Local monomialization of a system of first integrals

arXiv:1411.5333v1 [math.CV] 19 Nov 2014

Local monomialization of a system of first integrals A. Belotto University of Toronto [email protected] November 20, 2014

Abstract Given an analytic singular foliation ω with n first integrals (f1 , . . . , fn ) such that df1 ∧ . . . , dfn 6≡ 0, we prove that there exists a local monomialization of the system of first integrals, i.e. there exist sequences of local blowings-up such that the strict transform of ω has n monomial first integrals (xα1 , . . . , xαn ), α α where xαi = x1 i,1 · · · xmi,m and the set of multi-indexes (α1 , . . . , αn ) is linearly independent.

Contents 1 Introduction

2

2 Main Objects 2.1 Foliated Manifold . . . . . . . . 2.2 Compact Notation . . . . . . . 2.3 Monomial singular distribution 2.4 The analytic strict transform . 2.5 θ-Admissible Blowings-up . . . 2.6 Foliated Ideal sheaf . . . . . . .

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3 Strategy of Proof 11 3.1 Foliated sub-ring sheaf . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Main Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Proof of Theorem 1.1 (Assuming Theorem 3.7) . . . . . . . . . . 13 4 Dropping the Invariant 4.1 Basic Normal Form . . . . . 4.2 Preparation . . . . . . . . . 4.3 Combinatorial Blowings-up 4.4 Dropping the invariant from 4.5 Proof of Theorem 3.7 . . . .

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15 15 18 21 27 31

1

Introduction

To date, the existence of a reduction of singularities for general singular foliations is a difficult open problem. More precisely, given a couple (M, ω) where M is an algebraic or analytic variety and ω is a general singular foliation, a ref → M such that the strict duction of singularities of ω is a birational map σ : M transform of ω has only ’minimal’ singularities (introduced in [Mc], following the approach of the Mori program), called Log-Canonical singularities. In one hand, the problem is well-understood whenever the ambient variety has dimension smaller or equal than three (see [Ben, Se, P, McP, Ca] for a complete list of results). More precisely, the classical Bendixson-Seidenberg Theorem (see [Ben, Se]) gives a positive answer whenever the ambient dimension is two. For the three dimensional case, [P, McP] gives a positive answer for line foliations and [Ca] gives a positive answer for codimension one foliations. At the other hand, there is no result valid for dimension bigger or equal than four. In particular, there is not even a local result (such as a local uniformization) valid for dimension higher than three. In what follows, we consider singular foliations with first integrals and we present a local reduction of the system of first integrals. In particular, if ω is a totally integrable singular foliation (e.g. ω is given by the level curves of an analytic map), Theorem 1.1 below gives rise to a local reduction of singularities for ω. More precisely, our main result is the following: Theorem 1.1 (Main Theorem). Let M be a non-singular analytic manifold, E be a SNC (simple normal crossing) divisor on M , p a point of M and ω a singular distribution with n analytic first integrals (f1 , . . . , fn ) over p such that df1 ∧ ... ∧ dfn calculated at p is non-zero. Then, there exists a finite collection of morphisms Φi : (Mi , Ei ) → (M, E) such that: I ) The morphism Φi is a finite composition of adapted local blowing-ups (i.e local blowings-up with centers with SNC to E); II ) In each variety Mi , S there exists a compact set Ki ⊂ Mi such that the union of their images Φi (Ki ) is a compact neighborhood of p;

III ) The singular distribution ωi , given by the strict transform (or analytic strict transform) of ω by Φi , is n-monomial integrable, i.e. for every point qi in Mi , there exists an adapted coordinate system x = (x1 , . . . , xm ) centered at q (i.e. the exceptional divisor Ei is locally given by {x1 · · · xl = 0}) and n first integrals (xα1 , . . . , xαn ) of ωi .Oq such that α1 ∧...∧αn 6= 0. Notation 1.2 (Collection of local blowings-up). In order to simplify notation, a finite collection of morphisms Φi : (Mi , Ei ) → (M, E) that satisfies conditions [I] and [II] of Theorem 1.1 is called a collection of local blowings-up. When ω is a totally integrable singular foliations, the class of singularities of the transforms ωi is a sub-class of the Log-Canonical singularities that we call monomial singularities (see Definition 2.2 below). It comes as no surprise that monomial singularities are deeply related with monomial mappings (see [K, Cu1] for a definition). Indeed, the main idea to prove Theorem 1.1 is to 2

consider a local analytic mapping given by the first integrals (f1 , . . . , fn ) and to monomialize the level curves of this mapping, hence obtaining the final monomial form given in condition [III] of Theorem 1.1. Before going into further details, let us remark that the problem of monomialization of maps is a very delicate problem, which does not follow from resolution of singular varieties. The best results, to date, are set in the algebraic category and are given by the series of articles [Cu1, Cu2, Cu3, ADK]. In particular, in [Cu1, Cu2], Cutkosky proves the existence of a local monomialization of maps using an algebraic construction that follows Zariski ideas for local uniformization of varieties (see [Z]). This result can be used to prove Theorem 1.1 in the algebraic category, but its proof does not seem to extend in an easy way for the analytic category (although it might be possible to do so using the stars of Hironaka - see [Cu4]). We prove our main result using elementary geometrical arguments which are independent from [Cu1, Cu2]. The main idea is to adapt a global algorithm of monomialization of maps, valid only for low dimensions (see [Cu3]), for a higher dimensional case. In order to do that, we sacrifice the global property of the algorithm and we use techniques of [Bel] which allow one to preserve the monomiality of a singular foliation. To exemplify the idea of the proof, let us consider the case where there are only three first integrals (f1 , f2 , f3 ). By resolution of singular varieties, we can assume that f1 is a monomial function, i.e., f1 = xα1 . Now, we can adapt the algorithm given in [Cu3] in order to monomialize the second function f2 and obtain a map (f1 , f2 ) = (xα1 , xα2 ) where α1 and α2 are linearly independent multi-indexes. A further simple adaptation of the algorithm given in [Cu3] would lead to a monomialization of the function f3 which breaks the monomiality of the first pair (f1 , f2 ). Nevertheless, the techniques in [Bel, Bel] allow us to choose blowings-up which preserve the monomiality of an ambient singular foliation such as, in this case, df1 ∧ df2 . This is exactly what we need in order to monomialize the level curves of the map (f1 , f2 , f3 ) and prove Theorem 1.1. The manuscript is divided in four sections counting the introduction. In the second section we introduce the main objects used in the rest of the manuscript, including the definitions and techniques of [Bel, Bel] that are going to be necessary. In the third section we introduce the notion of foliated sub-ring sheaves and we present the main idea of the proof: first, we give a sense for monomialization of a sub-ring sheaf (see Lemma 3.6 below); second, we prove Theorem 1.1 assuming the technical Theorem 3.7. The forth and last section is completely devoted to prove Theorem 3.7. Finally, it is worth remarking that this manuscript is not completely self-contained: Theorems 2.11 and 2.12 below are proved in [Bel, Bel] and we do not reproduce their proofs in here.

3

2

Main Objects

2.1

Foliated Manifold

In what follows, a foliated analytic manifold is the triple (M, θ, E) where • M is a smooth analytic manifold of dimension m over a field K (where K is R or C); • E is an ordered collectionP E = (E (1) , ..., E (l) ), where each E (i) is a smooth divisor on M such that i E (i) is a reduced divisor with simple normal crossings; • θ is an involutive singular distribution defined over M and everywhere tangent to E. We briefly recall the notion of singular distribution following closely [BaBo]. Let DerM denote the sheaf of analytic vector fields over M , i.e. the sheaf of analytic sections of T M . An involutive singular distribution is a coherent sub-sheaf θ of DerM such that for each point p in M , the stalk θp := θ.Op is closed under the Lie bracket operation. Consider the quotient sheaf Q = DerM /θ. The singular set of θ is defined by the closed analytic subset S = {p ∈ M : Qp is not a free Op module}. A singular distribution θ is called regular if S = ∅. On M \ S there exists an unique analytic sub bundle L of T M |M\S such that θ is the sheaf of analytic sections of L. We assume that the dimension of the K vector space Lp is the same for all points p in M \ S (this always holds if M is connected). It is called the leaf dimension of θ and denoted by d. In this case θ is called an involutive d-singular distribution.

2.2

Compact Notation

In what follows, it will be useful to have a compact notation for denoting a collection of monomials. To that end, let u be a collection of k functions (u1 , . . . , uk ) and A be a t × k matrix:     α1 α1,1 . . . α1,k    ..  .. A =  ...  =  ... . .  αt

Then, we define: uA

αt,1

...

αt,k

 α   α1,1 α  u 1 u1 · · · uk 1,k     .. =  ...  =   . α

uαt

α

u1 t,1 · · · uk t,k

Lemma 2.1. Let u = (u1 , . . . , uk ) and x = (x1 , . . . xr ) be two collections of functions such that u = xB for some k × r matrix B. Then, for any t × k matrix A, we have that uA = xAB .

4



 β1   Proof. Indeed, let βi be the line vectors of B, i.e B =  ... . Notice that, by βk

definition ui = xβi , which implies that:

uA

Pk   α1,1  α β   r α1,j βj,i α  u1 · · · uk 1,k x 1,1 1 · · · xα1,k βk Πi=1 xi j=1       .. .. ..  = xC = = = . . .   α

α

where

Pk

α1,j βj,1  .. C = .  Pk j=1 αt,j βj,1

... .. . ...

j=1

which is clearly equal to AB.

2.3

Pk

xαt,1 β1 · · · xαt,k βk

u1 t,1 · · · uk t,k

Πri=1 xi

j=1

αt,j βj,i

 α1,j βj,r  ..  .  Pk j=1 αt,j βj,r

Pk

j=1

Monomial singular distribution

Definition 2.2 (Monomial singular distribution). Given a foliated manifold (M, θ, E), we say that the singular distribution θ is monomial at a point p if there exists set of generators {∂1 , ..., ∂d } of θ.Op and a coordinate system (u, w) = (u1 , . . . , uk , wk+1 , . . . wm ) centered at p such that: i) The exceptional divisor E is locally equal to {u1 · · · ul = 0} for some l ≤ k; ii) The singular distribution θ is everywhere tangent to E, i.e.,θ ⊂ DerM (−logE); iii) Apart from re-indexing, the vector-fields ∂i are of the form: ∂i = ∂wm+1−i for i from 1 to m − k, and ∂i =

k X

αi,j uj ∂uj (where αi,j ∈ Q) otherwise

j=1

iv) If ω ⊂ DerM (−logE) is a d-singular distribution such that θ ⊂ ω then θ = ω. In this case, we say that (u, w) is a monomial coordinate system and that {∂1 , ..., ∂d } is a monomial basis of θ.Op . Remark 2.3 (Geometrical Interpretation of (iv)). Assuming conditions [i − iii] above, it is clear that Property [iv] implies that the codimension one part of the singularity set of θ is contained in E. Notation 2.4 (An special monomial coordinate system). In what follows, we sometimes need to emphasis one of the coordinates in w. To that end, we will denote by (u, v, w) a monomial coordinate system and the vector field ∂v is contained in θ.Op .

5

Monomial singular distributions are deeply related with the existence of monomial first integrals: Lemma 2.5 (Monomial First Integrals). Given a foliated manifold (M, θ, E), the singular distribution θ is monomial if, and only if, for any monomial coordinate system (u, w) = (u1 , . . . , uk , wk+1 , . . . wm ) centered at p there exists m − d monomials uB = (uβ1 , . . . , uβm−d ), where B has maximal rank, such that θ.Op = {∂ ∈ Derp (−logE); ∂(uβi ) ≡ 0 for all i} In this case, we call uB a complete system of first integrals. Proof. First, let us assume that θ is a monomial singular distribution and let us fix a point p in M and a monomial coordinate system (u, w). It is clear that if f is a first integral of θ, then it can not depend on any coordinate w, since all the derivations ∂wi are contained in θ.Op . So, consider a monomial uβ and let us remark that: ∂i (uβ ) ≡ 0 if i ≤ m − k, and ∂i (uβ ) ≡ uβ

k X

αi,j βi , otherwise

j=1

So, the monomial uβ is a first integral of θ if, and only if: k X

αi,j βi = 0 for all m − k < i ≤ d

(2.1)

j=1

Thus, there exists a m − d linear subspace L of Qk that contain all vector β satisfying the equations 2.1. In particular, we can choose a system of generators {β1 , . . . , βm−d } of L such that βi ∈ Zk . So, the m − d monomials uB = (uβ1 , . . . , uβm−d ) are first integrals of θ an: θ.Op ⊂ {∂ ∈ Derp (−logE); ∂(uβi ) ≡ 0 for all i ≤ m − d} By the maximal condition [iv], we conclude that both singular distributions are equal. Now, let θ be the singular distribution over p given by {∂ ∈ Derp (−logE); ∂(uβi ) ≡ 0 for all i ≤ m − d} and let us prove that this is a monomial singular distribution. First, it is clear that the vector-fields Xi = ∂wm+1−i for i ≤ m − k are all contained in θ. So, Pk consider a vector-field of the form Y = j=1 αj uj ∂uj and let us notice that:   k X αj βi,j  Y (uβi ) = uβi  j=1

So, since Y is clearly tangent to E, it belongs to θ if and only if: k X

αj βi,j = 0 for all i

j=1

6

(2.2)

Consider the d + k − m linear subspace L of Qk that contain all vectors α satisfying the equations 2.2. In particular, for any fixed system of generators P {α1 , . . . , αd+k−m }, we define Yi = α u j i,j j ∂uj , which are vector-fields contained in θ. It rests to prove that there are no other vector-fields in θ. Indeed, let Z be any vector-field contained in θ: Z=

k X

Aj ∂uj +

j=1

m X

Bj ∂wj

j=k+1

Now Z(uβi ) ≡ 0 ⇐⇒ uβi

X

βi,j

j

Aj ≡0 uj

ej . Indeed, either uj divides Aj (and Thus, for each j ≤ k we can write Aj = uj A the result is trivial), or the terms βi,j are zero for all i. In this case, notice that the vector-field ∂uj is such that ∂uj (uβi ) ≡ 0 for all i ≤ m − d So, the coordinate uj either can be changed into a w coordinate, or it is an exceptional variable. In this case, the vector-field ∂uj is not tangent to E and Ai must be divisible by ui in order for the the vector-field Z to be tangent to E. Thus: m X X ej ∂uj + Bj ∂wj Z= uj A j=k+1

We now can take the Taylor expansion of Z(uβi ): βi

Z(u ) = u

βi

k X j=1

ej = uβi βi,j A

X

δ

u w

(δ,γ)

γ

k X

βi,j Aj,δ,γ

j=1

Pk which implies that j=1 βi,j Aj,δ,γ = 0 and, by the definition of the vector-fields Yi , there exists Ci,δ,γ such that: X Ci,δ,γ αi,j Aj,δ,γ = i

So, let Ci =

P

(δ,γ) Ci,δ,γ u

δ

wγ . It is clear that Z=

X

Ci Yi +

X

Bi X i

and, thus θ is equal to the singular distribution generated by the vector-fields {Xi , Yi }. Since θ clearly satisfies the maximal condition (iv), we conclude the Lemma. We also prove that the monomiality is an open property: Lemma 2.6 (Openness of monomiality). The monomiality is an open condition i.e. if θ is monomial at p in M , then there exists an open neighborhood U of p such that θ is monomial at every point q in U . 7

Proof. For a proof coming directly from the definition, see Lemma 2.2.1 in [Bel]. In here, we present a different proof (using Lemma 2.5) which is useful for the current manuscript. Indeed, let us fix a monomial coordinate system (u, w) = (u1 , . . . , uk , wk+1 , . . . , wm ) centered at p which is defined in a neighborhood U of p. Since θ is monomial, by Lemma 2.5, there exists m − d monomials uB = (uβ1 , . . . , uβm−d ), such that θ.Op = {X ∈ Derp (−logE); X(uβi ) ≡ 0 for all i} So, fix a point q of U and let (δ, γ) be its coordinate in the coordinate system (u, w). Apart from re-indexing, we can assume that δ = (0, . . . , 0, δt+1 , . . . , δk ) for some t ≤ k and let us consider the coordinate system (x, y, v) = (x1 , . . . , xt , yt+1 , . . . , yk , vk+1 , . . . vm ) where xi = ui y i = u i − δi vi = wi − γi which is a coordinate system centered at q. We can now write: uB = xB1 (y − δ)B2 where B1 is a k × t matrix and B2 is a k × (m − d − t) matrix such that   B = B1 B2

Furthermore, apart from re-ordering the lines of the matrix B, we can furhter write:  ′ ′  B1 B2 B= ′′ ′′ B1 B2  ′ ′ B1 where B1 = and the rank of B1 is maximal and equal to the rank of B1 . ′′ B1 So, there exists a change of coordinates (x(1), y(1), v(1)) such that: uB = x(1)C1 (y(1) − δ)C2 where:

 C = C1

 ′  C1 C2 = ′′ C1

 ′ ′  B1 C2 = ′′ ′′ C2 B1

0 Λ



where Λ is a maximal rank matrix of rational numbers. This implies that the ′ ′′ collection (x(1)B1 , x(1)B1 (y(1) − δ)Λ ) is a collection of first integrals of θ.Oq . ′ Since B1 has rank equal to B1 , we conclude that: ′

(x(1)B1 , (y(1) − δ)Λ ) is another collection of first integrals of θ.Oq . Furthermore, since Λ is of maximal rank, there exists a coordinate system (x(2), y(2), z(2), v(2)) where x(2) = x(1) and v(2) = v(1) such that: (y(1) − δ)Λ = y(2) − δ (2) 8

which finally implies that the monomial functions ′

(x(1)B1 , y(2)) are first integrals of θ.Oq . By the analyticity of θ and Lemma 2.5, we conclude that the singular distribution θ.Oq is monomial. Since q is an arbitrary point in U , we conclude that the monomiality property is open.

2.4

The analytic strict transform

f, E) e → (M, E) (i.e. the center of Given an admissible blowing-up σ : (M blowings-up C has SNC with E) let F be the exceptional divisor of the blowingDerM up. Consider the sheaf of OM f. Notice f-modules BlDerM f := O(F ) ⊗OM f f e that the morphism σ : (M , E) → (M, E) gives rise to an application: σ ∗ : DerM −→ BlDerM f

e Now, consider the coherent sub-sheaf DerM f(−log E) of DerM f composed by all e invariant. We notice that the derivations which leave the exceptional divisor E there exists a natural imersion: e ζ : DerM f(−log E) −→ BlDerM f

Finally, the analytic strict transform θe of θ is the singular distribution: θe = ζ −1 σ ∗ (θ)

e where ζ −1 (ω) stands for the coherent sub-sheaf of DerM f(−log E) generated by the pre-image of ω. For a relation betweent the analytic strict transform and the usual strict transform, see remark 2.9 below.

2.5

θ-Admissible Blowings-up

Given a foliated manifold (M, θ, E), the generalized k-Fitting operation (for k ≤ d) is a mapping Γθ,k that associates to each coherent ideal sheaf I over M the ideal sheaf Γθ,k (I) whose stalk at each point p in M is given by: Γθ,k (I).Op = h{det[Xi (fj )]i,j≤k ; Xi ∈ θp , fj ∈ I.Op }i where hSi stands for the ideal generated by the subset S ⊂ Op . Now, consider a regular analytic sub-manifold C of M and the reduced ideal sheaf IC that generates C. We say that C is a θ-admissible center if: • C is a regular closed sub-variety that has SNC with E; • There exists 0 ≤ d0 ≤ d such that the k-generalized Fitting-ideal Γθ,k (IC ) is equal to the structural ideal OM for all k ≤ d0 and is contained in the ideal sheaf IC otherwise. In particular, we say that a center is θ-invariant if the number d0 above is equal to 0. A θ-admissible blowing-up is a blowing-up with θ-admissible center. The following Theorem enlightens the interest of θ-admissible blowings-up: 9

Theorem 2.7 (θ-admissible blowings-up). (See Theorem 4.1.1 of [Bel]) Let (M, θ, E) be a monomial d-foliated manifold and: e E) f, θ, e → (M, θ, E) τ : (M

a θ-admissible blowing-up. Then θe is monomial.

Remark 2.8 (Proof of Theorem). In [Bel], a singular distribution is called monomial if it satisfies Properties [i-iii] of the Definition 2.2. So, Theorem 4.1.1 proves that θe also satisfies Properties [i − iii], but Claims nothing about Property [iv]. Nevertheless, by Remark 2.3, it is clear that if θ satisfies condition [iv], then θe also satisfies condition [iv], which gives rise to the formulation used on this work. Remark 2.9 (Strict Transform of a monomial singular distribution θ). As a e E) f, θ, e → (M, θ, E) is consequence of Theorem 2.7, if θ is monomial and τ : (M e θ-admissible, then the analytic strict transform θ coincides with the intersection e of the strict transform of θ with DerM f(−E). In particular, they are equal in e the non-dicritical case (i.e. if the strict transform is tangent to E).

Because of Theorem, 2.7, the θ-admissible blowings-up are going to be an essential tool in this work. So, let us present a couple of examples and an intuitive description of the Definition: Example 1: If C is an admissible and θ-invariant center (i.e if all leafs of θ that intersects C are contained in C) it is θ-admissible.

Example 2: If C is an admissible and θ-totally transverse center (i.e all vectorfields in θ are transverse to C) it is θ-admissible. Example 3: Let (M, θ, E) = (C3 , {∂x , ∂y }, ∅) and C = {x = 0}. Then C is a θ-admissible center, but it is neither invariant nor totally transverse. Indeed, Γθ,1 (IC ) = OM and Γθ,2 (IC ) ⊂ IC . Example 4: Let (M, θ, E) = (C3 , {∂x , ∂y }, ∅) and C = {x2 − z = 0}. Then C is not a θ-admissible center. Indeed, Γθ,1 (IC ) = (x, z). Remark 2.10 (Intuition of the Definition). (See Proposition 4.3.1 of [Bel]) If a center C is θ-admissible then, for each point p in C, there exists two singular distributions germs θinv and θtr such that: θp is generated by {θinv , θtr }; C is θinv -invariant; and C is θtr -totally transverse.

2.6

Foliated Ideal sheaf

A foliated ideal sheaf is a quadruple (M, θ, I, E) where I is a coherent and everye E) f, θ, e → where non-zero ideal sheaf of OM . Given an adapted blowing-up τ : (M e (M, θ, E), we define the transform I of I as the total transform I.OM f. We now recall two important results about foliated ideal sheaves. The first concerns θ-invariant global resolution of singularities:

10

Theorem 2.11 (θ-Invariant resolution of Ideal). (See Theorem 4.1.1 of [Bel]) Let (M, θ, I, E) be an analytic d-foliated ideal sheaf such that I is θ-invariant, i.e θ(I) ⊂ I. Then, for each point p in M and relatively compact open neighborhood U of p, there exists a sequence of θ-invariant blowings-up (i.e all blowings-up are θ-admissible) e I, e , θ, e E) e → (U, θ, I, E) τ : (U such that the ideal sheaf Ie is a principal ideal sheaf with support contained in Ei . In particular, if θ is monomial, then θe is monomial. The second concerns a θ-admissible local resolution of singularities:

Theorem 2.12 (θ-Resolution of Ideal). (See Theorem 1.1 of [Bel]) Let (M, θ, I, E) be an analytic d-foliated ideal sheaf. Then, for every point p in M , there exists a θ-admissible collection of local blowings-up (i.e all local blowings-up are θ-admissible) τi : (Mi , θi , Ii , Ei ) → (M, θ, I, E) such that the ideal sheaf Ii is a principal ideal sheaf with support contained in Ei . In particular, if θ is monomial, then θi is monomial.

3 3.1

Strategy of Proof Foliated sub-ring sheaf

A foliated sub-ring sheaf is a quadruple (M, θ, R, E) where R is a coherent and everywhere non-zero sub-ring sheaf of OM . Furthermore, we assume that at each point p in M there exists a finite system of generators (f1 , . . . , fn ) of R.Op such that df1 ∧ · · · ∧ dfn 6≡ 0 and, if g is another function of R.Op , then df1 ∧ · · · ∧ dfn ∧ dg ≡ 0 Remark 3.1. Notice that the above condition does not follow from coherence. For example, the sub-ring (x2 , xy, y 2 ) is coherent but does not satisfy the above condition. Remark 3.2. In the notation of Theorem 1.1, the sub-ring R stands for the ring of first integrals of the initial singular distribution ω. Remark 3.3. Since our goal is a local result, we work with a fixed system of generators of R. Finally, (M, θ, R, E) is said to be trivial at a point p if all functions f in R.Op are first integrals of θ, i.e X(f ) ≡ 0 for all X in θ.Op .

11

3.2

Main Invariant

Let (M, θ, R, E) be a monomial foliated sub-ring sheaf and consider a system of generators (f1 , . . . , fn ) of R and monomial coordinate systems (u, w) of p. Then, there exists a maximal multi-index δ such that: f i = g i + uδ T i

(3.1)

where gi are first integrals of θ and all monomials in the Taylor expansion of uδ Ti are not first integrals of θ. Definition 3.4 (Main Invariant). In the above notations, consider the coordinate dependent function λ ν(p, θ, R, (u, w)) := min{|λ| : ∂w Ti is a unit}. λ where we assume that ∂w is the identity if w is empty, and ν(p) = ∞ if there λ are no λ such that ∂w Ti is a unit. We define the tangency order of (M, θ, R, E) by: ν(p, θ, R) := min{ν(p, θ, R, (u, w)) : for all (u, w)}.

When there is no risk of confusion, we simply denote ν(p, θ, R) by ν(p). Lemma 3.5. The tangency order is independent of the set of generators of R and is upper-semicontinuous. Proof. At every point p, let I be the ideal generated by all the functions in θ[R].OU , where U is a neighborhood of p. Taking U sufficiently small, we can assume that I is generated by θ[(f1 , . . . , fn )], i.e by the set {X(fi ); X ∈ θ}. In particular, this implies that I is independent of the choice of generators of R and of coordinate systems of M . Now, consider the sequence of ideals: I1

=I

Ii+1 = Ii + θ[Ii ] Then, by Noetherianity, there exists an integer µ such that: Iµ = Iµ+1 Now, in the fixed (u, w) monomial coordinate system, let us divide in two cases: Assume that ν < ∞: In this case, we claim that µ = ν. Indeed, in one hand by the definition of µ and ν, we know that Iν = (uδ ) = Iµ . On the other hand, the ideal Iν−1 does not contain uδ yet (or it would contradict the definition of ν). This implies that µ = ν and, since µ is clearly an upper semi-continuous invariant of the pair (θ, I), we conclude that ν is upper semi-continuous. Assume that ν = ∞: In this case, it is clear that the invariant ν can only droop in a neighborhood of p (thus, satisfies the condition for semi-continuity) and we only need to prove that it is independent of the choice of generators of R. To that end, we claim that at every monomial coordinate system, the ideal sheaf Iµ is not monomial (which is is an intrinsic property of I). Indeed, if it were monomial, it would have to be equal to uγ for some γ. By another hand, this monomial would factor out from fi − gi (since it divides it), which implies that γ = δ. But this contratics the assumption that there exists no derivation of Ti which is a unit. Since this property in intrinsic of the ideal I, we conclude the Lemma. 12

Lemma 3.6 (Invariant zero or one). Let (M, θ, R, E) be a monomial d-foliated sub-ring sheaf and p a point of M where the invariant ν(p, θ, R) is 0 or 1. Then, for any system of generators (f1 , . . . , fn ) of R, there exists an index i0 and a monomial coordinate system (u, w) such that: fi0 = gi0 + uβ w1ǫ where we recall that gi0 is a first integral of θ and uβ w1ǫ is not a first integral (see equation 3.1). Moreover, the monomial uβ divides all functions fi − gi and the constant ǫ is 0 or 1. In particular, the (d − 1)-foliated sub-ring sheaf (M, ω, R, E) given by ω = {X ∈ θ; X(fi0 ) ≡ 0} is monomial. Proof. Fix a monomial system of coordinates (u, w) and recall that, by Lemma 2.5, there exists a complete system of first integrals uB = (uβ1 , . . . , uβm−d ) of θ. We now consider the cases where ν(p) is zero and one separately: First, assume that ν(p) = 0 (this is the case when ǫ = 0). Without loss of generality, we can assume that T1 is a unit. By the definition of the functions Ti , the multi-index δ has to be linearly independent with all the multi-indexes βi . Thus, apart from a change of coordinates (which preserves all monomials), we can assume that T1 = 1 . So, the singular distribution ω = {X ∈ θ; X(f1 ) ≡ 0} has a complete system of first integrals given by (uB , uβ ) which implies that it is monomial. Now, assume that ν(p) = 1 (this is the case when ǫ = 1). In this case, there exists a coordinate system (u, v, w) such that ∂v T1 is a unit. So, apart from a change of coordinates in the v coordinate, we can assume that T1 = v. Thus, the singular distribution ω = {X ∈ θ; X(f1 ) ≡ 0} has a complete system of first integrals given by (uB , uβ v) which implies that it is monomial. We now turn to the main technical result of the manuscript: Theorem 3.7 (Main Theorem 2). Let (M, θ, R, E) be a non-trivial monomial foliated sub-ring sheaf. Then, for each point p in M , there exists a θ-admissible collection of local blowings-up τi : (Mi , θi , Ri , Ei ) → (M, θ, R, E) such that, for every point qi in the pre-image of p, the invariant d(qi , θi , Ri ) is zero or one. The proof of the above Theorem is given in section 4. In the next subsection we show how this results proves the main Theorem 1.1.

3.3

Proof of Theorem 1.1 (Assuming Theorem 3.7)

Fixed the point p, recall that there exists n first integrals (f1 , . . . , fn ) of ω such that df1 ∧ · · · ∧ dfn 6= 0 So, let us consider a monomial m-foliated sub-ring sheaf (U, θ(0), R, E) where: • U is a sufficiently small neighborhood of p so that the first integrals (f1 , . . . , fn ) are everywhere defined and df1 ∧ · · · ∧ dfn 6= 0 everywhere; • R is the sub-ring generated by the first integrals (f1 , . . . , fn );

13

• θ(0) is the monomial singular distribution DerM (−logE) i.e the sheaf of derivations of M tangent to E In this case, let us notice that the singular distribution ω ∩ DerM (−logE) is obviously contained in θ(0). The proof follows a recursive argument: Claim 3.8. Let ω be a singular distribution with n first integrals (f1 , . . . , fn ) and (M, θ(k), R, E) be a monomial (m − k)-foliated sub-ring sheaf such that: i ) R is the sub-ring generated by global first integrals (f1 , . . . , fn ) of ω; ii ) Apart from re-indexing the functions (f1 , . . . , fn ), the singular distribution θ(k) is equal to {X ∈ DerM (−logE); X(fi ) ≡ 0 for all i ≤ k}. In particular ω ∩ DerM (−logE) ⊂ θ(k). Then, if k < n, there exists a collection of θ(k)-admissible local blowings-up: Φi : (Mi , θi (k), Ri , Ei ) → (M, θ(k), R, E) such that, for each point qi in the pre-image of q, there exists a monomial [m − (k + 1)]-foliated sub-ring sheaf (Mi , θi (k+1), Ri , Ei ) that satisfies properties [i] and [ii] in respect to the strict transform ωi of ω i.e: i ) Ri is the sub-ring generated by global first integrals τi∗ (f1 , . . . , fn ) = (f1∗ , . . . , fn∗ ) of ωi ; ii ) Apart from re-indexing the functions (f1∗ , . . . , fn∗ ), the singular distribution θ(k+1) is equal to {X ∈ DerMi (−logEi ); X(fi∗ ) ≡ 0 for all i ≤ k + 1}. In particular ω ∩ DerMi (−logEi ) ⊂ θi (k+1). Proof. Indeed, since df1 ∧ · · · ∧ dfn 6= 0 and k < n, the monomail folaited subring sheaf (M, θ(k), R, E) is non-trivial. Thus, by Theorem 3.7 there exists a collection of θ(k)-admissible local blowings-up: Φi : (Mi , θi (k), Ri , Ei ) → (M, θ(k), R, E) such that, for every point qi in the pre-image of q, the invariant ν(qi , θi (k), Ri ) is either zero or one. So, by Lemma 3.6 there exists a monomial (m−k−1)-foliated sub-ring sheaf (Mi , θi (k+1)(qi ), Ri , Ei ) where θi (k+1)(qi ) = {X ∈ θi (k); X(fi∗0 ) ≡ 0} for some index i0 > k (because θi (k+1)(qi ) is not equal to θi (k)). So, by the compacity of the pre-image of p and Lemma 2.6, after shirinking Mi if necessary, we can suppose that the singular distribution θi (k)(qi ) is monomial everywhere in Mi . Thus, it is independent of the point qi and we can simply denote it by θi (k). Moreover, since θi (k) is monomial, by Remark 2.9 we conclude that: θi (k) = {X ∈ DerMi (−logEi ); X(fi∗ ) ≡ 0 for all i ≤ k} So, apart from re-indexing, we conclude that: ∗ θi (k+1) = {X ∈ θi (k); X(fk+1 ) ≡ 0}

= {X ∈ DerMi (−logEi ); X(fi∗ ) ≡ 0 for all i ≤ k + 1} which proves the Claim. 14

So, it is clear we can recursively use the Claim over (U, θ(0), R, E) in order to get a collection of local blowings-up: Φi : (Ui , Ri , Ei ) → (U, R, E) where, for each point qi in the pre-image of p, there exists a trivial monomial (m − n)-foliated sub-ring sheaf (Ui , θ(m−n), Ri , Ei ) such that: θ(m−n) = {X ∈ DerUi (−logE); X(fi∗) ≡ 0 for all i ≤ n} Notice that the strict transform (or analytic strict transform) ωi of ω has first integrals in Ri , which implies that ωi ∩ DerUi (−logEi ) ⊂ θ(m−n). Now, by Lemma 2.5, given a point qi in Ui there exists a monomial coordinate system (u, w) centered at qi and n-monomial first integrals uB = (uβ1 , . . . , uβn ) of θ(m−n).Oqi where B is of maximal rank. Since ωi ∩ DerUi (−logEi ) ⊂ θ(m−n), it is clear that the monomials uB = (uβ1 , . . . , uβn ) are also first integrals of ωi , which finishes the proof.

4 4.1

Dropping the Invariant Basic Normal Form

Lemma 4.1 (Dealing with infinite invariant). Let (M, θ, R, E) be an adapted monomial foliated sub-ring sheaf and p a point where the invariant ν(p, θ, R) is infinite. Then, there exists a θ-admissible collection of local blowings-up τi : (Mi , θi , Ri , Ei ) → (M, θ, R, E) such that, for every point qi in the pre-image of p, the invariant ν(qi , θi , Ri ) is finite. Proof. Let {X1 , . . . , Xd } be a monomial system of generators of θ and (u, w) a monomial coordinate system at p. We prove the Lemma by strong induction on the number of singular vector-fields in {X1 , . . . , Xd }. Base Step: Suppose that all vector-fields Xi are regular, i.e. that there are zero singular vector-fields on {X1 , . . . , Xd }. By the definition of monomial coordinate system, this implies that (u, w) = (u1 , . . . , um−d , wm−d+1 , . . . , wm ) and we can assume that Xj = ∂wk with k = m − d + j. So, let us consider the Taylor expansion of Ti over p: X Ti = wλ Ti,λ (u) λ

where Ti,0 is equivalent to zero (otherwise, it would be a first integral of θ). Now, consider the ideal I generated by the functions: {Ti,λ (u); ∀ λ and i} Since I is θ-invariant, by Theorem 2.11 there exists a sequence of θ-invariant blowings-up: e R, e , θ, e E) e → (U, θ, R, E) τ : (U

that principalize I, where U is an open neighborhood of p where I is well-defined. Since the blowings-up are all θ invariant, for each point q in the pre-image of 15

p there exists a coordinate system (x, w) such that τ ∗ (Xi ) = ∂wk and I ∗ is generated by a monomial xβ . In particular, let (i0 , λ0 ) be an index such that Ti∗0 ,λ0 = xβ . Thus: Ti∗0 =

X λ

wλ Ti0 ,λ (u)∗



= xβ wλ0 U +

X

λ6=λ0



wλ Tei0 ,λ (x)

where U is a unit. Notice also that λ0 6= 0 (because Ti,0 ≡ 0 for all i). So, it is e R) e is finite and smaller or equal to kλ0 k. clear that the invariant ν(q, θ,

Induction Step: Suppose, by strong induction, that the Lemma is true if there are l0 vector-fields in {X1 , . . . , Xd } which are singular with l0 < l. We assume that there are l vector-fields over {X1 , ..., Xd } that are singular. So, we can rename this set as {Y1 , . . . , Yl , Zl+1 , . . . Zd }, where the vector-fields Yi are all singular and Zi are regular vector-fields. By the definition of monomial coordinate system, we have (u, w) = (u1 , . . . , um−d+l , wm−d+l+1 , . . . , wm ) and, apart from re-indexing, we can assume that: P • Yj = αi,j uk ∂uk for coefficients αi,j ∈ Q; • Zj = ∂wl with l = m − d + j.

So, let us consider the Taylor expansion of Ti over p: X Ti = wλ Ti,λ (u) λ

Now, notice that, given any monomial uγ : Yj (uγ ) = Kj,γ uγ where Kj,γ is a constant in Q. Let Kγ denote the vector (K1,γ , . . . , Kl,γ ). In this case, we have a notion of eigenvector associated to the vector-fields Yj : X Ti,λ (u) = Ti,λ,K (u) K

where all monomials uγ in the expansion of Ti,λ,K are such that Kγ = K. So, we can write: X X Ti = wλ Ti,λ,K (u) λ

K

Now, let I be the ideal generated by the functions: {Ti,λ,K ; ∀ i, λ and K} Since I is clearly θ-invariant, by Theorem 2.11 there exists a sequence of θinvariant blowings-up: e R, e , θ, e E) e → (U, θ, R, E) τ : (U 16

that principalize I, where U is an open neighborhood of p where I is well-defined. Since the blowings-up are all θ-invariant, for each point q in the pre-image of p there exists a coordinate system (x, w) such that τ ∗ (Zj ) = ∂wk and I ∗ is e q generated by a monomial xβ . In particular, the number of generators of θ.O which are singular must be smaller or equal than l. If the number of singular generators of q is strictly smaller than l, we can apply the strong induction hypothesis to obtain a θ-admissible collection of local blowings-up e R, eq , θeq , R eq , E eq ) → (U e , θ, e E) e τq : (U

where the invariant decreases to a finite value in the pre-image of a neighborhood of q. e q . In parSo, let us assume that there exists l singular vector-fields in θ.O ticular, it is clear that these vector-fields should be generated by Yj∗ (since Zj∗ are regular). Moreover, there exists an index (i0 , λ0 , K0 ) such that Ti∗0 ,λ0 ,K0 is a generator of I ∗ , i.e Ti∗0 ,λ0 ,K0 = xβ W , where W is a unit. Then: Ti∗0 =

X λ

wλ



X

Ti0 ,λ,K (u)∗

K



= xβ wλ0 W +

X



K6=K0

Tei0 ,λ0 ,K  +

X

λ6=λ0



wλ Tei0 ,λ (x)

We claim that all functions Tei0 ,λ0 ,K with K 6= K0 are not unities, which implies that X Tei0 ,λ0 ,K W+ K6=K0

e R) e is smaller or equal than kλ0 k. Indeed, let is a unit and the invariant ν(q, θ, us assume by absurd that Tei0 ,λ0 ,K is a unit for some K 6= K0 . In one hand, this implies that: Ti∗0 ,λ0 ,K = xβ V where V is a unit. By another hand, since K 6= K0 , there exists j0 such that the j0 entry of K and K0 are different. Furthermore, for any function H: Yj (H) = Kj H =⇒ Yj∗ (H ∗ ) = Kj H ∗ And, in particular: Yj∗0 (Ti∗0 ,λ0 ,K0 ) = Yj∗0 (xβ W ) =< K0 , ej0 > xβ W and Yj∗0 (Ti∗0 ,λ0 ,K ) = Yj∗0 (xβ V ) =< K, ej0 > xβ V which implies that:   Yj∗0 (W ) and = x < K0 , ej0 > + W   Yj∗ (V ) Yj∗0 (xβ ) = xβ < K, ej0 > + 0 V

Yj∗0 (xβ )

β

17

But, since Yj∗0 is singular: < K0 , ej0 > +

Yj∗ (V ) Yj∗0 (W ) 6=< K, ej0 > + 0 W V

which is clearly a contradiction. Thus, the invariant is finite in an open neighborhood of q, which concludes the Lemma. Lemma 4.2 (Basic normal form). Let p be a point of M where the invariant ν = ν(p, θ, R) is finite and bigger than one, i.e 1 < ν < ∞. Then, there exists a system of generators (f1 , . . . , fn ) of R and a monomial coordinate system (u, v, w) of p such that the functions Ti are given by: T1 = v ν U +

ν−2 X

a1,j (u, w)v j where U is an unity, and

j=0

Ti = v T¯i + ν

(4.1)

ν−1 X

ai,j (u, w)v

j

j=0

and the vector-field ∂v belongs to θ.Op . This Normal Form is called a Basic Normal Form. Proof. Since the invariant is finite, it is clear that there exists a coordinate system (u, v, w) of p such that the vector-field ∂v belongs to θ.Op and for and any set of generators (f1 , . . . , fn ) of R, apart from re-indexing, the function ∂vνν T1 is a unit. Furthermore, by the implicit function Theorem, there is a change of e = (u, V (u, v, w), w) such that ∂veν−1 e ≡ 0. Thus: coordinates (e u, e v , w) u, 0, w) ν−1 T1 (e T1 = veν U +

Ti = veν T¯i +

ν−2 X j=0

ν−1 X

e where U is an unity, and vej a1,j (e u, w)

j=0

e vej ai,j (e u, w)

e = u and w e = w, we have that ∂v = U ∂ve for some unit U . This Finally, since u clearly implies that ∂ve is contained in θ.Op , which proves the Lemma.

4.2

Preparation

Definition 4.3 (Prepared normal form). We say that (M, θ, R, E) satisfies the Prepared Normal Form at a point p, if the invariant ν(p, θ, R) is finite, and there exists a system of generators (f1 , . . . , fn ) of R and monomial coordinate systems (u, v, w) of p such that the functions Ti are given by: T1 = v ν U +

ν−2 X

v j ur1,j b1,j (u, w) + b1,0 (u, w)

j=1

Ti = v T¯i + ν

ν−1 X

(4.2) j

v u

ri,j

bi,j (u, w) + bi,0 (u, w)

j=1

where: 18

• U is an unity and T¯i are general functions; • For 0 < j < d the functions bi,j are either units or zero. Whenever they are units, the respective exponent ri,j is non-zero; • Either bi,0 = 0 for all i, or there exists an index i0 such that: bi0 ,0 (u, w) = uβ w1ǫ where the monomial uβ divides bi,0 for all i = 1, . . . , n and ǫ ∈ {0, 1}. Proposition 4.4 (Preparation). Let p be a point of M where the invariant ν = ν(p, θ, R) is finite and bigger than one, i.e 1 < ν < ∞. Furthermore, suppose that Theorem 3.7 is valid for any monomial foliated sub-ring sheaves (N, ω, S, F ) with dimN < dimM . Then, there exists a θ-admissible local sequence of local blowings-up τi : (Mi , θi , Ri , Ei ) → (M, θ, R, E) such that the foliated ideal sheaf (Mi , θi , Ri , Ei ) satisfies the Prepared Normal Form at every point qi in the preimage of p. Furthermore, ν(qi , θi , Ri ) ≤ ν(p, θ, R). Remark 4.5 (Triviality of the hypothesis). Notice that, if dimM = 1, the inductive hypothesis (Theorem 3.7 is valid for any monomial foliated sub-ring sheaves (N, ω, S, F ) with dimN < dimM = 1) is trivially true. Proof. By Lemma 4.2, the analytic d-foliated sub-ring sheaf (M, θ, R, E) satisfies the Basic Normal Form at p, i.e. there exists a system of generators (f1 , . . . , fn ) of R and a monomial coordinate system (u, v, w) of p such that the functions Ti are given by (4.1) and the vector-field ∂v belongs to θ.Op . The main idea of the proof is to modify the coefficients ai,j without changing the v-coordinate. This is obtained through two steps, where all blowings-up are not only θ-invariant, but also ∂v -invariant. First Step: Let us perform a θ-admissible collection of local blowings-up to get all necessary conditions over the coefficients ai,j with j > 0. Indeed, let π : M0 → N be the projection map given by π(u, v, w) = (u, w), where M0 is a small enough neighborhood of p, and let J be the principal ideal sheaf generated by the product of all non-zero ai,j with j > 0. Then, it is clear that there exist a d − 1 foliated ideal sheaf (N, ω, J , F ) such that: • The singular distribution θ is generated by the set {∂v , π ∗ ω}; • The inverse image of F is equal to E ∩ M0 . Now, by Theorem 2.12 there exists a ω-admissible collection of local blowings-up σi : (Ni , ωi , Ji , Fi ) → (N, ω, J , F ) such that the ideal sheaf Ji is monomial i.e. σi∗ J is a principal ideal sheaf with support contained in Fi . It is clear that we can extend σi to blowings-up at M0 by taking the product of the centers of τi by the v-coordinate: τi (1) : (Mi (1), θi (1), Ri (1), Ei (1)) → (M0 , θ0 , R0 , E0 ) 19

where all centers have SNC with the exceptional divisor and are invariant by the v-coordinate i.e. all centers are ∂v -invariant. Moreover, since all centers of σi are ω-admissible, we conclude that all centers of τi (1) are θ-admissible. Now, consider a point qi in the pre-image of p by τi (1) and let (u(1), v (1), w(1)) be a coordinate system at qi such that τi (1)∗ v = v (1). Since the pull-back (τi (1) ◦ π)∗ J is a principal ideal sheaf, we conclude that:

T1 = v (1)ν U +

ν−2 X

v (1)j u(1)r1,j (1) b1,j + b1,0 where U is an unity, and

ν−1 X

v (1)j u(1)ri,j (1) bi,j + bi,0

j=1

Ti = v (1)ν T¯i +

j=1

where: i) The functions bi,j are either zero or units for j > 0; ii) The monomials u(1)r1,j (1) have support in the exceptional divisor Ei (1) Notice also that ∂v(1) clearly belongs to θi .Oqi and, in particular, that ν(qi , θi , Ri ) ≤ ν(p, θ, R). Second Step: We now perform a θ-admissible collection of local blowings-up to get all necessary conditions over the coefficients bi,0 . Indeed, at each point qi in the pre-image of p, apart from taking smaller varieties Mi (1), there exists a projection map π : Mi (1) → Ni (1) given by π(u(1), v (1), w(1)) = (u(1), w(1)). Then, it is clear that there exist a d−1 foliated ideal sub-ring sheaf (Ni (1), ωi (1), Si (1), Fi (1)) such that: • The singular distribution θi (1) is generated by the set {∂v(1) , π ∗ ωi (1)}; • The inverse image of Fi (1) is equal to Ei (1); • The sub-ring Si (1) is generated by the restriction of all functions in Ri (1) to {v = 0}, i.e Si (i) is generated by fk |{v(1)=0} = gk + uδ bk,0 where we recall that gi are first integrals of θi (1) and, consequently, of ωi (1) (see equation 3.1). Notice that if all of the functions bi,0 = 0 we are done. Otherwise, the foliated sub-ring sheaf (Ni (1), ωi (1), Si (1), Fi (1)) is not trivial and, since dimNi (1) < dimMi (1), we can apply Theorem 3.7 to (Ni (1), ωi (1), Si (1), Fi (1)) in order to obtain a ω (1)-admissible collection of local blowings-up σi,j (2) : (Ni,j (2), ωi,j (2), Si,j (2), Fi,j (2)) → (Ni (1), ωi (1), Si (1), Fi (1)) such that, the invariant ν calculated for (Ni,j (2), ωi,j (2), Si,j (2), Fi,j (2)) is zero or one at every point. Furthermore, by Lemma 3.6, at each point in the pre-image of qi , there exists a coordinate (u(2), w(2)) and an index k0 such that:   σi,j (2)∗ gk0 + uδ bk0 ,0 = hk0 + uβ w1ǫ (4.3)   σi,j (2)∗ gk + uδ bk,0 = hk + uβebk,0 20

where hk is a first integral of ωi,j (2) for all k, uβ w1ǫ is not a first integral of ωi,j (2) and ǫ ∈ {0, 1}. It is clear that we can extend σi,j (2) to blowings-up at Mi (2) by taking the product of the centers of τi,j (2) by the v-coordinate: τi,j (2) : (Mi,j (2), θi,j (2), Ri,j (2), Ei,j (2)) → (Mi (1), θi (1), Ri (1), Ei (1)) where all centers have SNC with the exceptional divisor and are invariant by the v-coordinate i.e. all centers are ∂v -invariant. Moreover, since all centers of σi,j (2) are ω-admissible, we conclude that all centers of τi,j (2) are θ-admissible. Now, consider a point qi,j in the pre-image of qi and let (u(2), v (2), w(2)) be a monomial coordinate system of qi,j such that τi,j (2)∗ v (1) = v (2). So, by equation (4.3)   τi,j (2)∗ gk0 + uδ bk0 ,0 = hk0 + uβ w1ǫ   τi,j (2)∗ gk + uδ bk,0 = hk + uβebk,0

where hk is a first integral of θi,j (2) for all k, uβ w1ǫ is not a first integral of θi,j (2) and ǫ ∈ {0, 1}. Furthermore, since all blowings-up have SNC with the exceptional divisor, we conclude that: T1 = v (2)ν U +

ν−2 X

v (2)j u(2)r1,j (2) c1,j + c1,0 where U is an unity, and

ν−1 X

v (2)j u(2)ri,j (2) ci,j + ci,0

j=1

Ti = v (2)ν T¯i +

j=1

where: i) The functions ci,j are either zero or units for j > 0 (this follows from [i] of the First Step); ii) The monomials u(2)r1,j (2) have support in the exceptional divisor Ei,j (2) (this follows from [ii] of the Second Step); iii) We have that ck0 = uγ w1ǫ ck = uγ ebk,0

where ǫ ∈ {0, 1} and γ is equal to the multi-index β minus the multi-index that corresponds to the pull-back of uδ . To finish, notice that ∂v(2) clearly belongs to θi,j (2).Oqi and, in particular, that ν(qi,j , θi,j , Ri,j ) ≤ ν(p, θ, R).

4.3

Combinatorial Blowings-up

Definition 4.6 M , we say that respect to E) if of the divisor E

(Sequence of combinatorial blowings-up). Given a divisor E in f → M is a sequence of combinatorial blowings-up (with τ :M τ is a composition of blowings-up with centers that are strata and its total transforms. 21

Consider a monomial foliated manifold (M, θ, E) and suppose that (u, v, w) is a globally defined monomial coordinate system centered at a point p, where the vector-field ∂v belongs to θ. We remark that, by Lemma 2.5 there exists a collection of monomials uB = (uβ1 , . . . , uβm−d ) such that a vector field X oin DerM (−logE) belongs to θ if, and only if, X(uβi ) ≡ 0 for all i. e E) f, θ, e → We now consider a sequence of combinatorial blowings-up τ : (M (M, θ, E) with respect to the declared exceptional divisor F = {u1 · · · uk ·v = 0}. Notice that such a sequence is θ-admissible and, by Theorem 2.7, θe is monof by affine charts with a coordinate system mial. Moreover, we can cover M (x, w) satisfying: aj,l+1 a uj = x1 j,1 · · · xl+2 α

l+1 1 v = xα 1 · · · xl+1

(4.4)

wi = wi that we denote, to simplify notation, by: (u, v, w) = (xA , w) = (xA , xα , w)   A where A is a (k + 1)-square matrix given by: α   a1,1 . . . a1,k+1  ..  and α = α . . . .. A =  ... 1 . .  ak,1

αl+1

. . . ak,k+1



Notice that (x, w) is clearly a monomial coordinate system since (by Lemma 2.1) τ ∗ uB = xBA e Now, let q be another point in this affine which is a system of first integrals of θ. chart contained in the pre-image of p (recall that p is the origin of the original coordinate system). Then, apart from re-indexing, (x1 , . . . , xt , yt+1 + γt+1 , . . . , yl+1 + γl+1 , w), is a coordinate system centered at q, and γj 6= 0 for all the γj . We can also assume that t 6= 0, otherwise q would be outside the exceptional divisor (and, thus, outside the inverse image of p). In this case, we have a decomposition of the matrix A   A1 A2 A= α1 α2 where A1 is a k × t matrix, A2 is a k × (k + 1 − t) matrix, α1 is a 1 × t matrix and α2 is a 1 × (k − t + 1) matrix. We remark that, since q is a point on the e there exists at least one ui such that τ ∗ ui (q) = 0, which exceptional divisor E, clearly implies that A1 has to be a non-zero matrix. Lemma 4.7 (Claim 1). Assume A1 has maximal rank. Then, there exists a monomial coordinate system (x, y, z, w) = (x1 , . . . , xt , yt+1 , . . . , yl , z, w) centered at q such that e )Λ u = xA1 (y − γ (4.5) e) v = xα1 (z − γ w=w

22

where γ ej 6= 0 for all j and the matrix Λ = (λi,j ) of exponents has maximal rank, e q ). Moreover, if uξ is not with λi,j ∈ Q (in particular, ∂z is contained in θ.O e a first integral of θ.Op , then its total transform uξ = xξ U , where U is a unit, satisfies one of the following: e e • Either the monomial xξ is not a first integral of θ;

e q such that ∂y U is a unit. • Or, there exists a regular vector-field ∂yi ∈ θ.O i

Lemma 4.8 (Claim 2). Assume that A1 does not have maximal rank. Then, there exists a monomial coordinate system (x, y, w) = (x1 , . . . , xt , yt+1 , . . . , yl+1 , w) centered at q such that u = xA1 (y − λ)Λ v = xα1 w=w

(4.6)

where γej 6= 0 for all j and the matrix Λ = (λi,j ) of exponents has maximal rank and α1 doesn’t belong to the span of the rows of A1 . Moreover, if uξ is not e a first integral of θ.Op , then its total transform uξ = xξ U , where U is a unit, satisfies one of the following: e e • Either the monomial xξ is not a first integral of θ;

e q such that ∂y U is a unit. • Or, there exists a regular vector-field ∂yi ∈ θ.O i

Proof of Lemma 4.7. By hypothesis, apart from re-indexing of the uj ’s, we can write  ′  ′ A2 A1 , , A2 = A1 = ′′ ′′ A2 A1 ′

′

′

where det(A1 ) 6= 0, and A1 and A2 have the same height. So, we can write equations (4.4) in the compact form ′

′

u′ = xA1 (y − γ)A2 ′′

′′

u′′ = xA1 (y − γ)A2 v = xα1 (y − γ)α2 First change of coordinates:

′ −1

x(1) = x · (y − γ)(A1 ) y(1) = y

′

A2

After this change of coordinates we get (using Lemma 2.1) ′

u′ = x(1)A1 ′′

u′′ = x(1)A1 (y(1) − γ)Λ(1) v = x(1)α1 (y(1) − γ)λ(1)

23

(4.7)



  ′′ ′′ ′ ′  Λ(1) A2 − A1 (A1 )−1 A2 and the square matrix L := := has determinant ′ ′ λ(1) α2 − α1 (A1 )−1 A2 different from zero because the full matrix of exponents in (4.7) is obtained from A by a sequence of column elementary transformations. Notice also that the ′ ′ entries of (A1 )−1 A2 are rational numbers, not necessarily integers. Second change of coordinates: After reindexing the yi (1) −γi we can assume that the elements of the diagonal of L are different from zero. Thus, we consider y(2) − γ (2) = (y(1) − γ)Λ(1) (z (2) − γl+1 (2)) = (y(1) − γ)λ(1) x(2) = x(1)

(4.8)

so to get ′

u′ = x(2)A1 ′′

u′′ = x(2)A1 (y(2) − γ (2))Id v = x(2)α1 (z (2) − γl+1 (2)) where Id is the identity matrix. Third change of coordinates: We need to guarantee that the coordinate system is monomial. Thus, let us now recall the first integrals uB . Notice that the matrix B can be written as:   B = B1 B2

where B2 is a k × (k − t) matrix. With this notation, by Lemma 2.1: uB = x(2)BA1 (y(2) − γ (2))B2 = x(2)C1 (y(2) − γ (2))C2

where C1 = BA1 is a non-zero matrix (since B and A1 are of maximal rank) and C2 = B2 . Now, we perform a change of coordinates similar with the one given in Lemma 2.6 in order to obtain a monomial coordinate system. To that end, consider:  ′ ′  C1 C2 C= ′′ ′′ C1 C2  ′ ′ C1 where C1 = and the rank of C1 is maximal and equal to the rank of C1 . ′′ C1 So, there exists a change of coordinates (x(3), y(3), z (3), w(3)) where z (3) = z (2), such that: uB = x(3)D1 (y(3) − γ (3))D2 where:

 D = D1

 ′  D1 D2 = ′′ D1

 ′ ′  C1 D2 = ′′ ′′ D2 C1

 0 ∆

where ∆ is a maximal rank matrix of rational numbers. This implies that the ′ ′′ collection (x(3)D1 , x(3)D1 (y(3) − γ (3))∆ ) is a collection of first integrals of θ.Oq . Forth change of coordinates: Let r be the rank of ∆. Then, apart from reordering the y(3) coordinates, there exists a coordinate system (x(4), y(4), v(4), z (4), 24

w(4)) where x(4) = x(3), z (4) = z (3), w(4) = w(3) and v(4) = (yt+r+1 (3), . . . , yl+1 (3)) such that: (y(3) − γ (3))∆ = y(4) − γ y (4) where y(4) = (yt+1 (4), . . . , yr (4)), which implies that the monomial functions ′

(x(4)D1 , y(4)) are first integrals of θ.Oq , which guarantees that the coordinate system is monomial. Furthermore, since z (2) = z (4) we finally conclude that: u = x(4)A1 (y(4) − γ y (4))Λy (4) (v(4) − γ v (4))Λv (4) v = x(4)α1 (y(4) − γ y (4))λy (4) (v(4) − γ v (4))λv (4) (z (4) − γγl+1 (4) ) w=w where Λ(4) = [Λy (4), Λv (4)] is a maximal rank matrix of rational numbers. Fifth change of coordinates: It is now clear that we only need a change in the z (4) so that z (5) − γγl+1 (5) = (y(4) − γ y (4))λy (4) (v(4) − γ v (4))λv (4) (z (4) − γγl+1 (4) ) which clearly does not change the fact that the coordinate system is monomial. This is the coordinate system of the enunciate of the Lemma. Now, let uξ be a monomial which is not a first integral of θ.Op , i.e., the multiindex ξ doesn’t belong to the span of the rows of B. In this case, let us notice that: uξ = x(5)ξA1 (y(5) − γ y (5))ξΛy (5) (v(5) − γ v (5))ξΛv (5) Now, we claim that either ξA1 is a linearly independent vector with the span of the rows of D ′1 or ξΛv (5) 6= 0, which concludes the Lemma. Indeed, let us suppose, by contraction that ξA1 is spanned by the rows of D′1 and ξΛv (5) = 0. In this case, it is clear that the vector ξ[A1 , Λ(5)] is spanned by the rows of D, which clearly contradicts the fact that ξ is not in the span of the rows of B. This finishes the proof. Proof of Lemma 4.8. We have uB = xBA1 (y − γ)BA2 u = xA1 (y − γ)A2 v = xα1 (y − γ)α2 , Notice that, since A is of maximal rank but A1 does not have maximal rank, α1 doesn’t belong to the span of the rows of A1 . Thus, it does not belong to the span of the rows of BA1 . First change of coordinates: There exists a coordinate system (x(1), y(1), w(1)) where y(1) = y and w(1) = w such that uB = x(1)C1 (y(1) − γ)C2 u = x(1)A1 (y(1) − γ)Λ(1) v

= x(1)α1 25

   BA1 C1 and Λ(1) are matrices of maximal rank. = where C = C2 C2 

Second change of coordinates: We need to guarantee that the coordinate system is monomial. To that end, consider:  ′ ′  C1 C2 C= ′′ ′′ C1 C2 ′  ′ C1 and the rank of C1 is maximal and equal to the rank of C1 . where C1 = ′′ C1 ′ Since α1 does not belong to the span of the rows of C1 , there exists a change of coordinates (x(2), y(2), w(2)) where v = x(2)α1 , such that:



uB = x(2)D1 (y(2) − γ (2))D2 u = x(2)A1 (y(2) − γ (2))Λ(2) v = x(2)α1 where Λ(2) is a maximal rank matrix of rational numbers and  ′  ′  ′    D1 D2 C1 0 = D = D1 D2 = ′′ ′′ ′′ D1 D2 C1 ∆ where ∆ is a maximal rank matrix of rational numbers. This implies that the ′ ′′ collection (x(2)D1 , x(2)D1 (y(2) − γ (2))∆ ) is a collection of first integrals of θ.Oq . ′ Since D1 has rank equal to D1 , we conclude that: ′

(x(2)D1 , (y(2) − γ (2))∆ ) is another collection of first integrals of θ.Oq . Third change of coordinates: Since ∆ is of maximal rank, there exists a coordinate system (x(3), y(3), z(3), w(3)) where x(3) = x(2) and w(3) = w(2) such that: (y(2) − γ (2))∆ = y(3) − γ (3) which finally implies that the monomial functions ′

(x(3)D1 , y(3)) are fisrt integrals of θ.Oq . This implies that this coordinate system is monomial. Furthermore, since x(2)α1 is independent of the y(2) coordinate, we finally conclude that: u = x(3)A1 (y(3) − γ 1 (3))Λ1 (3) (z(3) − γ 2 (3))Λ2 (3) v = x(3)α1 w = w(3)   where Λ = Λ1 Λ2 is a maximal rank matrix of rational numbers and γ (3) = (γ 1 (3), γ 2 (3)) is a vector where no entry is zero. This proves that the coordinate system is monomial. 26

Now, let uξ be a monomial which is not a first integral of θ.Op , i.e., the multiindex ξ doesn’t belong to the span of the rows of B. In this case, let us notice that: uξ = x(3)ξA1 (y(3) − γ 1 (3))ξΛ1 (3) (z(3) − γ 2 (3))ξΛ2 (3) Now, we claim that either ξA1 is a linearly independent vector with the span of the rows of D ′1 or ξΛ2 (3) 6= 0, which concludes the Lemma. Indeed, let us suppose, by contraction that ξA1 is spanned by the rows of D′1 and ξΛv (3) = 0. In this case, it is clear that the vector ξ[A1 , Λ(3)] is spanned by the rows of D, which clearly contradicts the fact that ξ is not in the span of the rows of B. This finishes the proof.

4.4

Dropping the invariant from Prepared Normal Form

Lemma 4.9. Let (M, θ, R, E) be an analytic foliated sub ring sheaf that satisfies the Prepared Normal Form at a point p where the invariant ν = ν(p, θ, R) is finite and bigger than one, i.e 1 < ν < ∞. Then, for a small enough neighborhood M0 of p, there exists a sequence of θ-admissible blowings-up τ : (Mr , θr , Rr , Er ) → (M0 , θ0 , R0 , E0 ) such that, for all point q in the pre-image of p, the invariant ν(q, θr , Rr ) is strictly smaller than the initial invariant ν(p, θ, R). Proof. By hypothesis, there exists a local coordinate system (u, v, w) that satisfies the Prepared Normal Form at p with ν = νp (θ, R), i.e. that satisfies equations (4.2). Since θ is monomial, by Lemma 2.5, there exists m − d monomials uB = (uβ1 , . . . , uβm−d ), such that θ.Op = {X ∈ Derp (−logE); X(uβi ) ≡ 0 for all i} Let us now consider the ideal J generated by: v ν , and {v j uri,j bi,j }1≤j 0. Notice that we include only the monomial uβ and not uβ w1ǫ in the ideal. Now, consider a sequence of blowings-up: τ : (Mr , θr , Rr , Er ) → (M0 , θ0 , R0 , E0 ) that principalize J , where M0 is any fixed open neighborhood of p where J is well-defined. Since J is generated by monomials in the variables u and v, this sequence can be chosen to be combinatorial with respect to the divisor F := {u1 · · · ul v = 0} (see Definition 4.6). Furthermore, we know that the sequence τ is θ-admissible. Now, let q be a point of Mr in the pre-image of p. We claim that ν(q, θr , Rr ) < ν(p, θ, R), which is enough to conclude the Lemma. Indeed, since τ is a sequence of combinatorial blowings-up in respect to the divisor F , the point q satisfies the hypothesis of either Lemma 4.7 or 4.8. Thus, we have two cases to consider:

27

Case 1: We assume we are in conditions of Lemma 4.7. There exists a monomial system of coordinates (x, y, z, w) = (x1 , . . . , xt , yt+1 , . . . , yl , z, w) centered at q such that e )Λ u = xA1 (y − γ (4.9) el+1 ) v = xα1 (z − γ w=w

where e γj 6= 0 for all j and the matrix Λ = (λi,j ) of exponents has maximal rank, e q (this follows from the above with λi,j ∈ Q. In particular, ∂z is contained in θ.O coordinate change). So, after blowing-up we have the following expressions: el+1 )ν + τ ∗ T1 = U xSν (z − γ el+1 )ν + τ ∗ Ti = T˜i xSν (z − γ

ν−1 X =1

ν−1 X j=1

el+1 )i c1,j (x, y, w) + xS0 c1,0 (x, y, w) xS1,j (z − γ γl+1 )j ci,j (x, y, w) + xS0 ci,0 (x, y, w) xSi,j (z − e

(4.10)

where: e (x, y, w) + xα1 Ω(x, y, z, w), where • The function U is a unit of the form U e (x, y, w) is a unit and α1 6= 0 (because q is in the pre-image of p); U

• For j > 0 the functions ci,j are either zero or units (that don’t depend on z); • The term xS0 ci,0 is the pullback of bi,0 . In particular, either ci,0 = 0 for ci0 ,0 where e ci0 ,0 is a unit. all i, or the i0 -term xS0 ci0 ,0 is equal to xS0 w2ǫ e

We consider three cases depending on which generator of I pulls back to be a generator of the pull-back of I ∗ : Case 1.1:[The pull back of v ν generates J ∗ , i.e. Sν = min{Sν , S0 , Si,j }] In this case, by equation (4.10), we have:  h ez + U ee γl+1 ν + xα1 Ω2 z ν−1 + τ ∗ T1 = xSν U + terms where the exponent of z is < ν − 1]

e z+U e νe where α1 is a non-zero matrix and Ω2 = [z+e γl+1 ν]Ω. Since U γl+1 +xα1 Ω2 is a unit and the vector-field ∂z belongs to θr , it is clear that ν(q, θr , Rr ) ≤ ν −1. Case 1.2:[There is a maximum 0 < j1 < d such that the pull back of uri1 ,j1 v i generates J ∗ for some i1 , i.e Si1 ,j1 = min{Sν , S0 , Si,j }, Sν > Si1 ,j1 and Si,j > Si1 ,j1 for j > j1 ]. In this case, by equation (4.10), we have:   jX 1 −1 el+1 )j ci1 ,j + Ω(x, y, z, w) xSi1 ,j −Si1 ,j1 (z − γ el+1 )j1 ci1 ,j1 + τ ∗ Ti1 = xSi1 ,j1 (z − γ j=0

where Ω(0, y, z, w) ≡ 0. Since ci1 ,j1 is an unit and the vector-field ∂z belongs to θr , it is clear that ν(q, θr , Rr ) ≤ j1 < ν.

28

Case 1.3:[The pull-back of uβ is the only generator of J ∗ i.e. S0 = min{Sν , S0 , Si,j } and Sν > S0 , Si,j > S0 ] In this case, we recall that there exists i0 such that xS0 ci0 ,0 = xS0 w2ǫ W where W is a unit. We consider two cases depending on ǫ: Case 1.3a, ǫ = 1: Then τ ∗ Ti0 = xS0 [w2 W + Ω(x, y, z, w)] where W is a unit and Ω(0, y, z, w) ≡ 0. Since the vector-field ∂w2 clearly belongs to θr .Oq , we conclude that that ν(q, θr , Rr ) ≤ 1 < ν. Case 1.3b, ǫ = 0: In this case, notice that the monomial uβ+δ is not a first integral of θ.Op . Thus, Lemma 4.7 guarantees that the total transform uβ+δ = ef f = W (y − γ)δΛ is a unit, satisfies one of the following: xS0 +δ W , where W e e q , which implies that • Either xS0 +δ is not a first integral of θ.O e

τ ∗ [uδ Ti0 ] = xS0 +δ U

for some unit U . We conclude that that ν(q, θr , Rr ) = 0 < ν; e q such that ∂y W f is a unit. • Or, there exists a regular vector-field ∂yi ∈ θ.O i In particular: i h ef e f f (0) + Ω(x, y, z, w) −W (0) + xS0 +δ W τ ∗ [uδ Ti0 ] = xS0 +δ W e

where Ω(0, y, z, w) ≡ 0 and the monomial xS0 +δ W (0) is a first integral of e q . We conclude that that ν(q, θr , Rr ) ≤ 1 < ν. θ.O

Case 2: We assume we are in conditions of Lemma 4.8. There exists a monomial system of coordinates (x, y, w) = (x1 , . . . , xt , yt+1 , . . . , yl+1 , w) centered at q such that u = xA1 (y − λ)Λ (4.11) v = xα1 w=w where γej 6= 0 for all j and the matrix Λ = (λi,j ) of exponents has maximal rank and α1 doesn’t belong to the span of the rows of A1 . So, after blowing-up we have the following expressions: τ ∗ T1 = U xSν + τ Ti = T˜i x ∗

Sν

+

ν−1 X

=1 ν−1 X

xS1,j c1,j (x, y, w) + xS0 c1,0 (x, y, w) (4.12) Si,j

x

S0

ci,j (x, y, w) + x ci,0 (x, y, w)

j=1

where: • The function U is a unit and, for j > 0, the functions ci,j are either zero or units (that don’t depend on z); 29

• The term xS0 ci,0 is the pullback of bi,0 . In particular, either ci,0 = 0 for ci0 ,0 where e ci0 ,0 is a unit; all i, or the i0 -term xS0 ci0 ,0 is equal to xS0 w2ǫ e • We remark that:

Sν = να1 Si,j = jα1 + ri,j A1 , for i, j = 0, . . . , ν − 1. So, for a fixed i, each Sν and Si,j is a sum of an element of the span of the rows of A1 and a different multiple of the α1 . Since α1 is linearly independent with the rows of A1 , this means that the exponents Sν and Si,j are all distinct. Therefore, for each i fixed, all of the multi-indexes Si,j must be different. We consider three cases depending on which generator of I pulls back to be a generator of the pull-back of I ∗ : Case 2.1:[The pull back of v ν generates J ∗ , i.e. Sν = min{Sν , S0 , Si,j }, Sν < S0 and Sν < Si,j for all (i, j)] In this case, from equation (4.12), we have: h i e + Ω(x, y, z, w) τ ∗ [uδ T1 ] = xSν +δ U

e = U (y − γ)δΛ is a unit and Ω(0, y, z, w) ≡ 0. where δe = δA1 , the function U e Since xSν +δ is not a first integral of θe (which is clear from that fact that α1 doesn’t belong to the span of the rows of A1 ), we conclude that ν(q, θr , Rr ) = 0 < ν. (Must talk about the δ). Case 2.2:[There is a maximum 0 < j1 < ν such that the pull back of uri1 ,j1 v i is a generator of J ∗ for some i1 , i.e Si1 ,j1 = min{Sν , S0 , Si,j }, Sν > Si1 ,j1 and Si1 ,j > Si1 ,j1 for all j]. In this case, from equation (4.12), we have: e

ci1 ,j1 + Ω(x, y, z, w)] τ ∗ [uδ Ti1 ] = xSi1 ,j1 +δ [e where δe = δA1 , the function e ci1 ,j1 = ci1 ,j1 (y−γ)δΛ is a unit and Ω(0, y, z, w) ≡ e 0. Since xSi1 ,j1 +δ is not a first integral of θe (which is clear from that fact that α1 doesn’t belong to the span of the rows of A1 ), we conclude that ν(q, θr , Rr ) = 0 < ν. (Must talk about the δ). Case 2.3:[The pull-back of uβ is the generator of J ∗ i.e. S0 = min{Sν , S0 , Si,j } and Sν > S0 , Si,j > S0 ] In this case, we recall that there exists i0 such that xS0 ci0 ,0 = xS0 w2ǫ W where W is a unit. We consider two cases depending on ǫ: Case 2.3a, ǫ = 1: Then τ ∗ Ti0 = xS0 [w2 W + Ω(x, y, z, w)] where W is a unit and Ω(0, y, z, w) ≡ 0. Since the vector-field ∂w2 clearly belongs to θr .Oq , we conclude that that ν(q, θr , Rr ) ≤ 1 < ν. Case 2.3b, ǫ = 0: In this case, notice that the monomial uβ+δ is not a first integral of θ.Op . Thus, Lemma 4.8 guarantees that the total transform uβ+δ = ef f = W (y − γ)δΛ is a unit, satisfies one of the following: , where W xS0 +δ W 30

e e q , which implies that • Either xS0 +δ is not a first integral of θ.O e

τ ∗ [uδ Ti0 ] = xS0 +δ U

for some unit U . We conclude that that ν(q, θr , Rr ) = 0 < ν; e q such that ∂y W f is a unit. • Or, there exists a regular vector-field ∂yi ∈ θ.O i In particular: h i ef e f f (0) + Ω(x, y, z, w) −W (0) + xS0 +δ W τ ∗ [uδ Ti0 ] = xS0 +δ W e

where Ω(0, y, z, w) ≡ 0 and the monomial xS0 +δ W (0) is a first integral of e q . We conclude that that ν(q, θr , Rr ) ≤ 1 < ν. θ.O

4.5

Proof of Theorem 3.7

We suppose that we are in the hypothesis of Theorem 3.7. If ν = ν(p, θ, R) is infinite, then we apply Lemma 4.1 to obtain a θ-admissible collection of local blowings-up τi : (Mi , θi , Ri , Ei ) → (M, θ, R, E) such that, for every point qi in the pre-image of p, the invariant ν(qi , θi , Ri ) is finite. So, let us assume that the invariant ν := ν(p, θ, R) is finite. Then, by proposition 4.4, there exists a θ-admissible collection of local blowings-up σi : (Mi , θi , Ri , Ei ) → (M, θ, R, E) such that: at every point qi in the pre-image of p by σi , the dfoliated sub-ring sheaf (Mi , θi , Ri , Ei ) satisfies the Prepared Normal Form at qi with ν(qi , θi , Ri ) ≤ ν. Now, by Lemma 4.9 and compacity of the pre-image of p, there exists a θadmissible collection of local blowings-up σi,j : (Mi,j , θi,j , Ri,j , Ei,j ) → (Mi , θi , Ri , Ei ) such that: at every point qi,j in the pre-image of p by σi ◦ σi,j , the invariant ν(qi,j , θi,j , Ri,j ) is strictly smaller than the initial invariant ν. Thus, taking the finite collection of morphisms τi,j := σi ◦ σi,j , we obtain the necessary sequence of local blowings-up.

Acknowledgments I would like to express my gratitude to Professor Panazzolo, Professor Bierstone and Doctor Vera-Pacheco for the useful discussions on the subject.

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