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Anais da Academia Brasileira de Ciências
versão impressa ISSN 0001-3765versão On-line ISSN 1678-2690
An. Acad. Bras. Ciênc. vol.81 no.4 Rio de Janeiro dez. 2009
http://dx.doi.org/10.1590/S0001-37652009000400002
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
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 v_{p} (A), v_{p} (B) (note that v_{p} () 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 Sep_{p} () the set of separatrices of through p.
Let π_{1} : M_{1} → (M, p) be the blow-up of M with center at p and let F_{1}be the exceptional divisor . The blow-up π_{1} is called non-dicritical if E_{1} is invariant by the strict transform of by π_{1}; otherwise, E_{1} is generically transversal to and we say that π_{1} 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 S_{1} is the strict transform of S by π_{1} and p_{1}=S_{1} ∩ E_{1}. Moreover, we have that
The Camacho-Sad index I_{p} (, 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 ω_{j}are homogeneous polynomials of degree j. Then the projective class [ ω_{v}_{p}]of the v_{p}-jet of ωis well defined. We call the initial part of at p.
Let us write ω_{j} =A_{j}dx+B_{j}dy with and consider the homogeneous polynomial of degree v_{p}+1 given by P_{vp}+_{1}=xA_{vp}+yB_{vp}. 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 P_{vp}+_{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):f_{ij} ≠ 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} : M_{1} → (M, p) is the blow-up of p with and denotes the strict transform of C by π _{1 hT}, thenfor any q e E_{1}, 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 v_{p} (C) of C at p is equal to v +1. Therefore, we can write and with f_{i}, A_{i} and B_{i} being homogeneous polynomials of degree i.
Consider π _{1} :M_{1} → (M, p) the blow-up of p and denote by the exceptional divisor. Let us prove that
at each point q ∈ E_{1}.Then the result follows using similar arguments.
If q ∈ E_{1}is a non-singular point of , then . Since the singular points of and in E_{1}coincide, we also have that . Consequently, we only have to prove equality (3) at the singular points q_{1}, q_{2}, ..., q_{t} of in E_{1}. We can assume that all the singular points belong to the first chart of E_{1} and we take coordinates (x', y') with E_{1} =(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 q_{l} is given by (0, a_{l}) 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 (x_{1}, y_{1}) coordinates centered at q_{l} with x_{l }=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, r_{l}) belongs to Δ ( ω_{1} ; x_{i}, y_{i}). 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 f_{i} = 0 being a reduced equation of C_{i} and λ_{i} ≠ 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 separatrices, 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(y^{3} -x^{11}) =0 and 11(-x^{10} +yx^{7})dx + 3(y^{2} -x^{8})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}: M_{1} → (M, p) of p and let q_{1}, q_{2},... , q_{t}be the singular points of in . We can assume, without loss of generality, that all the points q_{i} belong to the first chart of E_{1}. Take (x', y') coordinates in the first chart of E_{1} with π _{1 }(x', y') = (x', x' 'y') and E_{1} = (x' = 0). Then, the foliations and are given by and respectively, where
Assume that the point q_{i} is given by (0, a_{i}) in the coordinates (x', y') and put . Thus, by Lemma 1 and equation (5), we have that
where k_{1} and k_{2} 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 = k_{1}/k_{2}. 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).
REFERENCES
CAMACHO C AND SAD P. 1982. Invariant Varieties Through Singularities of Holomorphic Vector Fields. Ann of Math 115: 579-595. [ Links ]
CAMACHO C, LINS NETO A AND SAD P. 1984. Topological Invariants and Equidesingularisation for Holomorphic Vector Fields. J Differential Geom 20: 143-174. [ Links ]
CORRAL N. 2003. Sur la topologie des courbes polaires de certains feuilletages singuliers. Ann Inst Fourier 53: 787-814. [ Links ]
ROUILLÉ P. 1999. Théorème de Merle: cas des 1-formes de type courbes généralisées. Bol Soc Bras Mat 30: 293-314. [ Links ]
Correspondence to:
Nuria Corral
E-mail: nuria.corral@unican.es
Manuscript received on August 29, 2008; accepted for publication on April 4, 2009; presented by Aron Simis