark 2.2. Given α, β ∈ A such that Γ ∩ U α ∩ U β = ∅ we can construct a germ f αβ ∈ Dif f (C 2 , 0) as follows: fix p ∈ Γ ∩ U α ∩ U β and a germ of plane From now on, we fix a collection of germs (f αβ ) Uα∩U β =∅ as above.
Notation. We will say that the separatrix γ of G generates the separatrix Σ of F .
Proof. Assume that X has a separatrix γ such that In fact, let (x α , y α ), (x β , y β ) and T be as before. Then This, of course, implies the assertion. In particular, the local separatrices Σ a glue together forming a global separatrix Σ along Γ such that We leave the converse to the reader.
Definition 4. Let G be a germ of foliation at (C 2 , 0) with an isolated singularity at 0. We say that a separatrix Lemma 2.5. Let G be a germ of foliation at (C 2 , 0) with an isolated singularity at 0 which is not equivalent to the radial foliation. Then G has a distinguished separatrix.
Proof. In the proof we use Seidenberg's resolution theorem [17]. Let S be a smooth complex surface and G be a foliation by curves on S. Given p ∈ Sing(G) ⊂ S we denote Dif f (S, p) the set of germs at p ∈ S of biholomorphisms f : (S, p) → S with a fixed point at p. Assume that the germ of G at p is defined by a germ of holomorphic vector field X with an isolated singularity at p. We use also the notations p). First of all, we observe that there are two possibilities for the foliation G: I. G has finitely many irreducible separatrices through p. This case is trivial and the details are left to the reader. II. G has infinitely many irreducible separatrices through p. Let us prove lemma 2.5 in this case. We will consider a blowing-up process used to resolve the foliation G (see [6]). The first case, is when G has a simple singularity at p and no blowing-ups are needed in the process. Let λ 1 and λ 2 be the eigenvalues of DX(p). The singularity is simple if: (a). λ 1 . λ 2 = 0 and λ2 λ1 / ∈ Q + . (b). λ 1 = 0 and λ 2 = 0 (or vice-versa). In this case, p is a saddle-node. In both cases G has one or two separatrices through p and so lemma 2.5 is true.
When the singularity is not simple, Seidenberg's theorem says that after a finite process of blowing-ups Π : (S, E) → (S, p) then all the singularities of the strict transform Π * (G) in the exceptional divisor E are simple. The blowing-up process Π can be considered as a composition of pontual blowing-ups (3) where in the j th step Π j : (S j , E j ) → (S j−1 , E j−1 ), j ≥ 2, we blow-up in a point p j−1 ∈ E j−1 . The exceptional divisor obtained in this step will be denoted as P 1 ≃Ẽ j ⊂ E j , so that Π j (Ẽ j ) = p j−1 . We use also the notationΠ j := Π 1 • ... • Π j . We will denote alsoG j :=Π * (G). The point p j−1 ∈ E j−1 is chosen between the non simple singularities ofG j−1 on E j−1 . Seidenberg's theorem can be stated as follows Theorem 2.1. It is possible to choose a blowing-up process as a above in such a way that all singularities of the strict transformG k =Π * k (G) are simple. Remark 2.4. There are two possibilities in each step Π j : (S j ,Ẽ j ) → (S j−1 , p j−1 ). We assume that p j−1 is a non simple singularity ofG j−1 . Let X j−1 be a germ at p j−1 of holomorphic vector field that represents the germ ofG j−1 at p j−1 . Let X ν = P ν (x, y) ∂ ∂x + Q ν (x, y) ∂ ∂y be the first non-zero jet of X j−1 at p j−1 , where P ν and Q ν are homogeneous polynomials of degree ν ≥ 1. Set F ν+1 (x, y) = x. Q ν (x, y) − y P ν (x, y).
(i). If F ν+1 ≡ 0 then F ν+1 is homogeneous of degree ν + 1 and the blowing-up is called non-dicritical. The divisorẼ j is invariant for the foliationG j and the singularities ofG j onẼ j are the directions correspondent to the directions defined by F ν+1 (x, y) = 0. (ii). If F ν+1 ≡ 0 then X ν = F ν−1 (x, y) R, where R = x ∂ ∂x + y ∂ ∂y is the radial vector field in C 2 and F ν−1 is homogeneous of degree ν − 1. In this case, the blowing-up is called dicritical. The divisorẼ j is non-invariant forG j and it is transverse toẼ j outside the set V j ⊂Ẽ j corresponding to the directions defined by the equation F ν−1 (x, y) = 0.
If ν = 1 thenG j−1 is equivalent to the radial foliation at p j−1 . We will say that p j−1 is a radial singularity ofG j−1 . If p j−1 is not radial forG j−1 then V j = ∅ and we can divide it into two disjoint subsets . We call τ j the set of tangencies ofG j withẼ j . Remark also that Sep(G) is finite if, and only if, all blowing-ups in the process are non-dicritical.
Since in the blowing-up process, in each step, 1 ≤ j ≤ k, we blow-up in some nonsimple singularity ofG j−1 , ifẼ j is dicritical, at the end the tangencies τ j "survive", in the sense that there exists a set τ ⊂ E k such that for any 1 ≤ j < k such that For each 1 ≤ j ≤ k denote by Dif f (S j , E j ) the set of germs of biholomorphisms f : (S j , E j ) → (S j , E j ).

Definition 5.
We say that f ∈ Dif f (S, p) can be lifted to Dif f (S j , E j ) if there exists a germ of biholomorphismf j ∈ Dif f (S j , E j ) such that the diagram below commutes: Remark 2.5. Observe that, if the liftf j of f exists then it is unique. When j = 1 (just one blowing-up) the lifting exists for any f ∈ Dif f (S, p), but if j ≥ 2 then there are germs f ∈ Dif f (S, p) that can not be lifted to Dif f (S j , E j ). However, we have the following: Claim 2.1. The blowing-up process can be done in such a way that any f ∈ Dif f G (S, p) can be lifted to the last step in an uniquef =f k ∈ Dif f (S k , E k ). Moreover,f preservesG k in the sense thatf * (G k ) =G k .
Proof. We say that the j th step of the blowing-up process is admissible if any f ∈ Dif f G (S, p) has a liftingf j ∈ Dif f (S j , E j ). We will obtain by induction a blowing-up process, as in (3), for which there are steps 1 = ℓ 1 < ℓ 2 < ... < ℓ r = k such that the ℓ th j step is admissible, for any 1 ≤ j ≤ r, andΠ k : (S k , E k ) → (S, p) is a resolution of the folition G.
First of all, the first step is admissible, because any f ∈ Dif f (S, p) admits a liftingf 1 ∈ Dif f (S 1 , E 1 ).
Assume that we have found some process for wich the ℓ := ℓ s step is admissible, ℓ ≥ 1, so that any f ∈ Dif f G (S, p) admits a liftingf =f ℓ ∈ Dif f (S ℓ , E ℓ ) satisfying f * (G ℓ ) =G ℓ . Given f ∈ Dif f G (S, p), with liftingf , and q ∈ E ℓ thenf is an equivalence between the two germs ofG ℓ at q and atf (q). In particular,f preserves the set of non simple singularities ofG ℓ .
IfG ℓ is not a resolution of G then it has at least one non simple singularity q 1 . Let Sat(q 1 ) = {f (q 1 ) | f ∈ Dif f G (S, p)} = {q 1 , ..., q m }. We then blow-up once at all points q j ∈ Sat(q 1 ), passing from the ℓ = ℓ s step to the ℓ s+1 := ℓ s + m step directly. Let E j be the divisor obtained by the blowing-up at q j .
Given f ∈ Dif f G (S, p) and its liftingf s , letf s (q j ) = q i(j) , 1 ≤ j ≤ m. Then we can obtain a liftingf s+1 off s such thatf s+1 ( E j ) = E i(j) , 1 ≤ j ≤ m. By Seidenberg's theorem this process must end at some step, when the final foliatioñ G k =Π * k (G) has all singularities simple. Let us finish the proof of lemma 2.5. We will consider two cases: (1). There is q 1 ∈ Sing(G k ) ∩ E k that has some separatrixγ not contained in E k . (2). All the separatrices of the singularities ofG k are contained in E k .
In the first case, let Sat( p), and γ is a distinguished separatrix of G.
In the second case necessarily there are dicritical irreducible divisors ofG k , saỹ E 1 , ...,Ẽ m , contained in E k (by Camacho-Sad theorem). This case will be divided into two sub-cases: (2.1). The set of tangencies τ is not empty.
In case (2.1) let q o ∈ τ andγ id be the leaf ofG k through q o . Then, for any f ∈ Dif f G (S, p) we have q f :=f (q o ) ∈ τ andγ f :=f (γ id ) is the leaf ofG k through f (q o ). Since q f ∈ E k , butγ f is not contained in E k , ∀f ∈ Dif f G (S, p), the immagẽ Π k (γ f ) := γ f is an irreducible separatrix of G through p. Therefore, if we define γ as in (4) then γ is a distinguished separatrix of G.
We will divide case (2.2) into two subcases: In case (2.2.1), let E be a connected component of E k \ jẼ j . Let r i=1 D i be decomposition of E into irreducible components, D i ≃ P 1 . Note that: (i). The graph formed by the divisors D i is a tree.
In this case, Sing(G k ) ∩ E = ∅ and contains a singularity q with a separatrixγ not contained in E. This is a consequence of Sebastiani's version of Camacho-Sad theorem (see [16]). In fact,γ is not contained in E k , for otherwise it would be contained in some irreducible divisor D of E k not contained in E, and D is non dicritical, which contradicts (iii). Therefore, we reduce the problem to case (1).
In case (2.2.2) all irreducible divisorsẼ j of E k are dicritical. We can assume that Sing(G k ) = ∅. In fact, if q o ∈ Sing(G k ) then q o is simple and any of their separatrices cannot be contained in E k , for otherwise some of the divisorsẼ j would be non-dicritical. Therefore, we are again in case (1). In particular, we can assume that all divisorsẼ j are radial, in the sense that for any q ∈Ẽ j the leaf ofG k through q is transverse toẼ j . Moreover, m ≥ 2 because otherwise p would be a radial singularity of G. In particular, we can assume thatẼ 1 ∩Ẽ 2 = {q o } = ∅. Let γ id be the leaf ofG k through q o . Note that, for any f ∈ Dif f G (S, p) then then A is finite. Therefore, we can construct a distinguished separatrix γ of G as in (4).
Finally, note that lemma 2.1 is a consequence of lemmae 2.4 and 2.5.

Proof of lemma 2.2. Since U is a tubular neighborhood of
Recall that the germ of F at any q ∈ Γ is equivalent to a product of a singular foliation by curves on (C 2 , 0) by a regular foliation of dimension n − 2. This implies that there exist a local coordinate system around q, z = (x, y) : U → C 2 ×C n−2 , x = (x 1 , x 2 ), y = (y 1 , ..., y n−2 ), and a holomorphic vector field X = P (x) ∂ ∂x1 + Q(x) ∂ ∂x2 , with an isolated singularity at 0 ∈ C 2 , such that • F | U is generated by the n − 1 commuting vector fields X, Y 1 := ∂ ∂y1 ,..., Y n−2 := ∂ ∂yn−2 . Moreover, the separatrix Σ of F along Γ is induced by a separatrix γ = (f (x 1 , x 2 ) = 0) of X, such that X(f ) = h. f , where we have assumed h(0) = 0.
Proof. Let U α ∩ U β ∩ Σ = ∅. We assert that there exists a (n − 1) × (n − 1) as in (5). But since cod(Σ) = 2 the entries of A αβ can be extended to U α ∩ U β by Hartog's theorem. Now, from (c) we get and from (c) and (5) Y This finishes the proof of lemma 2.2.

2.4.
Proof of lemma 2.3. The case in which Σ is smooth was proved in [12].
Here we give a more general proof (suggested by J. V. Pereira). Let us consider first the case n = 3: M = P 3 . In this case, Γ is a compact algebraic curve so that H 2 Dr (Γ) ≃ R and the map is an isomorphism. In fact, we will prove that We will see that Γ C 1 (N Σ ) represents the intersection number of a small deformation Γ t of Γ with Σ. Let X (P 3 ) be the vector space of holomorphic vector fields on P 3 : dim(X (P 3 )) = 15. Given Z ∈ X (P 3 ) we will denote by (t, q) ∈ C × P 3 → Z t (q) ∈ P 3 its flow and Γ t := Z t (Γ). Let U be a tubular neighborhood U of Γ with U ⊂ α U α . Remark 2.6. There exist Z ∈ X (P 3 ) and ǫ > 0 with the following properties: We leave the proof of remark 2.6 for the reader. Let us finish the proof of lemma 2.3 in the case of P 3 .
The idea is to prove that, if t ∈ D * ǫ \ B then Γ C 1 (N Σ ) = #(Γ t ∩ Σ), the intersection number of Γ t with Σ. By (d) of remark 2.6 #(Γ t ∩ Σ) > 0 and so First of all, note that On the other hand, Z −t (Σ) can be defined in the covering U t := (Z −t (U α )) α∈A by the divisor (f α • Z t ) α∈A , with associated cocycle g t : This divisor can be interpreted as a holomorphic section of the line bundle induced by g t | Γ on P icc(Γ). In particular, if C 1 (g t | Γ ) is its first Chern class then its degree is done by Since the map This finishes the proof of lemma 2.3 in the case of P 3 .
The case of P n , n ≥ 4, can be reduced to the previous by taking sections by generic 3-planes linearly embedded in P n . We leave the details to the reader.
2.5. Proof of corollary 2.1. Recall that X(f ) = h. f , where X represents the normal type G of F along Γ and f ∈ O 2 is reduced. By lemma 2.3 we have h(0) = 0. Let f µ and X ν be the first non-zero jets of f and X at 0 ∈ C 2 , respectively. Then and X ν = X 1 is not nilpotent; has at least one non-zero eigenvalue. On the other hand, we have seen that Γ is a Kupka component of F if, and only if, tr(X 1 ) = 0. If tr(X 1 ) = 0 and X 1 has a non-zero eigenvalue, then we can assume that X 1 = λ x 1 ∂ ∂x1 − x 2 ∂ ∂x2 , λ = 0. In this case, X has exactly two separatrices through 0 ∈ C 2 which are smooth and tangent to x 1 = 0 and x 2 = 0. We can assume that these separatrices have equations x 2 ) = 0. Therefore, we must have tr(X 1 ) = 0 and Γ is a Kupka component of F .

Corollaries 2 and 3
3.1. Proof of corollary 2. A codimension one foliation G on P n of degree zero has a rational first integral of degree one. It is defined in some coordinate system (x 1 , ..., x n+1 ) ∈ C n+1 by a the form ω = x 1 dx 2 − x 2 dx 1 . In particular, Π −1 (Sing(G)) = (x 1 = x 2 = 0), which is a l.t.p component.
Conversely, let F be a codimension one foliation on P n , n ≥ 3. It is known that F has at least one irreducible component of codimension two [13]. Assume that all components of Sing(F ) are l.t.p. Let Ω be a 1-form on C n+1 that represents F in homogeneous coordinates: F Ω = Π * (F ). Then (a). i R Ω = 0, where R is the radial vector field on C n+1 . (b). The coefficients of Ω are homogeneous of degree d + 1, where d = deg(F ).
Proof. Let ω be a holomorphic 1-form that represents F in a neighborhood of q. The hypothesis and theorem 1 imply that dω(p) = 0.
On the other hand, Π * (ω) represents F Ω in a neighborhood, say U , of p. It follows that Π * (ω) = ϕ. Ω on U , where ϕ ∈ O * (U ). Therefore, Since Π is a submersion, it follows that dΩ p = 0. Therefore, the coefficients of Ω must be of degree one and F has degre zero, as asserted in corollary 2.
3.2. Proof of corollary 3. The idea is to use corollary 2. Assume that there exists an integrable 2-form η on C n , n ≥ 4, with homogeneous coefficients of degree d ≥ 1 and such that Sing(η) = {0}. Denote by F η the holomorphic codimension two foliation of C n generated by η. By assumption Sing(F η ) = {0}. Note also that the codimension two distribution of C n \ {0} tangent to F η is given by where i v denotes the interior product. The fact that ker(η) has codimension two is equivalent to (6) η ∧ η = 0 .
Let ω = i R η, where R = n j=1 z j ∂ ∂zj is the radial vector field on C n . We have two possibilities: either ω ≡ 0, or ω ≡ 0.
In the first case, η generates a codimension two foliation on P n−1 : there exists a codimension two foliation F on P n−1 such that Π * (F ) = F η , where Π : C n \ {0} → P n−1 denotes the canonical projection. However, any codimension two foliation on P n−1 , n ≥ 4, has at least one singularity: if q ∈ Sing(F ) then the line Π −1 (q) ⊂ C n is contained in the singular set of η.
In the second case ω is a 1-form on C n with homogeneous coefficients of degree d + 1. Proof. The following is equivalent to the integrability of the distribution ker(η): (I). for any p ∈ C n \{0} there exists a germ coordinate system (x, y) : (C n , p) → (C 2 , 0) × (C n−2 , 0), with x = (x 1 , x 2 ), such that η p = ϕ(x, y) dx 1 ∧ dx 2 , where η p is the germ of η at p and ϕ ∈ O * p . Since the coefficients of η are homogeneous of degree d we have L R η = (d + 2) η, where L R denotes the Lie derivative in the direction of R. From this we get Now, from (6) we get If we consider a coordinate system as in (I) we have η p = ϕ dx 1 ∧ dx 2 and dη p = dϕ ∧ dx 1 ∧ dx 2 and this implies that i R η p ∧ i R dη p = 0, as the reader can check.