Abstracts
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.
singular holomorphic foliations; complex dynamics
Este trabalho fornece uma condição necessária e suficiente a fim de que duas folheações singulares curva generalizada admitam mesma redução de singularidades e mesmo índice de Camacho-Sad em cada ponto infinitamente vizinho.
folheações holomorfas singulares; dinâmica complexa
MATHEMATICAL SCIENCES
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
Correspondence to Correspondence to: Nuria Corral E-mail: nuria.corral@unican.es
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.
RESUMO
Este trabalho fornece uma condição necessária e suficiente a fim de que duas folheações singulares curva generalizada admitam mesma redução de singularidades e mesmo índice de Camacho-Sad em cada ponto infinitamente vizinho.
Palavras-chave: folheações holomorfas singulares, dinâmica complexa.
1 INTRODUCTION
The germs of holomorphic foliations over (, 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 introduce 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 and
defined in a neighbourhood of a point p ∈ M. Let
,
∈
be 1-forms defining
and
respectively in a neighbourhood of p. It is clear that if the n-jets of
and
coincide for n big enough, then the foliations
and
will share all the properties mentioned above. But this condition is not necessary as we show with examples.
2 LOCAL INVARIANTS
Let M be an analytic complex manifold of dimension two and let be a singular holomorphic foliation on M. In a neighbourhood of any point p ∈M, the foliation
is defined by a 1-form
with A, B ∈and gcd( A, B) = 1. The multiplicity
of
at p is the minimum of the multiplicities vp (A), vp (B) (note that vp () is the vanishing order at p). The point p is called a singular point of
if Vp(
) > 1.
Let S be a germ of an irreducible analytic curve at p. We say that S is a separatrix of through p if f divides ω ^ df, where f=0 is a reduced equation of S. Denote by Sepp (
) the set of separatrices of
through p.
Let π1 : M1→ (M, p) be the blow-up of M with center at p and let F1be the exceptional divisor . The blow-up π1 is called non-dicritical if E1 is invariant by the strict transform
of
by π1; otherwise, E1 is generically transversal to
and we say that π1 is dicritical.
Consider now a non-singular invariant curve S=(y=0) of the foliation through p. The multiplicity of
at p along S is given by
(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
where S1 is the strict transform of S by π1 and p1=S1∩ E1. Moreover, we have that
The Camacho-Sad index Ip (, S) of
relative to S at p is given by
where A(x, y)=ya(x, y) (see (Camacho and Sad 1982)).
It is important to notice that, if are 1-forms defining
in a neighbourhood of p, then
where
is a unit. In particular, this implies that the jets of order
of ω and ω' coincide up to multiplication by a constant. We can write
where the coefficients of ωjare homogeneous polynomials of degree j. Then the projective class [ ωvp]of the vp-jet of ωis well defined. We call
the initial part of
at p.
Let us write ωj =Ajdx+Bjdy with and consider the homogeneous polynomial of degree vp+1 given by Pvp+1=xAvp+yBvp. The tangent cone
of
at p is given by
If then
is a finite union of lines and we denote by
the projective class of the polynomial Pvp+1. Notice that two foliations with different initial parts, may have the same tangent cone. Moreover, the blow-up π 1 is non-dicritical if and only if
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 of a function at p can be written as a convergent power series
We denote Δ ( f;x, y)={(i, j):fij ≠ 0}. The Newton polygon
is the convex hull of
If C is the germ of curve at p defined by f=0, then
Consider now a germ of foliation
given by ω =A(x, y)dx+B(x, y)dy in a neighbourhood of p. The Newton polygon
of
is the convex hull of
where Δ ( ω; x, y)= Δ (xA; x, y) ∪ Δ (yB;x, y).
Finally, let us recall the desingularization process of a foliation. We say that p is a simple singularity of if there are coordinates (x, y) centered at p so that
is defined by a 1-form of the type
with μ ≠ 0 and If λ =0, the singularity is called a saddle-node. A reduction of singularities of
is a morphism
n:M' → (M, p)
composition of a finite number of blow-ups of points such that the strict transform of
by π satisfies that
- each irreducible component of the exceptional divisor D= π -1(p) is either invariant by
or transversal to
;
- all the singular points of
are simple and do not belong to a dicritical component of the exceptional divisor D.
The minimal morphism , in the sense that it cannot be factorized by another morphism with the above properties, is called the minimal reduction ofsingularities of
The centers of the blow-ups of any reduction of singularities of
are called infinitely near points of
'. In particular, all the centers of the blow-ups to obtain
and the final singularities in
are infinitely near points of
. This notion extends the well known notion of infinitely near points of a curve.
3 GENERALIZED CURVE FOLIATIONS
Let be the space of singular holomorphic foliations of (M, p). We denote by
the subspace of
composed by generalized curve foliations, that is, foliations without saddle-node singularities in their reduction of singularities (see (Camacho et al. 1984)). Given a germ
of analytic curve in (M, p), we denote by
the subspace of
composed by foliations whose curve of separatrices is C, and we put
. Notice that foliations in
are non-dicritical. Particular elements of
are the foliations given by df=0, where f=0 is a reduced equation of C; let
be one of such foliations. It is known (see (Camacho et al. 1984, Rouillé 1999)) that, for an element
, we have that:
(i) The minimal reduction of singularities of
and C coincide.
(ii) The multiplicities satisfy
, where
.
(iii) For any local coordinates (x, y) in a neighbourhood of p, the Newton polygons
and
coincide.
Moreover, we have the following result
LEMMA 1. Let be a morphism composition of a finite number of punctual blow-ups. Let
be the exceptional divisor and take
an irreducible component with
. Then, for any foliation
, the strict transforms
and
satisfy the following
-
;
-
If
are local coordinates at
, then
;
-
.
REMARK 1. If π 1 : M1→ (M, p) is the blow-up of p with and
denotes the strict transform of C by π 1 hT, thenfor any q e E1, where (·)q denotes the intersection multiplicity at q.
PROOF OF LEMMA 1. Assertions 1 and 2 are a direct consequence of the above properties (ii) and (iii) applied at each infinitely near point of . Let us prove assertion 3.
Let (x, y) be coordinates in a neighbourhood of p and take a 1-form ω= A(x, y)dx + B(x, y)dy defining in a neighbourhood of p. Consider a reduced equation f = 0 of C and let
be the foliation defined by df = 0. If we denote
, then the multiplicity vp (C) of C at p is equal to v +1. Therefore, we can write
and
with fi, Ai and Bi being homogeneous polynomials of degree i.
Consider π 1 :M1→ (M, p) the blow-up of p and denote by the exceptional divisor. Let us prove that
at each point q ∈ E1.Then the result follows using similar arguments.
If q ∈ E1is a non-singular point of , then
. Since the singular points of
and
in E1coincide, we also have that
. Consequently, we only have to prove equality (3) at the singular points q1, q2, ..., qt of
in E1. We can assume that all the singular points belong to the first chart of E1 and we take coordinates (x', y') with E1 =(x' = 0) and π 1 (x', y') = (x', x'y'). In these coordinates, the foliation
is given by
Let us write with
. We can assume that the point ql is given by (0, al) in the coordinates (x', y'), for l = 1, ..., t .In particular, we have that
.
In the coordinates (x', y'), the foliation is given by
with
By definition, we have that .
Take (x1, y1) coordinates centered at ql with xl =x' and yi =y'-at. From expression (4), we deduce that
and
Then the equality of the Newton polygons and
implies that the point (1, rl) belongs to Δ ( ω1 ; xi, yi). Taking into account that
we deduce that
Thus for each l = 1, ..., t, and the following equality
gives the result.
COROLLARY 1. Given two foliations and
of
, the polynomials
and
coincide at each infinitely near point p of C.
4 LOGARITHMIC FOLIATIONS
Given a germ of plane curve , a logarithmic foliation
in
is given by a closed meromorphic 1-form η= 0 where
with fi = 0 being a reduced equation of Ci and λi ≠ 0. Notice that if η as above defines a logarithmic foliation in , then the foliation given by η + α = 0, with α holomorphic and dα = 0, is also logarithmic and belongs to
. In fact, we can write
for suitable equations
of C i. Thus, logarithmic foliations in
are characterized by their exponent vector
. We denote by
one of these foliations. Observe that not all the foliations defined by a 1-form of the type (*) belong to
since they could be dicritical foliations.
Recall that a logarithmic foliation is a logarithmic model of a foliation
if they have the same separatric
es, the same reduction of singularities and the same Camacho-Sad indices at the final singularities 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 foliation
, we denote by
the exponent vector of the logarithmic model of
. We denote by
the set of foliations
such that
. Thus, the set
is equal to
Notice that if and only if the foliation given by
is non-dicritical. The goal of this article is to study the properties which characterize the foliations in one of such sets
Consider now a meromorphic 1-form ηof the type (*) and assume that the foliation given by η =0 belongs to . Thus all the foliations defined by η + α= 0, with αholomorphic, belong to
but in general they are not logarithmic. However, it is not true that all foliations of
are defined by a 1-form of the type η + α =0 with a holomorphic.
5 INFINITESIMAL INITIAL PART
Given a non-dicritical foliation , we define the infinitesimal initial part
of
to be the family
where q varies among the infinitely near points of
. We wonder under what conditions two foliations
and
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 of
and
having the same n-jet at the point p, with n being big enough. However, this condition is not necessary as shown by d(y3 -x11) =0 and 11(-x10 +yx7)dx + 3(y2 -x8)dy = 0. Then, the result is
THEOREM 1. Let and
be two foliations in
. The foliations
and
have the same infinitesimal initial part if and only if
.
PROOF. Take and
two foliations in
and assume that
. Consider the minimal reduction of singularities
of C (notice that
). By hypothesis, the initial parts
and
coincide at each point
. In particular, this implies that the Camacho-Sad indices
and
coincide for a component
through q and consequently
.
Reciprocally, take a foliation and let us show that
, where
is a logarithmic foliation in
Put
and assume that
and
are given by the holomorphic 1-forms
and
respectively, where
with being homogeneous polynomials of degree i. Consider the blow-up π 1: M1→ (M, p) of p and let q1, q2,... , qtbe the singular points of
in
. We can assume, without loss of generality, that all the points qi belong to the first chart of E1. Take (x', y') coordinates in the first chart of E1 with π 1 (x', y') = (x', x' 'y') and E1 = (x' = 0). Then, the foliations
and
are given by
and
respectively, where
Assume that the point qi is given by (0, ai) in the coordinates (x', y') and put . Thus, by Lemma 1 and equation (5), we have that
where k1 and k2 are non-zero constants. The second inequality of (5) implies that
with and
being polynomials of degree t - 1. Now, since
is a logarithmic model of
, we have the equality of the Camacho-Sad indices
Computation of the indices gives that
The equalities of the Camacho-Sad indices imply that with k = k1/k2. Then, we have that
. Moreover, since
, we get that
. It follows that
and
have the same initial parts
and
at p.
The same arguments prove the equality at each infinitely near point q of C, and the result is straightforward.
ACKNOWLEDGMENTS
The author was partially supported by the research projects MTM2007-66262 (Ministerio de Educación y Ciencia), MTM2006-15338-C02-02 (Ministerio de Educación y Ciencia), VA059A07 (Junta de Castilla y León) and PGIDITI06PXIB377128PR (Xunta de Galicia).
Manuscript received on August 29, 2008; accepted for publication on April 4, 2009; presented by Aron Simis
- CAMACHO C AND SAD P. 1982. Invariant Varieties Through Singularities of Holomorphic Vector Fields. Ann of Math 115: 579-595.
- CAMACHO C, LINS NETO A AND SAD P. 1984. Topological Invariants and Equidesingularisation for Holomorphic 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.
- ROUILLÉ P. 1999. Théorème de Merle: cas des 1-formes de type courbes généralisées. Bol Soc Bras Mat 30: 293-314.
Correspondence to:
Publication Dates
-
Publication in this collection
16 Nov 2009 -
Date of issue
Dec 2009
History
-
Accepted
04 Apr 2009 -
Received
29 Aug 2008