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Singularities of Logarithmic Foliations Compositio Math. 142 (2006) 131–142 doi:10.1112/S0010437X05001545 Singularities of logarithmic foliations Fernando Cukierman, Marcio G. Soares and Israel Vainsencher Abstract A logarithmic 1-form on CPn canbewrittenas m m dFi ω = Fj λi = λ0F0 dF0 + ···+ λmFm dFm 0 0 Fi m with Fi =( 0 Fj)/Fi for some homogeneous polynomials Fi of degree di and constants λi ∈ C such that λidi = 0. For general Fi,λi, the singularities of ω consist of a schematic union of the codimension 2 subvarieties Fi = Fj = 0 together with, possibly, finitely many isolated points. This is the case when all Fi are smooth and in general position. In this situation, we give a formula which prescribes the number of isolated singularities. 1. Introduction and statement of result The search for numerical invariants attached to algebraic foliations goes back to Poincar´e[Poi91]. He was interested in determining bounds for the degree of curves left invariant by a polynomial vector field on C2. Recent work has treated the question by establishing relations for the number of singularities of the foliation and certain Chern numbers and then using positivity of certain bundles. For a survey of recent results, see [Bru00, CL91, Est02, Soa00]. A foliation of dimension r on a smooth variety X of dimension n is a coherent subsheaf F of the tangent sheaf TX of generic rank r, locally split in codimension 2. 1 If r = n − 1 (codimension one foliations), the foliation corresponds to a global section of ΩX ⊗L for some line bundle L. n Suppose X = CP , with homogeneous coordinates x0,...,xn. Recall Euler’s sequence, 1 ⊕n+1 ΩCPn (1) →O →O(1). A global section ω of 1 ⊕n+1 ΩCPn (d) ⊂O (d − 1) canbewrittenas n ω = Fi dxi 0 where Fi is a homogeneous polynomial of degree d − 1, subject to the condition Fixi =0 (contraction by the radial vector field on Cn+1). Received 31 July 2004, accepted in final form 3 January 2005. 2000 Mathematics Subject Classification 32S65, 14C17. Keywords: holomorphic foliations, characteristic classes, excess intersection. The authors were partially supported by CAPES-Brasil, CNPq-Brasil and SPU-Argentina. This journal is c Foundation Compositio Mathematica 2006. F. Cukierman, M. G. Soares and I. Vainsencher The degree of a codimension one foliation F, deg F, is the number of tangencies of the leaves of F with a generic one-dimensional linear subspace of CPn. A simple calculation shows that deg F = d−2 if the 1-form defining F has components Fi of degree d − 1. The form ω is integrable if ω ∧ dω =0. Integrable 1-forms make up a Zariski closed subset of P(H0(Ω1(d)). We denote by Fol(CPn; d) the space of codimension one integrable holomorphic foliations of degree d − 2ofCPn. Not much is known about the dimensions nor the number of irreducible components of Fol(CPn; d) (but see [CL96]and[CEL01]). When ω can be written as m m dFi ω = Fj λi = λ0F0 dF0 + ···+ λmFm dFm 0 0 Fi for some homogeneous polynomials Fi of degree di and λi ∈ C such that λidi =0,wesayω is m logarithmic of type d = d0,...,dm. Given positive integers d0,...,dm,setd = i=0 di and consider the hyperplane m CP(m − 1,d)={(λ0,...,λm) ∈ CP | Σdiλi =0}. Define a rational map Ψ by m 0 n Ψ n CP(m − 1,d) × P(H (CP , O(di))) −→ Fol(CP ; d) i=0 m m dFi ((λ0,...,λm), (F0,...,Fm)) → Fj λi . j=0 i=0 Fi The closure of the image of Ψ is the set Log n(d) of logarithmic foliations of type d (of degree d − 2) of CPn. Recall the following result. Theorem 1 (Calvo-Andrade [Cal94]). For fixed di and n 3, logarithmic foliations form an n irreducible component of the space of codimension one integrable holomorphic foliations of CP of degree d − 2 (with d = di). The singular scheme of the foliation defined by ω ∈ H0(Ω1(d)) is the scheme of zeros of ω.This is the closed subscheme with ideal sheaf given by the image of the co-section ω∨ :(Ω1(d))∨ →O. For ω general in H0(Ω1(d)), there are just finitely many singularities, to wit (cf. Jouanolou [Jou79, Theorem 2.3, p. 87] setting, in his notation, m = d − 1,r = n), n 1 i n +1 n−i cn(Ω (d)) = (−1) d . n CP 0 i On the other hand of course, a general ω is not integrable. Theorem 2 (Jouanolou [Jou79]). For integrable ω, the singular set must contain a codimension 2 component. It is easy to see that, for logarithmic (hence, integrable) forms ω = λ0F0 dF0 + ···+ λm Fm dFm the singular set contains the union of all codimension two subsets Fi = Fj =0,i= j. It is worth mentioning that Jouanolou describes examples of integrable 1-forms with singular schemes containing positive dimensional components of ‘wrong’ positive dimension. We found no hint as to the existence of isolated singularities for general enough foliations. 132 Singularities of logarithmic foliations Let Di be the divisor associated to Fi. We assume that the following genericity conditions hold: the Di,i=0,...,m,are smooth and in general position, (1) λi =0 ,i=0,...,m. Remark that (1) defines a Zariski open subset of m 0 n CP(m − 1,d) × P(H (CP , O(di))). i=0 Before stating our main result recall that the complete symmetric function σ,ofdegree in the variables X1,...,Xk is defined by: σ0 = 1 and, for 1, i1 ik σ(X1,...,Xk)= X1 ...Xk . i1+···+ik= We then have the following. n Theorem 3. Let F be a logarithmic foliation on CP of type d = d0,...,dm,givenby ω = λ0F0 dF0 + ···+ λmFm dFm and satisfying (1). Then the singular scheme S(F) of F can be written as a disjoint union S(F)=Z ∪ R where Z = Di ∩ Dj i<j and R is finite, consisting of n i n +1 N(n, d)= (−1) σn−i(d) i=0 i points counted with natural multiplicities. Moreover: (i) N(n, d)=0if n m and di =1for all i; m (ii) N(n, d)= n+1 if n<mand di =1for all i; (iii) N(n, d) > 0 whenever di 2 for some i. It will be shown below, see formula (8)in§ 4.3,that n+1 n (1 − h) N(n, d)=thecoefficientofh in m 0 (1 − dih) from which we deduce the following example. 1.1 Example n+1 m n m If di =1foralli then (1 − h) / 0 (1 − dih) reduces to (1 − h) /(1 − h) and we have items (i) and (ii) of the theorem. n m (i) n m.Inthiscase(1− h) /(1 − h) is a polynomial of degree n − m<nand hence the coefficient of hn vanishes, so that there are no isolated zeros. n m m−n (ii) n<m.Inthiscase(1 − h) /(1 − h) reads 1/(1 − h) and it is easily seen that the n m coefficient of h is n+1 . 133 F. Cukierman, M. G. Soares and I. Vainsencher 2. Proof of the theorem We will show that, if a point is non-isolated in S(F), then it lies in Di ∩ Dj for some i<j. Indeed, let C be an irreducible component of S(F)ofdimension1 dim C n − 2. By ampleness and general position, we may pick a point p ∈ C lying in the intersection of precisely k of the divisors Di,1 k min{n, m +1}.Letfi be a local equation for Di at p.Nearp, the foliation F is given by the 1-form m df i = f0 ···fm λi . i=0 fi Renumbering the indices we may assume p ∈ D0 ∩···∩Dk−1. The local defining equations fi =0 of the Di,fori =0,...,k − 1, are part of a regular system of parameters, i.e. df 0,...,dfk−1 are linearly independent at p.Writeg = fk ···fm.Sincep ∈ Dj, k j m, we may assumeg ˜ vanishes nowhere around p and write as k−1 m k−1 df j df i df j = f0 ···fk−1g˜ λj + λi = f0 ···fk−1g˜ λj + η , j i j j=0 f i=k f j=0 f m where η = i=k λi(df i/fi) is a holomorphic closed form near p.Sinceη is closed, by the formal Poincar´e lemma it is exact near p,sayη = dξ.Setϑ = /g .ThenF is defined around p by k−1 k−1 df j d(exp[ξ/λ0]f0) df j ϑ = f0 ···fk−1 λj + dξ = f0 ···fk−1 λ0 + λj . j=0 fj exp[ξ/λ0]f0 j=1 fj Set z0 =exp[ξ/λ0]f0 and z1 = f1,...,zk−1 = fk−1.Sinceu =exp[ξ/λ0] is a unit, we also have that z0,...,zk−1 are part of a regular system of parameters at p.Nowϑ canbewrittenas k−1 z0 dz0 dzj ϑ = z1 ···zk−1 λ0 + λj . u z0 j=1 zj Thus, F is defined around p by the 1–form k−1 k−1 dz0 dzj ϑ = z0z1 ···zk−1 λ0 + λj = λjz0 ···zj ···zk−1 dzj. (2) z0 j=1 zj j=0 If k =1,(2) shows that the foliation is defined near p by dz0 and then is non-singular at p. Hence, we necessarily have k 2. Note that the ideal of the scheme of zeros of ϑ (aswellasofω)nearp is generated by the k monomials z0 ···zj ···zk−1 with 0 j k − 1.
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