2. Proof of Theorem 1.9

A L C I D E S L I N S - N E T O

L O C A L T R A N S V E R S E LY P R O D U C T SI N G U L A R I T IE S

F E UIL L E TA G E S À S T R U C T U R E P R O D UI T L E L O N G D U L IE U SI N G U L IE R

Abstract. — In the main result of this paper we prove that a codimension one foliation ofPⁿ, 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.

Résumé. — Nous démontrons qu’un feuilletage de codimension un dePⁿqui est localement un produit autour de tous les points d’une composante de codimension 2 de l’ensemble singulier, a une composante de Kupka. En particulier, nous obtenons une généralisation d’un résultat déjà connu de Calvo Andrade et Brunella sur les feuilletages avec une composante de Kupka.

1. Basic definitions and results

It is known that a holomorphic codimension one foliation on Pⁿ, n > 3, with a Kupka component in the singular set has a rational first integral, which in homoge- neous coordinates is of the form P^k/Q<sup>`</sup>, where P and Q are generic homogeneous polynomials with k. deg(P) = `. deg(Q). The Kupka component for this specific example is the set Π(P = Q = 0), where Π : Cⁿ⁺¹ \ {0} → Pⁿ is the canonical projection (cf. [Bru09, CA16, CA99]). The aim of this paper is to generalize this

Keywords:foliation, locally product.

2020Mathematics Subject Classification:37F75, 34M15.

DOI:https://doi.org/10.5802/ahl.78

result for codimension one foliations with alocal transversely product component in the singular set. We will define this concept in a more general situation.

Let F be a holomorphic foliation of dimension k >2 on a complex manifold M of dimensionn>k+ 1, with singular set Sing(F). We say thatF is a transversely product at a pointp∈Sing(F) if the germF_pofF atpis holomorphically equivalent to a product of a germ of singular foliation of dimension one with an isolated singularity by a regular foliation of dimension k −1. In other words, we can say that there exists a germ of submersionϕ: (M, p)→(C^n−k+1,0) and a germ of a one dimensional foliation G at 0∈C^n−k+1, with Sing(G) ={0}, such thatF_p =ϕ^∗(G).

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 1.1. — 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.

Remark 1.2. — If Γ is a l.t.p. component of Sing(F) then it follows from the definition that:

(a) Γ is smooth. Let dim_C(Γ) =m.

(b) There exists a singular one dimensional foliation G, on a polydisc V ofC^n−m, with an isolated zero at 0∈ V, such that for any p∈ Γ there exists a local chart (U, z) around p∈U satisfying the following conditions:

(b1) z = (x, y) : U →C^n−m×C^m with x(U) =V. (b2) 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, Γ∩U =x⁻¹(0).

The germ ofG at 0∈Q is called thenormal type ofF 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 holo- morphic vector field X =^Pn−m_j=1 A_j(x)_∂x^∂

j, or by the (n−m−1)-form (1.1) η=i_Xdx₁∧ · · · ∧dxn−m =

n−m

X

j=1

(−1)^j−1A_j(x)dx₁∧ · · · ∧dx^d_j∧ · · · ∧dxn−m, where in (1.1) dx^d_j means omission of dx_j in the product. The formη, considered as a form onU in the coordinates (x, y), defines F |U.

Let us see some examples:

Example 1.3. — Recall that a pointp∈M is a Kupka singularity of the foliation Fifp∈Sing(F) andF is represented in a neighborhood ofpby an integrable (n−k)- form η such that dη(p)6= 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_qM|i_v(dη(q)) = 0} .

The distribution D is integrable and defines a regular foliation of dimension k −1 in a neighborhood of p. There exists a local chart (U, z = (x, y)), where

x = (x₁, . . . , xn−k+1) : U → C^n−k+1, y: U → C^k−1 and z(p) = (0,0), such that dη = dx1∧ . . . ∧dxn−k+1. In this case, the form η can be written as η = iXdx1∧ . . . ∧dxn−k+1, where

X =

n−k+1

X

j=1

A_j(x) ∂

∂x_j defines the normal type ofF_p.

Note thatdη= ∆(X)dx₁∧ · · · ∧dxn−k+1, where

∆(X) =div(X) =

n−k+1

X

j=1

∂Aj

∂x_j , so that ∆(X)≡1.

Definition 1.4. — 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 1.5. — LetP andQbe homogeneous polynomials onCⁿ⁺¹,n >3, where deg(P) = p and deg(Q) = q. The levels of the rational function f = ^P_Q^qp define a singular foliation ofPⁿ, that will be denoted by F(P, Q). We say that P and Q are transverse if the set

nz ∈Cⁿ⁺¹

P(z) = Q(z) = 0 and dP(z)∧dQ(z) = 0^o

is either {0}, or empty (if p=q= 1). If P and Q are transverse then the subset Γ of Pⁿ 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. [Bru09, CA16, CA99, CLN94]):

Theorem 1.6. — Let F be a holomorphic foliation of codimension one on Pⁿ, n > 3. If F has a Kupka component then F = F(P, Q), where P and Q are transverse polynomials.

Example 1.7. — Example 1.5 admits the following generalization: let P₁, . . . , P_m be homogeneous polynomials onCⁿ⁺¹ with deg(P_j) =d_j, 16j 6m. Assume that n>m+ 1 >4 and thatP₁, . . . , P_m are transverse; i.e. the set

nz ∈Cⁿ⁺¹

P1(z) =· · ·=Pm(z) = 0 and dP1(z)∧ . . . ∧dPm(z) = 0^o

is either {0}, or empty (if d₁ = · · · = d_m = 1). Let (k₁, . . . , k_m) ∈ N^m be such thatgcd(k₁, . . . , k_m) = 1 and k₁. d₁ =· · ·=k_m. d_m. The levels of the rational map P: Pⁿ →P^m−1, defined by

P :=^hP₁^k1 : . . . :P_m^kmi ,

define a foliation of codimension m − 1 on Pⁿ, denoted by F(P₁, . . . , P_m). If P₁, . . . , P_m are transverse then the set Γ ⊂ Pⁿ, defined in homogeneous coordi- nates by (P₁ = · · · = P_m = 0), is a Kupka component of F(P₁, . . . , P_m). The normal type of F(P₁, . . . , P_m) at the points of Γ is given by the linear vector field S=^Pm_j=1d_jx_j∂x^∂

j .

A natural problem is the following:

Problem 1.8. — Let F be a holomorphic foliation of codimensionm−1on Pⁿ, wheren >m+1>4. Assume thatF has a Kupka componentΓ. Are there transverse homogeneous polynomials P1, . . . , Pm on Cⁿ⁺¹ such that F = F(P1, . . . , Pm) and Γis defined by (P₁ =· · ·=P_m = 0)?

Some partial results about this problem were proved (see for instance [CA09, CJCAFP14]).

In this paper we generalize Theorem 1.6:

Theorem 1.9. — Let F be a holomorphic foliation of codimension one on Pⁿ, n>3. Assume thatF has a l.t.p. component Γ. Then Γ is a Kupka component of F. In particular, F is like in Example 1.5.

Let us state some consequences of Theorem 1.9.

Corollary 1.10. — Let F be a codimension one holomorphic foliation on Pⁿ, n>4. Assume that there is a linear embeddingi: P³ →Pⁿsuch thati^∗(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 Qare homogeneous polynomials on Cⁿ⁺¹ with

deg(P) = p and deg(Q) =q.

The proof of Corollary 1.10 is based in the fact that if there exists a linear embed- ding i: P³ →Pⁿ such that i^∗(F) has a first integral then F has also a first integral (see [CLN96]).

Corollary 1.11. — LetF be a codimension one foliation onPⁿ,n>3. Assume that all components of its singular set are l.t.p. Then F has degree zero: the first integral of Corollary 1.10 is of the formL₂/L₁, where L₁ and L₂ are linear.

Corollary 1.12. — Letη be an integrable 2-form on Cⁿ, n >4, with homoge- neous coefficients of the same degree d>1. Thendim_C(sing(η))>1.

Remark 1.13. — Corollary 1.12 was proved in [CLN19] in the case n = 4. We would like to observe that the assertion is not true in the case of distributions of C⁴. The following example, due to Krishanu and Nagaraj [DN13]: define a 2-form θ on C⁴ by

θ =x²₃dx2∧dx3−x²₁dx3∧dx1+ (x₁x2+x3x4) dx1∧dx2+

hx²₄dx₁+x²₂dx₂+ (x₁x₂−x₃x₄) dx₃ⁱ∧dx₄

has Sing(θ) = {0} and satisfies θ∧ θ = 0. Hence, it generates a distribution of codimension two onC⁴\ {0}. This distribution is not integrable.

Theorem 1.9 motivates the following problem:

Problem 1.14. — Let F be a holomorphic foliation on Pⁿ of codimension >2 and dimension>2. Assume thatF has a l.t.p. componentΓ. IsΓa Kupka component of F?

A crucial point of our proof of Theorem 1.9 is the Camacho–Sad theorem on the existence of a separatrix for germs of holomorphic vector fields on (C²,0) [CS82].

The same type of argument cannot be used in the general case: there are examples of germs of vector fields on (C^m,0), m >3, without separatrices [GML92].

The proof of Theorem 1.9 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 1.11 and 1.12.

Acknowledgement

I would like to thank J. Vitório Pereira for helpful conversations that suggested me a simplification of the proof of Lemma 2.5.

2. Proof of Theorem 1.9

2.1. Preliminaries and idea of the proof

LetF be a codimension one foliation on Pⁿ, 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 points of Γ is a germ of singular foliation at (C²,0) with an isolated singularity at 0∈C² (see Remark 1.2 and Example 1.3). We can assume that this transversal type is given by germ at 0∈C² of vector field X =X₁(x, y)_∂x^∂ +X₂(x, y)_∂y^∂ , where X₁, X₂ ∈ O₂ and X₁(0,0) = X₂(0,0) = 0. Recall that Γ is a Kupka component if, and only if, we have T r(DX(0))6= 0, where

T r(DX(0)) := ∂X₁

∂x (0) +∂X₂

∂y (0)

is the trace of the linear partDX(0) ofX at 0 ∈C². In this case, as we have pointed out before,F is like in Example 1.5 (see Theorem 1.6).

Another useful ingredient is the normal Baum–Bott index of the component Γ, that we will denote asBB(F,Γ). Since Γ is a l.t.p. component ofSing(F) thenBB(F,Γ) coincides with the Baum–Bott index of X at the singularity 0 of X, denoted by BB(X,0) (see [CLN13]) (for the definition of BB(X,0) see [Bru00]). In [CLN13, Lemma 3.4, Section 3.2] it is proven that ifBB(F,Γ)6= 0 andDX(0) 6≡0 then Γ is a Kupka component and we are done.

One of the tools used in the proof of [CLN13, Lemma 3.4] 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 1.9.

Definition 2.1. — Let F be a holomorphic foliation of dimension k on a n dimensional compact complex manifold, 2 6 k < n, and Γ be l.t.p. component of F (recall that dim(Γ) =k−1). Aseparatrix Σ of dimension ` along Γof F, where k6` < n, is a germ of` analytic manifold alongΓwhich isF-invariant in the sense that:

(a) Σ sup Γ.

(b) Σ\Γ is contained in an union of leaves of F.

Remark 2.2. — Let Σ be a separatrix of F of dimension ` along Γ, as in Defini- tion 2.1. Fixp∈Γ and (x, y) : U →C^n−k+1×C^k−1, a local coordinate system around pas in Remark 1.2. It follows from the definition that x⁻¹(x(Σ∩U)) = Σ∩U.

Let T be 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 1.9, whereF has codimension one, then dim(T) = 2 and the normal type is a germ G of one dimensional foliation on (C²,0). In this case Σ∩T is a finite number of analytic separatrices of G as considered in [CS82]. The next result will be used in proof of Theorem 1.9.

Lemma 2.3. — Let F be a holomorphic codimension one foliation on a compact complex manifold M, where dim(M) = n > 3, and Γ be a l.t.p. component of Sing(F). If the normal type of F alongΓis not equivalent to the radial foliation of (C²,0) then F admits an irreducible separatrix Σalong Γ with dim(Σ) =n−1.

Recall that the radial foliation of C² is defined by the form x dy−y dx and its leaves are the straight lines through 0. Lemma 2.3 will be proved in Section 2.2.

From now on, in this section, we will assume thatF is a codimension one holomor- phic foliation on the compact manifoldM,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 thenormal bundle of Σ along Γ.

Since dim(Σ) = n−1 we can find a Leray covering U = (U_α)_α∈A of Γ by open sets and two collections f = (f_α)_α∈A and g = (g_{α β})_{Uα∩Uβ6=∅} with the following properties:

(a) f_α ∈ O(U_α), ∀α∈A, and f_a = 0 is a reduced equation of Σ∩U_α. (b) g_{α β} ∈ O^∗(U_α∩ U_β) and f_α =g_{α β}. f_β on U_α ∩ U_β 6= ∅.

Of course g = (g_{α β})_{Uα∩Uβ6=∅} is a multiplicative cocycle. We define the normal bundle of Σ along Γ as the line bundle onP ic(Γ) induced on a tubular neighborhood U ⊂^S_αU_α by the cocycle g = (g_{α β})_Uα∩Uβ6=∅.

It will be denoted byNΣ. Letc1(N_Σ) be the first Chern class of NΣ, considered as an element of H²(U,R) via the homomorphism H²(U, Z) → H²(U, R) ' H_DR² (U) induced by the inclusion Z→R.

As we have seen in Remark 1.2, the normal type ofF along Γ can be represented by a germ at 0∈C² of holomorphic vector fieldX =A₁(x, y)_∂x^∂ +A₂(x, y)_∂y^∂ with an isolated at 0. When we intersect Σ with a germ of transversal sectionT '(C²,0) we obtain a separatrix of X, say γ := Σ∩T (in general γ is not irreducible). Let f ∈ O₂ be a reduced analytic equation of γ. Sinceγ is X-invariant we can write

(2.1) X(f) =h. f, where h ∈ O₂.

Lemma 2.4. — In the above situation, if h(0) = 0then c₁(N_Σ) = 0.

On the other hand, we have the following:

Lemma 2.5. — If the ambient space is M = Pⁿ, n > 3, then c₁(N_Σ) 6= 0. In particular, if X(f) =h. f then h(0)6= 0.

As a consequence of Lemma 2.5 we get the following:

Corollary 2.6. — IfM =Pⁿ, n>3, then Σis a Kupka component of F. In particular, Theorem 1.6 will imply Theorem 1.9. Lemma 2.3 will be proved in the next section.

2.2. Proof of Lemma 2.3.

LetF be a holomorphic codimension one foliation on a compact complex manifold M with dim(M)>3. Assume thatF has a l.t.p. component Γ with normal typeG, where G is a germ of foliation on (C²,0) with an isolated singularity at 0∈C². As before, we will assume thatG is the foliation defined by a germ at (C²,0) of vector fieldX =X_1∂x^∂ +X_2∂y^∂ with an isolated singularity at the origin ofC². The germ of foliationG can be defined also by the 1-form

ω =iX(dx∧dy) =X1dy−X2dx ,

so that, dω(0) = T r(DX(0))dx∧dy. We can assume that ω has a representative, denoted byω, defined in the polydisce Q=D² with an isolated singularity at 0∈D². By the definition of l.t.p. component, we can find a coveringU = (Uα)α∈A of Λ by open sets biholomorphic to polydiscs, a collection of local charts ((z_α, Uα))_α∈A and a multiplicative cocycle (k_{α β})_{Uα∩Uβ6=∅} with the following properties:

(1) z_α = (x_α, y_α) : U_α → C²×Cⁿ⁻², where x_α(U_α) = Q and Γ∩U_α = x⁻¹_α (0),

∀α∈A.

(2) F |_Uα is defined by the integrable 1-form ωe_α :=x^∗_α(ω). The germ ofe ωe_α along Γ∩U_α will be denoted by ω_α.

(3) ωea=kα β.ωeβ onUα∩Uβ 6=∅.

We will assume that U satisfies the following:

(4) If U_α∩U_β 6=∅then Γ∩U_α∩U_β 6=∅ and connected.

Remark 2.7. — Given α, β ∈ A such that Γ∩U_α ∩U_β 6= ∅ we can construct a germ f_{α β} ∈ Diff(C²,0) as follows: fix p ∈ Γ∩ U_α ∩ U_β and a germ of plane T =T_{α, β} '(C², p) transverse to Γ at p. Note that x_α|_T, x_β|_T: (T, p) →(C²,0) are biholomorphisms. Therefore, we define

f_{α β} =x_α◦(x_β|T)⁻¹ =x_α|T ◦(x_β|T)⁻¹ ∈DiffC²,0.

Sinceω_α|_T = (x_α|T)^∗(ω),ω_β|_T = (x_β|_T)^∗(ω) andω_a =k_αβ. ω_βwe getf_αβ^∗ (ω) =h_αβ. ω, where

h_{α β} =k_{α β}|_T ◦(x_β|_T)⁻¹ ∈ O^∗₂.

The biholomorphism fα β can be interpreted as the glueing map of F |U_β with F |U_β. From now on, we fix a collection of germs (f_αβ)_{Uα∩Uβ6=∅} as above.

Lemma 2.8. — F admits a separatrix Σ along Γ if, and only if,X (or ω) has a separatrixγ (not necessarily irreducible) such thatf_{α β}(γ) =γ for allΓ∩U_α∩U_β 6=∅.

Terminology

We will say that the separatrixγ of G generates the separatrix Σ ofF.

Proof. — Assume thatX has a separatrixγ such thatf_{α β}(γ) =γ for all Γ∩U_α∩ U_β 6= ∅. Given α ∈ A define Σ_α := x⁻¹_α (γ). We assert that if U_α ∩U_β 6= ∅ then Σ_α∩U_β = Σ_β ∩U_α. In fact, let (x_α, y_α), (x_β, y_β) and T be as before. Then

Σ_α∩T =x⁻¹_α (γ)∩T = (x_α|_T)⁻¹(γ) = (x_α|_T)⁻¹(f_αβ(γ)) = (x_β|_T)⁻¹(γ) = Σ_β∩T . This, of course, implies the assertion. In particular, the local separatrices Σ_a glue together forming a global separatrix Σ along Γ such that Σ∩U_α = Σ_α, ∀α∈A.

We leave the converse to the reader.

Definition 2.9. — Let G be a germ of foliation at (C²,0) with an isolated singularity at 0. We say that a separatrix γ of G is distinguished if for any f ∈ Diff(C²,0)such that f^∗(G) = G then f(γ) =γ.

Lemma 2.10. — LetGbe a germ of foliation at(C²,0)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 [Sei68]. Let S be a smooth complex surface andG be a foliation by curves onS. Givenp∈Sing(G)⊂S we denote Diff(S, p) the set of germs atp∈Sof biholomorphismsf: (S, p)→S with a fixed point atp. Assume that the germ ofGatpis defined by a germ of holomorphic vector field X with an isolated singularity at p. We use also the notations

Diff_G(S, p) ={f ∈Diff(S, p)|f^∗(G) =G} . and

Diff⁰_G(S, p) ={f ∈DiffG(S, p)|f preserves the leaves of G}. Remark 2.11. — Note that:

(1) Given f ∈DiffG(S, p), thenf^∗(X) =h_X. X, where h_X ∈ O_p^∗. (2) Diff_G(S, p) is a sub-group of Diff(S, p).

(3) Given f ∈ Diff_G(S, p) and an irreducible separatrix γ of G through p then f(γ) is also a separatrix ofG through p.

Let Sep(G) be the set of irreducible separatrices of G through p. By (3) of Re- mark 2.11, DiffG(S, p) acts in Sep(G) as (f, δ) ∈ DiffG(S, p)× Sep(G) → f(δ) ∈ Sep(G). The idea of the proof is to find a finite subsetG_o :={γ₁, . . . , γ_k} ⊂Sep(G) such thatf(Go)⊂Go for all f ∈DiffG(S, p). In this case, the set γ :={f(γ1)|f ∈ Diff_G(S, p)} ⊂ Go contains finitely many irreducible separatrices of G through p and can be considered as a germ of curve through p such that f(γ) = γ for all f ∈Diff_G(S, p), and so γ is a distinguished separatrix ofG through p. Let us prove the existence of the finite setG_o.

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 throughp. Let us prove Lemma 2.10 in this case.

We will consider a blowing-up process used to resolve the foliationG (see [CS82]).

The first case, is whenG has a simple singularity atpand no blowing-ups are needed in the process. Letλ1 and λ2 be the eigenvalues ofDX(p). The singularity is simple if:

(a) λ₁. λ₂ 6= 0 and ^λ_λ²

1 ∈/ Q+.

(b) λ₁ 6= 0 andλ₂ = 0 (or vice-versa). In this case, p is a saddle-node.

In both casesG has one or two separatrices through p and so Lemma 2.10 is true.

When the singularity is not simple, Seidenberg’s theorem says that after a finite process of blowing-ups Π : (S, E)^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 blowing-ups of points

(2.2) S, E^e

:=S^e_k, E_k−→^Πk (S^e_k−1, E_k−1)^Π−→^k−1 . . . −→^Π2 S^e₁, E₁−→^Π1 S^e₀, E₀= (S, p) where in the j^th step Πj: (S^ej, Ej) → (S^ej−1, Ej−1), j > 2, we blow-up in a point pj−1 ∈Ej−1. The exceptional divisor obtained in this step will be denoted as P¹ ' Ee_j ⊂ E_j, so that Π_j(E^e_j) = pj−1. We use also the notation Π^e_j := Π₁ ◦ . . . ◦Π_j. We will denote also G^e_j :=Π^e∗(G). The point p_j−1 ∈E_j−1 is chosen between the non simple singularities ofG^e_j−1 on E_j−1. Seidenberg’s theorem can be stated as follows Theorem 2.12. — It is possible to choose a blowing-up process as above in such a way that all singularities of the strict transform G^ek =Π^e∗_k(G) are simple.

Remark 2.13. — There are two possibilities in each step Π_j: S^e_j,E^e_j→S^ej−1, pj−1

.

We assume that pj−1 is a non simple singularity of G^ej−1. Let Xj−1 be a germ at pj−1 of holomorphic vector field that represents the germ of G^ej−1 at pj−1. Let X_ν

= P_ν(x, y)_∂x^∂ +Q_ν(x, y)_∂y^∂ be the first non-zero jet of Xj−1 at pj−1, where P_ν and Qν are homogeneous polynomials of degree ν > 1. Set Fν+1(x, y) = x. Qν(x, y)− y Pν(x, y).

(i) If F_ν+1 6≡0 then F_ν+1 is homogeneous of degreeν+ 1 and the blowing-up is called non-dicritical. The divisorE^e_j is invariant for the foliation G^e_j and the singularities of G^e_j on E^e_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 = Fν−1(x, y)R, where R = x_∂x^∂ +y_∂y^∂ is the radial vector field in C² and Fν−1 is homogeneous of degree ν−1. In this case, the blowing-up is called dicritical. The divisor E^e_j is non-invariant for G^e_j and it is transverse to E^e_j outside the set V_j ⊂ E^e_j corresponding to the directions defined by the equation Fν−1(x, y) = 0.

If ν = 1 thenG^ej−1 is equivalent to the radial foliation at pj−1. We will say that pj−1 is a radial singularity of G^ej−1. If pj−1 is not radial for G^ej−1 then V_j 6=∅and we can divide it into two disjoint subsets V_j =τ_j∪σ_j, where

• σ_j =Sing(G^e_j)∩E^e_j.

• τ_j =V_j \Sing(G^e_j). We call τ_j the set of tangencies of G^e_j with E^e_j. Remark also thatSep(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, 16j 6k, we blow-up in some non- simple singularity ofG^ej−1, ifE^e_j is dicritical, at the end the tangenciesτ_j “survive”, in the sense that there exists a set τ ⊂ E_k such that for any 1 6 j < k such that τ_j 6=∅ then

τ_j ⊂Π_k◦ · · · ◦Π_j+1(τ)

For each 1 6 j 6 k denote by Diff(S^e_j, E_j) the set of germs of biholomorphisms f: (S^e_j, E_j)→(S^e_j, E_j).

Definition 2.14. — We say that f ∈Diff(S, p) can be lifted to Diff(S^e_j, E_j) if there exists a germ of biholomorphismf^e_j ∈Diff(S^e_j, E_j)such that the diagram below commutes:

Se_j, E_j

f˜j //

Πe_j

Se_j, E_j

Πe_j

(S, p) ^{f //}(S, p)

Remark 2.15. — Observe that, if the lift f^e_j of f exists then it is unique. When j = 1 (just one blowing-up) the lifting exists for any f ∈ Diff(S, p), but if j > 2 then there are germs f ∈ Diff(S, p) that cannot be lifted to Diff(S^ej, Ej). However, we have the following:

Claim 2.16. — The blowing-up process can be done in such a way that any f ∈ DiffG(S, p) can be lifted to the last step in an unique f^e= f^e_k ∈ Diff(S^e_k, E_k).

Moreover, f^epreserves G^e_k in the sense that f^e∗(G^e_k) = G^e_k.

Proof. — We say that the j^th step of the blowing-up process is admissible if any f ∈DiffG(S, p) has a liftingf^e_j ∈Diff(S^e_j, E_j). We will obtain by induction a blowing- up process, as in (2.2), for which there are steps 1 =`1 < `2 < · · · < `r = k such that the `^th_j step is admissible, for any 1 6 j 6 r, and Π^e_k: (S^e_k, E_k) → (S, p) is a resolution of the foliation G.

First of all, the first step is admissible, because anyf ∈Diff(S, p) admits a lifting fe₁ ∈Diff(S^e₁, E₁).

Assume that we have found some process for which the` :=`_s step is admissible,

`>1, so that anyf ∈DiffG(S, p) admits a lifting f<sup>e</sup>=f<sup>e</sup><sub>`</sub> ∈Dif f(S^e_{`</sub>, E<sub>`}) satisfying fe^∗(G^e`) =G<sup>e</sup>`. Givenf ∈DiffG(S, p), with liftingf^e, andq∈E`thenf<sup>e</sup>is an equivalence between the two germs of G<sup>e</sup><sub>`</sub> at q and at f(q). In particular,^e f^epreserves the set of non simple singularities ofG^e_`.

If G^e<sub>`</sub> is not a resolution of G then it has at least one non simple singularity q<sub>1</sub>. Let Sat(q<sub>1</sub>) ={f<sup>e</sup>(q<sub>1</sub>)|f ∈Diff<sub>G</sub>(S, p)}={q<sub>1</sub>, . . . , q<sub>m</sub>}. We then blow-up once at all pointsq<sub>j</sub> ∈Sat(q<sub>1</sub>), passing from the `=`<sub>s</sub> step to the`_s+1 := `_s+m step directly.

LetE^bj be the divisor obtained by the blowing-up at qj.

Givenf ∈DiffG(S, p) and its liftingf^e_s, letf^e_s(q_j) =q_i(j), 16j 6m. Then we can obtain a lifting f^e_s+1 of f^e_s such that f^e_s+1(E^b_j) = E^b_i(j), 1 6j 6m. By Seidenberg’s theorem this process must end at some step, when the final foliation G^e_k = Π^e∗_k(G)

has all singularities simple.

Proof of Lemma 2.10. — Let us finish the proof of the lemma. We will consider two cases:

(1) There is q₁ ∈Sing(G^e_k)∩E_k that has some separatrix γ_e not contained in E_k. (2) All the separatrices of the singularities of G^e_k are contained in E_k.

In the first case, let Sat(q₁) = {f(q^e ₁)|f ∈ DiffG(S, p)} = {q₁, . . . , q_m}. Given f ∈DiffG(S, p) and q_j =f^e(q₁) then f(^eγ) :=e γe_f is a separatrix of G^e_k not contained in E_k. Since γ_ef is not contained in E_k, its image γ_f := Π^e_k(γ_ef) is a separatrix of G through p. Moreover, since q₁, . . . , q_m are simple singularities of G^e_k the set {_eγ_f|f ∈Diff_G(S, p)} is finite. Therefore, if we set

(2.3) γ = ^[

f∈DiffG(S, p)

γ_f

then f(γ) =γ,∀f ∈DiffG(S, p), andγ is a distinguished separatrix of G.

In the second case necessarily there are dicritical irreducible divisors of G^e_k, say Ee₁, . . . , E^e_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.

• (2.2) τ =∅.

In case (2.1) let q_o ∈ τ and γ_eid be the leaf of G^e_k through q_o. Then, for any f ∈DiffG(S, p) we have q_f :=f(q^e _o)∈ τ andγe_f := f(^eγe_id) is the leaf of G^e_k through f(qe _o). Since q_f ∈ E_k, but γ_ef is not contained in E_k, ∀ f ∈ DiffG(S, p), the image Πe_k(γ_ef) := γ_f is an irreducible separatrix of G through p. Therefore, if we define γ as in (2.3) then γ is a distinguished separatrix of G.

We will divide case (2.2) into two subcases:

(2.2.1) E_k\^[

j

Ee_j 6=∅.

(2.2.2) E_k=^[

j

Ee_j.

In case (2.2.1), let E^b be a connected component of Ek \^S_jE^ej. Let ^Sr_i=1Di be decomposition ofE^b into irreducible components, D_i 'P¹. Note that:

(i) The graph formed by the divisors D_i is a tree.

(ii) The intersection matrix (D_i. D_j)_{16i, j6r} is negative.

(iii) If D is an irreducible divisor of E_k such that D 6⊂ E^b but D∩E^b 6= ∅ then D=E^e_j for some j. In particular, D is dicritical.

In this case, Sing(G^ek)∩E^b 6= ∅ and contains a singularity q with a separatrix γ_e not contained inE. This is a consequence of Sebastiani’s version of Camacho–Sad^b theorem (see [Seb97]). In fact, γ_e is not contained in E_k, for otherwise it would be contained in some irreducible divisor D of Ek not contained in E, and^b D is non dicritical, which contradicts (iii). Therefore, we reduce the problem to case (1).

In case (2.2.2) all irreducible divisors E^e_j of E_k are dicritical. We can assume that Sing(G^e_k) = ∅. In fact, if q_o ∈ Sing(G^e_k) then q_o is simple and any of their separatrices cannot be contained inE_k, for otherwise some of the divisorsE^e_j would be non-dicritical. Therefore, we are again in case (1). In particular, we can assume that all divisors E^e_j are radial, in the sense that for any q ∈ E^e_j the leaf of G^e_k through q is transverse to E^e_j. Moreover, m > 2 because otherwise p would be a radial singularity ofG. In particular, we can assume thatE^e1∩E^e2 ={qo} 6=∅. Let

eγ_id be the leaf of G^e_k through q_o. Note that, for any f ∈DiffG(S, p) then qf :=f(q^e o) =f^e(E^e₁)∩f(^eE^e₂)∈Ek,

so that γef := f^e(γeid) is the leaf of G^ek through qf. For each f ∈ DiffG(S, p) the projectionγ_f :=Π^e_k(γ_ef) is an irreducible separatrix of G. Since

A:={q_f|f ∈DiffG(S, p)} ⊂ ^[

i6=j

Ee_i∩E^e_j

then A is finite. Therefore, we can construct a distinguished separatrix γ of G as

in (2.3).

Finally, note that Lemma 2.3 is a consequence of Lemmas 2.8 and 2.10.

2.3. Proof of Lemma 2.4.

SinceU is a tubular neighborhood of Γ the map Θ∈H_Dr² (U)7→Θ|_Γ ∈H_Dr² (Γ) is an isomorphism. Therefore, it is sufficient to prove that c₁(N_Σ)|_Γ = 0.

Recall that the germ of F at any q ∈ Γ is equivalent to a product of a singular foliation by curves on (C²,0) by a regular foliation of dimension n−2. This implies that there exist a local coordinate system aroundq,z = (x, y) :U →C²×Cⁿ⁻²,x= (x₁, x₂), y= (y₁, . . . , yn−2), and a holomorphic vector fieldX =P(x)_∂x^∂

1 +Q(x)_∂x^∂

2, with an isolated singularity at 0∈C², such that

• F |U is generated by the n−1 commuting vector fieldsX,Y1 := _∂y^∂

1, . . . , Yn−2

:= _∂y^∂

n−2.

Moreover, the separatrix Σ ofF along Γ is induced by a separatrixγ = (f(x₁, x₂) = 0) ofX, such thatX(f) =h. f, where we have assumedh(0) = 0. It follows that we can find a Leray coveringU = (U_α)_α∈A of Γ by open sets with the following properties:

(a) For each α∈ A, there exists a coordinate system z_α = (x_α, y_α) : U_α →C²× Cⁿ⁻², where Σ∩U_α = (x_α = 0), x_α = (x_α1, x_α2) and y_α = (y_α1, . . . , yα n−2).

(b) For each α∈A,F |_Uα is generated by the n−1 holomorphic vector fields X_α=P(x_α) ∂

∂x_α1 +Q(x_α) ∂

∂x_α2 , Y_αj = ∂

∂y_αj, j = 1, . . . , n−2 .

(c) Σ∩U_α has the reduced equation f_α= 0, where f_α =f(x_α). In particular, if we set h_α =h(x_α) then

X_α(f_α) =h_α· f_a, Y_αj(f_α) = 0, ∀ 16j 6n−2.

Consider the multiplicative cocycle g = (g_{α β})_Uα∩Uβ6=∅ such that f_α = g_{α β}. f_β on U_α ∩ U_β 6= ∅.

Claim 2.17. — If U_α ∩U_β 6= ∅ then g_{α β} is locally constant on U_α ∩U_β ∩Γ : dg_{α β}|_Uα∩Uβ∩Σ ≡0. In particular c₁(N_Σ) = 0.

Proof. — Let U_α∩U_β ∩Σ 6= ∅. We assert that there exists a (n−1)×(n−1) matrix A_αβ, with entries in

O(Uα∩Uβ), A_αβ =a^ij_αβ

06i, j6n−2 , such that

(2.5)















X_α =a⁰⁰_{α β}. X_β +

n−2

X

j=1

a^0j_{α β}. Y_{β j} Y_αi =aⁱ⁰_{α β}. X_β +

n−2

X

j=1

a^ij_{α β}. Y_{β j}, 16i6n−2

In fact, since F |_Uα∩Uβ is generated by both systems hX_α, Y_{α i}|16i6n−2i and hX_β, Y_{β i}|16i6n−2i, we can find a matrix A_{α β} with entries in O(U_α ∩ U_β \ Σ) as in (2.5). But sincecod(Σ) = 2 the entries of Aα β can be extended toUα ∩ Uβ by Hartog’s theorem.

Now, from (c) we get

0 =Y_αi(f_a) = Y_αi(g_αβ. f_β) = Y_αi(g_αβ). f_β +g_αβ. Y_αi(f_β) and from (c) and (2.5)

Y_{α i}(f_β) =aⁱ⁰_{α β}. X_β(f_β) +

n−2

X

j=1

a^ij_αβ. Y_{β j}(f_β) =aⁱ⁰_{α β}. h_β. f_β

=⇒ Y_{α i}(g_{α β}) +aⁱ⁰_{α β}. h_β f_β = 0 =⇒ Y_{α i}(g_{α β}) =−aⁱ⁰_{α β}. h_β. Now,h_β|_Uα∩Uβ∩Σ =h(0) = 0 and so

Y_{α i}(g_{α β})|_{Uα∩Uβ∩Σ} = ∂g_{α β}

∂y_αi(0, y_α) = 0,16i6n−2 =⇒ dg_{α β}|_{Uα∩Uβ∩Σ} = 0.

This finishes the proof of Lemma 2.4.