Infinitesimal initial part of a singular foliation

(Annals of the Brazilian Academy of Sciences) ISSN 0001-3765

www.scielo.br/aabc

Infinitesimal initial part of a singular foliation

NURIA CORRAL

Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria Avda. de los Castros s/n, 39005 Santander, Spain

Manuscript received on August 29, 2008; accepted for publication on April 4, 2009; presented byARONSIMIS

ABSTRACT

This work provides a necessary and sufficient condition to assure that two generalized curve singular foliations have the same reduction of singularities and same Camacho-Sad indices at each infinitely near point.

Key words: singular holomorphic foliations, complex dynamics.

1 INTRODUCTION

The germs of holomorphic foliations over(C2,0) are dynamical objects much more general than curves or even levels of meromorphic functions. In (Camacho et al. 1984), Camacho, Lins Neto and Sad intro-duce a class of foliations that share the reduction of singularities with their curve of separatrices. A more accurate approximation to a foliation (that assures coherent linear holonomies) is to compare them with levels of multivalued functions (logarithmic foliations); this has been done in (Corral 2003) by adding a control of the Camacho-Sad indices. In this paper we show that the necessary and sufficient condition for two foliations to share reduction of singularities, separatrices and Camacho-Sad indices is to have the same initial parts up to blow-up.

Let M be an analytic complex manifold of dimension two and consider two germs of foliations F andG defined in a neighbourhood of a point p ∈ M. LetωF, ωG ∈ 1_M,p be 1-forms definingF and G respectively in a neighbourhood of p. It is clear that if the n-jets ofωF and ωG coincide for n big

enough, then the foliationsF andG will share all the properties mentioned above. But this condition is not necessary as we show with examples.

2 LOCAL INVARIANTS

LetM be an analytic complex manifold of dimension two and letF be a singular holomorphic foliation onM. In a neighbourhood of any point p∈ M, the foliationF is defined by a 1-form

ω= Ad x+Bd y

with A,B ∈OM,pand gcd(A,B)=1. The multiplicityνp(F)ofFat pis the minimum of the

multiplic-itiesνp(A), νp(B)(note thatνp( )is the vanishing order at p). The point pis called asingular pointofF

ifνp(F)≥1.

Let S be a germ of an irreducible analytic curve at p. We say that S is aseparatrix ofF through

pif f dividesω∧d f, where f =0 is a reduced equation ofS. Denote by Sep_p(F)the set of separatrices ofF throughp.

Letπ1 : M1 → (M,p)be the blow-up of M with center at pand let E1 be the exceptional divisor π₁−1(p). The blow-up π1 is called non-dicritical if E1 is invariant by the strict transform π1∗F of F byπ1; otherwise, E1is generically transversal toπ1∗F and we say thatπ1isdicritical.

Consider now a non-singular invariant curve S =(y =0)of the foliationF through p. The multi-plicity ofF at p along Sis given by

μp(F,S)=ordx(B(x,0))

(see (Camacho et al. 1984)). Notice that it coincides with the multiplicity of the restriction ξ|S = B(x,0)∂/∂x of the vector field ξ = B(x,y)∂/∂x − A(x,y)∂/∂y to the curve S. Its behaviour under

blow-up is given by

μp1(π

∗

1F,S1)=μp(F,S)−νp(F)+1 if π1is non-dicritical, (1)

μp1(π

∗

1F,S1)=μp(F,S)−νp(F) if π1is dicritical, (2)

whereS1is the strict transform ofSbyπ1andp1= S1∩E1. Moreover, we have that X

q∈E1

μq(π1∗F,E1)=νp(F)+1.

TheCamacho-Sad index Ip(F,S)ofFrelative toSat p is given by

Ip(F,S)= −Resx=0a(x,0)

B(x,0)

where A(x,y)=ya(x,y)(see (Camacho and Sad 1982)).

It is important to notice that, if ω, ω′ ∈ 1_M,p are 1-forms defining F in a neighbourhood of p,

thenω=h∙ω′_{whereh ∈ O}

M,pis a unit. In particular, this implies that the jets of orderνp =νp(F)of

ωandω′coincide up to multiplication by a constant. We can writeω=P_j≥νpωj, where the coefficients

of ωj are homogeneous polynomials of degree j. Then the projective class [ωνp] of the νp-jet of ω is well defined. We callp(F)= [ωνp]the initial part of Fat p.

Let us writeωj = Ajd x +Bjd ywith Aj,Bj ∈OM,p and consider the homogeneous polynomial of

degreeνp+1 given by Pνp+1=x Aνp +y Bνp. The tangent cone T Cp(F)ofFat pis given by

Pνp+1(x,y)=0.

Let us now recall the construction of the Newton polygon of a foliation. Taking local coordinates (x,y)in a neighbourhood of p∈ M, a germ f ∈OM,p of a function at pcan be written as a convergent

power series f =P_i+j≥νp(f) fi jxiyj. We denote1(f;x,y) = {(i, j) : fi j 6=0}. TheNewton polygon

N(f;x,y)is the convex hull of1(f;x,y)+(R≥0)2. IfC is the germ of curve at pdefined by f =0, thenN(C;x,y)=N(f;x,y). Consider now a germ of foliationFgiven byω= A(x,y)d x+B(x,y)d y

in a neighbourhood ofp. TheNewton polygonN(F;x,y)ofFis the convex hull of1(ω;x,y)+(R≥0)2 where1(ω;x,y)=1(x A;x,y)∪1(y B;x,y).

Finally, let us recall the desingularization process of a foliation. We say that pis asimple singularity

ofF if there are coordinates(x,y)centered at pso thatF is defined by a 1-form of the type

λyd x−μxd y+h.o.t.

withμ 6= 0 andλ/μ 6∈ Q_>0. Ifλ = 0, the singularity is called asaddle-node. Areduction of

singula-ritiesofF is a morphism

π :M′→(M,p)

composition of a finite number of blow-ups of points such that the strict transform π∗F ofF byπ sat-isfies that

– each irreducible component of the exceptional divisor D = π−1_{(p) is either invariant by π}∗_{F or} transversal toπ∗F;

– all the singular points of π∗F are simple and do not belong to a dicritical component of the excep-tional divisorD.

The minimal morphismπF, in the sense that it cannot be factorized by another morphism with the above

properties, is called the minimal reduction of singularities of F. The centers of the blow-ups of any reduction of singularities ofF are calledinfinitely near points ofF. In particular, all the centers of the

blow-ups to obtainπF and the final singularities in π_F−1(p)are infinitely near points of F. This notion

extends the well known notion of infinitely near points of a curve.

3 GENERALIZED CURVE FOLIATIONS

LetF=F(M,p)be the space of singular holomorphic foliations of(M,p). We denote byGthe subspace ofFcomposed bygeneralized curve foliations, that is, foliations without saddle-node singularities in their

reduction of singularities (see (Camacho et al. 1984)). Given a germC = ∪r

i=1Ci of analytic curve in (M,p), we denote byFCthe subspace ofFcomposed by foliations whose curve of separatrices isC, and

we putGC = G∩FC. Notice that foliations in FC are non-dicritical. Particular elements ofGC are the

foliations given byd f = 0, where f = 0 is a reduced equation ofC; letGC be one of such foliations.

It is known (see (Camacho et al. 1984, Rouillé 1999)) that, for an elementF ∈GC, we have that:

(i) The minimal reduction of singularities ofF andC coincide.

(iii) For any local coordinates (x,y) in a neighbourhood of p, the Newton polygons N(F;x,y) and N(C;x,y)coincide.

Moreover, we have the following result

LEMMA1. Letπ :(Me,p˜) → (M,p)be a morphism composition of a finite number of punctual blow-ups. Let D = π−1(p)be the exceptional divisor and take E ⊂ D an irreducible component with p˜ ∈ E. Then, for any foliationF ∈GC, the strict transformsπ∗_{F andπ}∗_G

C satisfy the following

1. νp˜(π∗F)=νp˜(π∗GC);

2. If(x˜,y˜)are local coordinates at p, then˜ N(π∗_{F; ˜x,y˜)=N(π}∗_G

C; ˜x,y˜);

3. μp˜(π∗F,E)=μp˜(π∗GC,E).

REMARK 1. Ifπ1 : M1 → (M,p)is the blow-up of p with E1 = π1−1(p)andπ1∗C denotes the strict transform ofC byπ1, thenμq(π∗GC,E)=(π∗C∙E)qfor anyq ∈ E1, where(∙)q denotes the

intersec-tion multiplicity atq.

PROOF OF LEMMA1. Assertions 1 and 2 are a direct consequence of the above properties (ii) and (iii)

applied at each infinitely near point ofF. Let us prove assertion 3.

Let(x,y)be coordinates in a neighbourhood of p and take a 1-form ω = A(x,y)d x +B(x,y)d y

definingF in a neighbourhood of p. Consider a reduced equation f =0 ofC and letGC be the foliation

defined by d f = 0. If we denote ν = ν0(F), then the multiplicityνp(C)ofC at p is equal to ν+1.

Therefore, we can write f = P_i≥ν+1 fi, A = P_i≥ν Ai and B = P_i≥ν Bi with fi, Ai and Bi being

homogeneous polynomials of degreei.

Considerπ1 : M1 → (M,p)the blow-up of p and denote by E1 =π1−1(p)the exceptional divisor. Let us prove that

μq π1∗F,E1=μq π1∗GC,E1 (3)

at each pointq ∈ E1. Then the result follows using similar arguments.

If q ∈ E1 is a non-singular point of π1∗F, then μq(π1∗F,E1) = 0. Since the singular points of π₁∗F andπ₁∗G_C in E1 coincide, we also have that μq(π1∗GC,E1) = 0. Consequently, we only have to

prove equality (3) at the singular points q1,q2, . . . ,qt of π₁∗F in E1. We can assume that all the sin-gular points belong to the first chart of E1 and we take coordinates(x′,y′) with E1 = (x′ = 0) and π1(x′,y′)=(x′,x′y′). In these coordinates, the foliationπ₁∗G_Cis given by

(ν+1)fν+1(1,y′)+x′(. . .)

d x′+x′

∂fν+1

∂y′ (1,y ′

)+x′(. . .)

d y′=0. (4)

Let us write fν+1(1,y) = k ∙Qtl=1(y−al)rl withk ∈ C∗. We can assume that the pointql is given by (0,al)in the coordinates(x′,y′), for l=1, . . . ,t. In particular, we have thatμql(π

∗

1GC,E1)=rl.

In the coordinates(x′,y′), the foliationπ₁∗Fis given byω1=π₁∗ω/x′ν =0 with

ω1= Pν+1(1,y′)+x′(∙ ∙ ∙)

d x′+x′ Bν(1,y′)+x′(∙ ∙ ∙)

By definition, we have thatμql(π ∗

1F,E1)=ordy=al(Pν+1(1,y)).

Take (xl,yl) coordinates centered at ql with xl = x′ _{and yl = y}′ _{−al. From expression (4), we} deduce that

N(π₁∗G_C;xl,yl)⊂ {(i, j) : i ≥1}

and N(π₁∗GC;xl,yl)∩ {i=1} = {

}

{

≥

}

j

i

(1, rl)

N

(

π

∗

1

G

;

x

, y

)

Then the equality of the Newton polygons N(π₁∗G_C;xl,yl) and N(π₁∗F;xl,yl) implies that the point(1,rl)belongs to1(ω1;xl,yl). Taking into account that

1 ω1;xl,yl∩ {i =1} =1 xlPν+1(1,yl +al);xl,yl

∪1 xlylBν(1,yl+al);xl,yl

we deduce that

ordy=al Pν+1(1,y)

≥rl; ordy=al Bν(1,y)

≥rl−1. (5)

Thus μql(π ∗

1F,E1)≥μql(π ∗

1GC,E1) for each l=1, . . . ,t, and the following equality

ν+1=

t

X

l=1 μql π

∗

1F,E1=

t

X

l=1 μql π

∗

1GC,E1.

gives the result.

COROLLARY 1. Given two foliations F and G of GC, the polynomials pp(F) and pp(G) coincide at each infinitely near point p of C.

4 LOGARITHMIC FOLIATIONS

Given a germ of plane curveC = ∪r_i=1Ci in(M,p), a logarithmic foliationLinGC is given by a closed

meromorphic 1-formη=0 where

η=

r

X

i=1 λi

d fi

fi , (∗)

with fi = 0 being a reduced equation ofCi andλi 6= 0. Notice that ifηas above defines a logarithmic

foliation inG_C, then the foliation given byη+α =0, withαholomorphic anddα=0, is also logarithmic and belongs toGC. In fact, we can writeη+α = Pri=1λid fi∗/fi∗ for suitable equations fi∗ = 0 ofCi.

We denote by L_λ one of these foliations. Observe that not all the foliations defined by a 1-form of the type (∗) belong toGC since they could be dicritical foliations.

Recall that a logarithmic foliation L is alogarithmic model of a foliationF if they have the same separatrices, the same reduction of singularities and the same Camacho-Sad indices at the final singulari-ties after desingularization. The existence of logarithmic models for non-dicritical generalized curves was proved in (Corral 2003) and is unique once a reduced equation of the separatrices is fixed. Then, for each foliationF ∈ GC, we denote byλ(F)the exponent vector of the logarithmic model ofF. We denote by

GC,λthe set of foliationsF ∈GCsuch thatλ(F)=λ. Thus, the setGCis equal to

GC = ∪_λ∈Pr_C−1GC,λ.

Notice thatGC,λ 6= ∅if and only if the foliation given byPri=1λid fi/fi = 0 is non-dicritical. The goal

of this article is to study the properties which characterize the foliations in one of such setsGC,λ.

Consider now a meromorphic 1-formηof the type (∗) and assume that the foliation given byη =0 belongs toGC. Thus all the foliations defined by η+α =0, withα holomorphic, belong toGC,λbut in

general they are not logarithmic. However, it is not true that all foliations ofGC,λare defined by a 1-form

of the typeη+α =0 withαholomorphic.

5 INFINITESIMAL INITIAL PART

Given a non-dicritical foliation F ∈ F, we define the infinitesimal initial part J(F) of F to be the family{q(F)}whereq varies among the infinitely near points ofF. We wonder under what conditions

two foliations F andG have the same infinitesimal initial part. It is clear that having the same curve of separatrices and the same reduction of singularities are necessary conditions. But these conditions are not enough even if we work with generalized curve foliations. Notice also that a sufficient condition is that ofF andGhaving the samen-jet at the point p, withn being big enough. However, this condition is not

necessary as shown byd y3−x11=0 and 11 −x10+yx7d x+3 y2−x8d y =0. Then, the result is

THEOREM1. LetF andGbe two foliations in GC. The foliationsF andGhave the same infinitesimal initial part if and only if λ(F)=λ(G).

PROOF. Take F and G two foliations in GC and assume that J(F) = J(G). Consider the minimal

reduction of singularitiesπC :Me →(M,p)ofC(notice thatπC =πF =πG). By hypothesis, the initial

partsq(F)andq(G)coincide at each pointq ∈πC−1(p). In particular, this implies that the Camacho-Sad

indices Iq(π∗F,F)andIq(π∗G,F)coincide for a component F ⊂ π_C−1(p)throughq and consequently

λ(F)=λ(G).

Reciprocally, take a foliation F ∈ GC,λ and let us show that p(F) = p(L), where L = Lλ is a

logarithmic foliation in GC,λ. Put ν = νp(F) and assume that F andL are given by the holomorphic

1-formsωF =0 andωL=0 respectively, where

ωF = AF_ν (x,y)+AF_ν+1(x,y)+ ∙ ∙ ∙d x+ B_νF(x,y)+B_νF₊₁(x,y)+ ∙ ∙ ∙d y;

with A−_i ,B_i−being homogeneous polynomials of degreei. Consider the blow-upπ1: M1 → (M,p)of

p and letq1,q2, . . . ,qt be the singular points ofπ∗

1F inE1 = π1−1(p). We can assume, without loss of generality, that all the pointsqi belong to the first chart of E1. Take(x′,y′)coordinates in the first chart of E1 withπ1(x′,y′) = (x′,x′y′)and E1 = (x′ = 0). Then, the foliationsπ₁∗F andπ₁∗Lare given by ω₁F =0 andω₁L=0 respectively, where

ωF₁ = P_νF₊₁(1,y′)+x′(∙ ∙ ∙)d x′+x′(B_νF(1,y′)+x′(∙ ∙ ∙))d y′;

ωL1 = PνL+1(1,y′)+x′(∙ ∙ ∙)

d x′+x′(B_νL(1,y′)+x′(∙ ∙ ∙))d y′.

Assume that the pointqi is given by(0,ai)in the coordinates(x′_,y′_{)and putri =μ}

qi(π ∗

1F,E1). Thus, by Lemma 1 and equation (5), we have that

P_νF₊₁(1,y)=k1

t

Y

i=1

(y−ai)ri_{; P}L

ν+1(1,y)=k2

t

Y

i=1

(y−ai)ri_, (6)

wherek1andk2are non-zero constants. The second inequality of (5) implies that

B_νF(1,y)= HF(y)∙

t

Y

i=1

(y−ai)ri−1_{; B}L

ν(1,y)= H

L

(y)∙

t

Y

i=1

(y−ai)ri−1_,

with HF(y)and HL(y)being polynomials of degreet −1. Now, sinceLis a logarithmic model ofF, we have the equality of the Camacho-Sad indices

Iq_i π1∗F,E1=Iqi π ∗

1L,E1 for i =1, . . . ,t.

Computation of the indices gives that

Iqi(π ∗

1F,E1) = −Resy=ai

B_νF₊₁(1,y)

P_νF₊₁(1,y) = −

HF(ai)

k1∙Qtj=1

j6=i

(ai −aj)

Iq_i(π₁∗L,E1) = −Resy=ai

B_νL₊₁(1,y)

P_νL₊₁(1,y) = −

HL(ai)

k2∙Qtj=1

j6=i

(ai −aj)

The equalities of the Camacho-Sad indices imply that HF(y) = k HL(y) with k = k1/k2. Then, we have that B_νF(1,y) = k B_νL(1,y). Moreover, since P_νF₊₁(1,y) = k P_νL₊₁(1,y), we get that AF_ν (1,y) =

k AL_ν(1,y). It follows thatF andLhave the same initial partsp(F)andp(L)atp.

The same arguments prove the equality q(F) = q(G)at each infinitely near point q ofC, and the

result is straightforward.

ACKNOWLEDGMENTS

RESUMO

Este trabalho fornece uma condição necessária e suficiente a fim de que duas folheações singulares curva generaliza-da admitam mesma redução de singularigeneraliza-dades e mesmo índice de Camacho-Sad em cageneraliza-da ponto infinitamente vizinho.

Palavras-chave: folheações holomorfas singulares, dinâmica complexa.

REFERENCES

CAMACHOCANDSADP. 1982. Invariant Varieties Through Singularities of Holomorphic Vector Fields. Ann of Math 115: 579–595.

CAMACHO C, LINSNETOAAND SADP. 1984. Topological Invariants and Equidesingularisation for Holomor-phic Vector Fields. J Differential Geom 20: 143–174.

CORRAL N. 2003. Sur la topologie des courbes polaires de certains feuilletages singuliers. Ann Inst Fourier 53: 787–814.