Dicritical logarithmic foliations

Publ. Mat. 50 (2006), 87–102 DICRITICAL LOGARITHMIC FOLIATIONS Felipe Cano and Nuria Corral Abstract We show the existence of weak logarithmic models for any (dicritical or not) holomorphic foliation Fof (C2 ,0) without saddlenodes in its desingularization. The models are written in terms of a representative set of separatrices, whose equisingularity types are controlled by the Milnor number of the foliation. 1. Introduction In this paper we show the existence of weak logarithmic models for any holomorphic foliation Fof (C2,0) without saddle-nodes in its desingularization (these foliations are called generalized curves in [2]). If F is non-dicritical, the existence of a logarithmic model has been proved in [5] and it gives an approximation of Fby a (logarithmic) foliation with linear projective holonomies given by the linear part of the projective holonomies of F. The first difference in the dicritical case is the choice of the separatrices. Actually, the logarithmic model should be of the form η= 0 for η= r X i=1 λi dfi fi that exhibits the separatrices fi= 0. But there are several possible ways to do this. For instance the foliation d((y−x2)/(y+x2)) = 0 can also be written down as d(y/x2) = 0. We choose a representative set S of separatrices of Fto get explicitly the closed logarithmic form η. In Theorem 5 it is proved that the possible equisingularity types of Sare bounded by a function of the Milnor number of F. Note also that the 2000 Mathematics Subject Classification. Primary: 32S65. Key words. Singular foliations, dicritical foliations, logarithmic forms, pencil of curves, singularities. Partially supported by the Spanish research project MTM 2004-07978 and by the Junta de Castilla y Le´on (VA123/04). 88 F. Cano, N. Corral results in this paper would provide a normalized presentation of a pencil of curves, in terms of multivalued functions. To have a weak logarithmic model Lfor Froughly speaking means that the reduction of the singularities of Lis longer than the one of F and coincides with it outside a “escape set” of non-singular points for F placed at dicritical components (domination of foliations). The escape set can be chosen to be a singleton at such dicritical components and this determines the residues λi. We prove (Theorem 14) that this concept depends in fact only on the C-divisor (S, λ) and not on the particular logarithmic foliation we take. This is key in the proof of the existence of weak logarithmic models (Theorems 3 and 16) since it serves to “glue” the models by induction. We end the paper by a list of examples showing the difficulties in order to get a logarithmic model without escape set. 2. Domination of foliations. Representative sets of separatrices We consider holomorphic singular foliations Fdefined on ambient spaces Mof dimension two, equipped with a normal crossing divisor D⊂M. As usual, we say that a component Fof Dis dicritical if it is generically transversal to F; if Fis invariant, we say that it is non-dicritical. The reduction of singularities of Fat P∈Mcan be done relatively to Das follows. Denote Sing(F, D) the set of points Q∈Mwhere Fand Ddo not have normal crossings; that is, either Q∈Sing For Q6∈ Sing Fbut the only invariant curve of Fat Qdoes not have normal crossings with D. In the case D=∅, note that Sing(F,∅) = Sing F. We say that P∈Sing(F, D) is simple for (F, D) iff P∈D, it is a simple singularity in the sense of Seidenberg [10], [12] and each component of Dthrough Pis non-dicritical. Then we have Theorem (Reduction of singularities [3], [12]).Let (M, P)be the germ of Mat P. There is a finite composition of blowing-ups π= π(F,D,P ):M′−→ (M, P)such that, if F′is the transform of Fby π and D′=π−1(D∪ {P}), then any Q∈Sing(F′, D′)is simple and πis minimal in the sense that it cannot be factorized by another morphism with the above property. We call π(F,D,P )the minimal reduction of singularities of (F, D) at P. To get it it’s enough to blow-up successively the non-simple points in Sing(F, D) (the processus stops by similar arguments to the ones Dicritical Logarithmic Foliations 89 in [12]). Note that if D=∅the reduction π(F,∅,P )is maybe longer than Seidenberg’s reduction; the main difference appears because of the tangencies with the created dicritical components. If Fis non-dicritical (each succesive blowing-up is non-dicritical) and D=∅, we get essentially the same reduction of singularities as Seidenberg. Even if Fis non-dicritical, the dicritical components of the starting divisor Dmay produce a longer reduction of singularities; unfortunately, we have to keep them in this paper in order to assure coherence in our induction statements. Notation. Since F, D and its reduction of singularities will be fixed, in all this paper we keep the notations π=π(F,D,P ),E′=π−1(P), D′= π−1(D∪ {P}) and we denote F′,L′,G′the transforms of the foliations F,L,Gby π. For a germ of analytic curve Γ at P, we will denote by Γ′the strict transform of Γ by π. Also we shall keep the notation π1:M1→(M, P ) for the blowing-up of (M, P) with center Pand we will denote E1=π−1 1(P) the exceptional divisor, D1=π−1 1(D∪ {P}) and F1,L1,G1, Γ1the transforms of objects as above. Let Gbe another singular foliation on Mand assume that Pis a simple singularity for both (F, D) and (G, D). We say that Phas same linear type of singularity for (F, D) and (G, D) if IP(F, F) = IP(G, F) for any component Fof Dthrough P, where IP(F, F) is the Camacho-Sad index (see [1]). In this paper we exclude saddle-nodes at the end of the desingularization, that is, we restrict ourselves to generalized curves as defined in [2]; in this case, the indices at simple singularities correspond to the two quotients of eigenvalues. Definition 1. We say that (G, D)dominates (F, D) at Pif: (1) The components of D′are simultaneously dicritical or non-dicritical for the transforms F′and G′of F,Gby π. (2) For any non-dicritical component F′of D′we have that Sing(F′, D′)∩F′= Sing(G′, D′)∩F′ and each Q∈Sing(F′, D′) is a simple singularity for (G′, D′) with same linear type and same separatrices as for (F′, D′) . Note that in the case D=∅and Fnon-dicritical, we get that π(F,∅,P )=π(G,∅,P )and F,Ghave the same set of separatrices. Definition 2. Let Γ be a separatrix for Fat Pnot contained in D. Following [2], we say that Γ is isolated for (F, D) if the strict transform Γ′does not cut E′in a F′-dicritical component. We say that Γ is an 90 F. Cano, N. Corral F′−curvette for Fif Γ′cuts E′in a F′-dicritical component F′. A set Sof separatrices of Fat Pis representative for (F, D) if it contains the isolated separatrices, the non-dicritical components of Dat Pand the number of F′-curvettes in Sfor each dicritical component F′of E′is equal to max{0,2−vD′F′}, where the valence vD′F′is the number of components of D′that intersect F′(excluding F′). Note that if Fis non-dicritical the only representative set of separatrices for (F, D) is the set of all separatrices. In the case of a dicritical Fwe know that we have infinitely many separatrices. In order to get a logarithmic model we have to choose finitely many of them. The representative sets of separatrices give our choice. The idea is to take not too much “curvettes” to assure that the equisingularity types of the chosen set of separatrices remain bounded by the Milnor number of the foliation (Theorem 5). This paper is devoted to prove the following theorem: Theorem 3. Let Fbe a generalized curve of (C2,0) and S={Γi}r i=1 a representative set of separatrices for F. Take a local equation fi= 0 of Γifor i= 1,2,...,r. Then there are λi∈C∗,i= 1,2,...,r, such that if η= r X i=1 λi dfi fi the logarithmic foliation η= 0 dominates F. Let us explain with two examples the definition of a representative set of separatrices. Consider the foliation (Suzuki’s example, see [4]) Fdefined by ω= (2y2+x3)dx −2xydy = 0. It has only one isolated separatrix given by y2−x3= 0. Thus it is not possible to construct a logarithmic foliation written only in terms of the isolated separatrices and dominating Fsince it will be non-dicritical. To obtain a logarithmic foliation that dominates Fit is enough to consider the foliation d((y2−x3)/x2) = 0. Note that S={y2−x3= 0, x = 0}is a representative set of separatrices of F. The second example is given by the foliations Fp,q defined by pxdy −qydx = 0 with p, q ∈N. If p, q ≥2 and gcd(p, q) = 1, then the only dicritical component in the reduction of singularities of Fp,q has valence 2. The curvettes of this dicritical component are the curves yp−cxq= 0 with c∈C. However, the foliation Fp,q is given by d(yp/xq) = 0 where S={x= 0, y = 0}are the isolated separatrices and Sis a representative set of separatrices. Note also that the type of equisingularity of the curves yp−cxq= 0 is not bounded by a function of the Milnor number since µ(Fp,q) = 1. Let us give now some properties of a representative set of separatrices. Dicritical Logarithmic Foliations 91 Proposition 4. Let Fbe a generalized curve at P∈M,Da normal crossings divisor and S={Γi}r i=1 a representative set of separatrices for (F, D). Denote by CPS ⊂ E1the tangent cone of S. Then we have (1) S 6=∅, that is r≥1. (2) If Shas only one element and it is non-singular at P, then P /∈ Sing F. (3) If π1is non-dicritical for F, then CPS= Sing F1= Sing(F1, D1). (4) If π1is F-dicritical, then Sing(F1, D1)⊂CPS. (5) If π1is F-dicritical, then CPShas max{2, vD′E1}points. Proof: (1) If Fis non-dicritical, the result follows from [1]. Note also that the arguments in [1] imply that any connected component of the union of non-dicritical components of the exceptional divisor after reduction of singularities will support at least one isolated separatrix of F, which is necessarily an element of S. Thus, it remains to consider the case that all the components of the exceptional divisor are dicritical; since two such components do not intersect, there is only one of them, we are in the radial case and Shas two elements. (2) If Fis non-dicritical, the result follows from [2], since Fhas the same reduction of singularities as the set of separatrices. Moreover, if S has only one element, then Fmust be non-dicritical. To see this, note first that Fis not radial, and hence, by the above arguments, the only element of Sis an isolated separatrix of F, thus there are no “curvettes” in S. Then, there is a dicritical component F′with valence greater or equal than two. Now F′cuts at least two connected components of the union of non-dicritical components. This implies the existence of at least two elements in Sin view of the arguments in (1). (3) It is evident that CPS ⊂ Sing F1, since F1has at least two separatrices at the points in CPS: the exceptional divisor E1and the strict transform of an element of S. To see that Sing F1= Sing(F1, D1), take a point Q /∈Sing F1, then either D1=E1locally at Qor D1=E1∪F and in both cases we have the normal crossings property, since E1is invariant. It remains to show that if Q /∈CPS, then Q /∈Sing F1, but this follows from (2) since the only separatrix of F1at Qis the exceptional divisor E1. (4) Take Q∈Sing(F1, D1). Since E1is dicritical, then Qis not a simple singularity for (F1, D1) and thus the reduction of singularities continues through Q. By (1), we get an element of Spassing through Q. (5) We have that CPS=CP{E1-curvettes in S} ∪ CPD∪Sing(F1, D1). 92 F. Cano, N. Corral Moreover, since the points in Sing(F1, D1) are not simple singularities, we get that the valence vD′E1is equal to the number of points in CPD∪ Sing(F1, D1). The result follows noting that if vD′E1≥2 there are no E1-curvettes in S. Theorem 5. There is a finite set E(µ)for each integer number µ≥0 such that the following property holds. Denote by µPFthe Milnor number of F. Then the equisingularity type of Sis in E(µPF), for any generalized curve Fand a normal crossings divisor Dat P∈M, where S={Γi}r i=1 is any representative set of separatrices for (F, D). Proof: By a theorem in [6], the equisingularity types of the union S∗of isolated separatrices and non-dicritical components of Dare bounded (in the sense of the statement) by a function of the Milnor number. Now, to get the equisingularity type of a representative set of separatrices we have just to add at most one (or two arrows in the radial case) arrow to the terminal vertices of the dual graphs representing each equisingularity type for S∗. 3. Closed logarithmic forms and C-divisors Consider a non-empty finite set S={Γi}r i=1 of germs of irreducible curves at (M, P ) and take a list λ={λi}r i=1 with λi∈C∗. The pair D= (S, λ) is called a C-divisor at Pand its support is Supp D=∪r i=1Γi. Write mi=νPΓithe multiplicity of Γiat P. We say that Pis nonsingular for Dif r= 1 and m1= 1. We say that Pis a simple singularity for Dif r= 2 and m1=m2= 1, the curves Γ1and Γ2have distinct tangents and λ1/λ2/∈Q<0. More generally, these definitions can be made relative to a normal crossings divisor D⊂Mas follows. We say that Pis non-singular for (D, D) if it is non-singular for Dand Γ1 has normal crossings with Dand we say that it is a simple singularity for (D, D) if it is a simple singularity for Dand ∅ 6=D⊂Γ1∪Γ2. Let us define the transform DQ= (SQ, λQ) of Dat a point Q∈E1 under the blowing-up π1:M1→(M, P). Denote IQthe set of indices defined by the property that Q∈Γi 1for i∈IQand put λ0=Pr i=1 miλi. Then SQ={Γi 1}i∈IQ, λQ={λi}i∈IQ,if λ0= 0. SQ={Γ0 1=E1} ∪ {Γi 1}i∈IQ, λQ={λ0} ∪ {λi}i∈IQ,if λ06= 0. Note that if SQ=∅then Q /∈Supp DQ. This construction can be iterated by blowing-up at points in the support. To be precise, we say that a finite composition of blowing-ups σ:M∗→(M, P) is a D-admissible Dicritical Logarithmic Foliations 93 morphism if the centers of the blowing-ups are in the support of the corresponding transformed C-divisors. Moreover, we have a reduction of singularities as follows Proposition 6. Let D= (S, λ)be a C-divisor and Da normal crossings divisor on (M, P). There is a D-admissible morphism ˜π=π(D,D,P ):˜ M→ (M, P )such that any point Q∈Supp D˜πin the support of the transformed C-divisor is non-singular or simple for (D˜π,˜ D), where ˜ D= ˜π−1({P} ∪ D)and ˜πis minimal in the sense that it cannot be factorized by another morphism with the above property. The proof is a consequence of the reduction of singularities of curves and Euclides’ algorithm, very similar (but simpler) to the proof in [12]. Consider reduced equations fi= 0 of Γifor i= 1,2,...,r and put f=f1f2···fr. We say that a closed meromorphic 1-form ηis (S, λ)- logarithmic if η= r X i=1 λi dfi fi +α, where αis holomorphic. In particular αis closed and hence α=dh. In fact, we can write η=Pr i=1 λidf∗ i/f∗ ifor suitable local equations f∗ i= 0 of each Γi(to see this, put α=du/u, where u(P)6= 0). Note that if η is (S, λ)-logarithmic, then π∗ 1ηis (SQ, λQ)-logarithmic at Q∈E1. Definition 7. We say that D= (S, λ) is a 1-faithful C-divisor at Pif there is a D-logarithmic 1-form ηsuch that νP(fη) = m−1, where m=νPf=Pr i=1 mi. (A direct computation shows that this will be then true for any D-logarithmic 1-form). We say that Dis faithful if its transform is 1-faithful at any point of the support after any D-admissible morphism. The non-singular points and the simple singularities are persistent under blowing-up and also the C-divisor is faithful at this kind of points. Thus in order to verify the faithfulness of Dis enough to verify the 1-faithfulness at the intermediary points in the reduction of singularities. Lemma 8. Consider the C-divisor D=(S, λ)and let ηbe a D-logarithmic 1-form. Denote by Lηthe foliation η= 0. Put λ0=Pr i=1 miλi, as above. a) If λ06= 0, then π1is non-dicritical for Lηand Dis 1-faithful. b) If λ0= 0 and Dis 1-faithful, then π1is dicritical for Lη. 94 F. Cano, N. Corral Proof: First, let us recall that if a foliation Gis given at Pby ω= 0, where ω=a(x, y)dx+b(x, y)dy is a germ of holomorphic form of order t, then π1is dicritical for Gif and only if xInta+yIntb= 0, where Inta, Intbdenote the parts of degree tof a,b(here aand bare not necessarily without common factor). Now the part a) of the lemma is an immediate computation, since fη is a holomorphic 1-form that gives Lη. To prove part b), it is enough to note that the fact that Dis 1-faithful means that the initial part of fη has degree m−1 and then the condition λ0= 0 is equivalent to the dicriticalness of π1. As a consequence, if λ0= 0 and π1is non-dicritical, the divisor Dfails to be 1-faithful. Example 9. Taking S={y−x2= 0, y +x2= 0}and λ= (1,−1) we get a non 1-faithful D= (S, λ). In fact λ0= 0 and π1is non-dicritical for the foliation d((y−x2)/(y+x2)) = 0. Note that the same foliation is given by d(y/x2) = 0 and this presentation is faithful. More generally, if the curves of Shave all the same tangent and λ0= 0, we have a non-dicritical blowing-up, as we show next. Proposition 10. Assume that the curves of S={Γi}r i=1 have all the same tangent at Pand that λ0= 0. Take any (S, λ)-logarithmic form η. Then π1is non-dicritical for the logarithmic foliation Lgiven by η= 0. Proof: Put fi(x, y) = ymi+hi(x, y) with νP(hi)> mifor i= 1,2,...,r. Take coordinates (u, v) in the second chart of the blowing-up such that π1(u, v) = (uv, v). We can write fi=fi(uv, v) = vmi(1 + Fi(u, v)) with Fi(u, 0) = 0 and then we have dfi fi =mi dv v+dFi 1 + Fi , π∗ 1η= r X i=1 λi dFi 1 + Fi =A(u, v)du u+B(u, v)dv v. Let us put M= max{k:vkdivides Aand B},A(u, v) = vM˜ A(u, v) and B(u, v) = vM˜ B(u, v). To show that v= 0 is invariant by π∗ 1η= 0 is enough to verify that ˜ B(u, 0) 6= 0. Since π∗ 1ηis a closed form, we have v∂A ∂v =u∂B ∂u . If vkdivides B, it must also divide u∂B/∂u and consequently vkdivides v∂A/∂v. The hypothesis over the Fiimply that A(u, 0) = 0 and thus vkdivides A. We conclude that M= max{k:vkdivides B}and then ˜ B(u, 0) 6= 0. Dicritical Logarithmic Foliations 95 Remark 11 (Mattei).This proof corresponds to the fact that a multivalued or meromorphic function in P1 Cwith at most either one zero or one pole is constant. Corollary 12. Let D= (S, λ)be 1-faithful. Then Supp Dπ1has at least two points in the exceptional divisor E1of the first blowing-up π1. As a consequence, if Dis faithful and σ:M∗→(M, P)is any D-admissible morphism, then each component F∗of E∗=σ−1(P)has at least two points in Supp Dσ. In the next proposition we recover the relationship between nondicritical components and support components. Proposition 13. Let D= (S, λ)be a faithful C-divisor, consider a Dlogarithmic form η, and the corresponding foliation Lη. Let σ:M∗→ (M, P )be any D-admissible morphism. Then a component F∗of E∗= σ−1(P)is non-dicritical for the transform Lη σif and only if F∗is contained in the support of the transformed C-divisor Dσ. Proof: It follows just applying the properties of 1-faithfulness at each center in the sequence of blowing-ups. 4. Existence of weak logarithmic models In this section, we consider a generalized curve Fat P, a normal crossings divisor D⊂Mand a representative set of separatrices S= {Γi}r i=1 for (F, D) at P. We say that a C-divisor D= (S, λ) is a weak logarithmic model for (F, D) at P, relatively to Sif and only if we have the following properties: (1) Dis faithful. (2) π(D,D,P )=π(F,D,P ). (3) Put π=π(F,D,P ):M′→(M, P). Then any component F′of E′ is dicritical for F′if and only if F′is not contained in the support Supp Dπ. (4) Any point Q∈Sing(F′, D′) is in the support Supp Dπand the quotient of residues −λ2 Q/λ1 Qis the quotient of the eigenvalues of F′ at Q. The fact that D= (S, λ) is a weak logarithmic model for (F, D) depends only on the projective class [λ]∈Pr−1 C. Note that P∈Supp D. Moreover, the definition is compatible with the blowing-ups in the following sense. Take any point Q∈E1in the support Supp Dπ1, then Dπ1is a weak logarithmic model for (F1, D1) at Q. 102 F. Cano, N. Corral (Mexico, 1986), Lecture Notes in Math. 1345, Springer, Berlin, 1988, pp. 192–232. [10] J.-F. Mattei and R. Moussu, Holonomie et int´egrales premi`eres, Ann. Sci. ´ Ecole Norm. Sup. (4) 13(4) (1980), 469–523. [11] P. Rouill´ e, Courbes polaires et courbure, Th`ese, Univ. Bourgogne, Dijon (1996). [12] A. Seidenberg, Reduction of singularities of the differential equation A dy =B dx,Amer. J. Math. 90 (1968), 248–269. Felipe Cano: Departamento de ´ Algebra, Geometr´ıa y Topolog´ıa Universidad de Valladolid Facultad de Ciencias 47005-Valladolid Spain E-mail address:[email protected] Nuria Corral: Departamento de Matem´atica Aplicada I Universidad de Vigo Escuela U. de Ingenier´ıa T´ecnica Forestal. Campus A Xunqueira. 36005 Pontevedra Spain E-mail address:[email protected] Primera versi´o rebuda el 7 de febrer de 2005, darrera versi´o rebuda el 17 d’octubre de 2005.