Bounding the degree of solutions to Pfaff equations

Publicacions Matem`atiques, Vol. 44 (2000), 593–604 BOUNDING THE DEGREE OF SOLUTIONS TO PFAFF EQUATIONS Marco Brunella and Lu´ ıs Gustavo Mendes Abstract We study hypersurfaces of complex projective manifolds which are invariant by a foliation, or more generally which are solutions to a Pfaff equation. We bound their degree using classical results on logarithmic forms. 1. Introduction Starting with Poincar´e[Po] and Painlev´e[Pa], many mathematicians considered the following problem: given a foliation Fon CPnof degree d and a hypersurface V⊂CPninvariant by F, is it possible to bound the degree of Vby a number h(d) which depends only on d(and not on F)? In such a generality, the answer is clearly negative: for example, the curve {xp=yq}in CP2has degree max{p, q}and it is invariant by the foliation given by pydx −qxdy = 0, whose degree is one (see also [LN] for other interesting examples). However several positive results have been obtained by those authors and, recently, by [CL], [Ca], [Ba], [Br] and [So]. The philosophy behind these results is that the failure of an uniform bound h(d) is due to the existence of “bad singularities” of Vor of F. For instance, one finds the bound h(d)=d+2inCP2, provided that either Vhas only normal crossing singularities [CL], or F has only nondicritical singularities along V[Ca]. Our aim is to attract the attention of the reader to the relation between this problem and these results and some basic properties of logarithmic forms, discovered by Deligne and Bogomolov [De], [Bo]. We shall work in the context of Pfaff equations, more general than foliations. For our purposes, the simplest definition is the following: Definition 1.1. Given a complex manifold Xand a holomorphic line bundle Non X,aPfaff equation of codimension p,1≤p≤dimC(X)−1, is a nontrivial global section σof Ωp X⊗N, where Ωp Xdenotes the sheaf of holomorphic p-forms on X. 594 M. Brunella, L. Gustavo Mendes Let us consider the corresponding generalization of the notion of leaf of a foliation. Given a hypersurface V, let iVσdenote the restriction of σto V, i.e. the section of Ωp V⊗N|Vobtained by projecting σ|Vvia (Ωp X⊗N)|V→Ωp V⊗N|V. Definition 1.2. Given a Pfaff equation σ∈H0(X, Ωp X⊗N) and a hypersurface V⊂X, we shall say that Vis a solution to σif iVσ≡0. Some remarks on these definitions may be useful. Usually one requires that the zero set (σ)0of a Pfaff equation σ∈H0(X, Ωp X⊗N) has codimension ≥2, but this is not really important. Anyway, if σ∈ H0(X, Ωp X⊗N) vanishes on a hypersurface Z⊂X, then we can replace σby σ:= σ h∈H0(X, Ωp X⊗N), where N=N⊗O X(−Z) and h∈ H0(X, OX(Z)) vanishes on Z, and then σ|Z≡ 0. This division of σby h has no significant consequences, because in the study of Pfaff equations one is more interested in the saturated subsheaf of Ωp Xgenerated by σ than in σitself, and this subsheaf is unchanged by the division. A foliation Fof codimension pgives rise to a Pfaff equation of codimension p, because locally such a foliation can be seen as the kernel of a holomorphic p-form; in this case, Ncorresponds to the determinant line bundle of the rank pnormal sheaf of F. But the converse is not generally true, except when p= dimC(X)−1, because no integrability assumption is done. When the Pfaff equation arises from a foliation F, Definition 1.2 reduces to say that the hypersurface Vis saturated by the leaves of F. In order to state our result in a simple form, let us suppose now that Xis a projective manifold with Picard group Pic(X)=Z, although a more general fact, without restrictions on Pic(X), will be found in the course of the proof. Let Hbe the positive generator of Pic(X). We naturally define the degree of a holomorphic line bundle Las the integer d(L) such that L=d(L)Hin Pic(X). For a hypersurface V, the degree d(V) is defined as d(OX(V)). We recall that projective manifolds with Pic(X)=Zare quite abundant: for instance, the classical NoetherLefschetz theorem states that a generic hypersurface Xin CPnof high degree has Pic(X)=Z. We refer to Section 2 for the basic properties of logarithmic forms. Theorem. Let Xbe a complex projective manifold with Pic(X)=Z and let σ∈H0(X, Ωp X⊗N)be a Pfaff equation. Let V⊂Xbe a normal crossing hypersurface, which is solution to σ. Then d(V)≤d(N) and the inequality is strict if Vis smooth. Moreover, if d(V)=d(N)then σis given by a global closed logarithmic p-form on Xwith poles along V. Degree of Solutions to Pfaff Equations 595 When dimC(X) = 2, we recover results of [CL] and [Ba]. Let us consider the case when a Pfaff equation on CPnarises from a foliation with codimCSing(F)≥2 (and codimC(σ)0≥2). One usually defines the degree d(F) as the degree of the tangency set between the (n−p)-dimensional leaves of Fand a generic linear subspace Π ≃CPp(remark that this tangency set is a hypersurface of Π). In Lemma 3.2 we show that d(N)=d(F)+p+ 1 and we recover the result of [CL]inCP2, since our bound becomes d(V)≤d(F)+p+1. At the end of the paper we shall discuss to which extent the normal crossing hypothesis on Vmay be weakened. Acknowledgements. We thank the hospitality of the C.R.M., Barcelona, where this work was completed. The second author is supported by a grant of “Conseil R´egional de Bourgogne”, France, and he thanks the attention of P. Sad and M. G. Soares. 2. Logarithmic forms In this section we recall basic facts on logarithmic forms (see, for instance, [Sa]). Let Xbe a complex manifold and Va hypersurface with at most normal crossing singularities. A logarithmic p-form on Xwith poles along Vis a meromorphic p-form ωwith polar set (ω)∞⊂Vsuch that ωand dω have at most simple poles along V. Equivalently, if f= 0 is a local reduced equation of V, then fω and fdω are holomorphic. Obviously, this condition is equivalent to fω and df ∧ωbeing holomorphic. This definition can be localized on open sets of X; therefore we obtain a (coherent, analytic, locally free) sheaf on X, denoted Ωp X(log V). Remark that everything makes sense even if p=0orp=n= dimC(X), and Ω0 X(log V)=OX,Ω n X(log V)≃Ωn X⊗O X(V). If (z1,...,z n) is a local coordinate system around x=(0,...,0) ∈V such that Vis locally expressed by V={z1·... ·zk=0}, then n p=0 Ωp X(log V) is locally generated by holomorphic forms and {dz1 z1,...,dzk zk}. More precisely, every section ωof Ωp X(log V) can be locally written as ω=ω0+ k  j=1 ωj∧dzj zj ,(1) where ω0is a holomorphic p-form and each ωjis a local section of Ωp−1 X(log V). Remark that the exterior product of two logarithmic forms 596 M. Brunella, L. Gustavo Mendes is still a logarithmic form. We also note that the existence of a decomposition like (1) is strongly dependent on the hypothesis that Vhas only normal crossing singularities. In particular, for every j=1,...,k, we can locally decompose ωas ω=γj+ηj∧dzj zj ,(2) where γjis a local section of Ωp X(log V), ηjis a local section of Ωp−1 X(log V) and moreover both γjand ηjdo not contain Vj:= {zj=0} in their polar set. The decomposition (1), as well as (2), is not unique; however, a simple computation shows that the restriction of ηjto Vjis intrinsically defined by ω, i.e. does not depend on the involved choices. Setting Γj:= Vj∩(∪k i=1,i=jVi), which is a normal crossing hypersurface of Vj, we therefore have a well defined map ResVj:ω→ ηj|Vj, and ηj|Vj∈Ωp−1 Vj(log Γj) is called local residue of ωalong Vj. Summing on jand patching together these local constructions, we finally obtain the residue map Res: Ωp X(log V)→Ωp−1 ˆ V(log Γ), where ˆ Vis the normalization of Vand Γ ⊂ˆ Vis the normal crossing hypersurface induced by Von ˆ V. Note that the kernel of the residue map is exactly Ωp Xand that there is an exact sequence 0→Ωp X→Ωp X(log V)→Ωp−1 ˆ V(log Γ) →0. The next lemma was proved in [De], as a by-product of a logarithmic Hodge decomposition, but an elementary proof was later found in [No]. For sake of completeness, we give a proof, which is even simpler than that of [No]. Lemma 2.1. [De]Let Xbe a complex projective manifold, V⊂X a normal crossing hypersurface and ωa global logarithmic p-form with polar set contained in V. Then ωis closed. Proof: In order to prove that a p-form is closed, it is sufficient to prove that the restriction of the p-form to a generic (p+ 1)-dimensional submanifold is closed. Hence we may assume that n= dimC(X)=p+1, and the proof will be by induction on p. Degree of Solutions to Pfaff Equations 597 The case p= 0 is trivial: Ω0 X(log V)=OXand a global holomorphic function is constant. Assume now that the lemma has been proved for (p−1)-forms. If ω∈H0(X, Ωp X(log V)), then, as in [No], we may consider the current Tωof bidegree (p, 0) defined by Tω(φ)=X ω∧φ, for every smooth (1,p+ 1)-form φ. This is well defined, i.e. the integral is convergent, precisely because ωhas logarithmic poles along V: the 2-form 1 zdz ∧d¯zis integrable on the disc. Similarly, we may associate to dω, which is still logarithmic, a current Tdω of bidegree (p+1,0). We then have, in the sense of currents, ∂Tω=Tdω, which follows from the fact that if ψis any smooth (0,p+ 1)-form then Xd(ω∧ψ) = 0 by Stokes theorem (here ω∧ψis again a current, and the integral is its value on d1≡0). In particular, we have ∂Tω≡0⇔dω =0. On the other hand, ∂Tωis not zero: a simple computation, based on ∂(dz z)=2πiδ0(where δ0is the Dirac distribution), shows that [No] ∂Tω=2πi TRes(ω), where, with a negligible abuse of notation, we identify TRes(ω)(a current on ˆ Vof bidegree (p−1,0)) with its direct image in X(a current of bidegree (p, 1)). By induction hypothesis, Res(ω) is closed, i.e. ∂TRes(ω)≡0, and so ∂∂Tω≡0. By regularity theory, the current ∂Tωis in fact a holomorphic (p+ 1)-form, because it is of bidegree (p+1,0) and ∂-closed. By Stokes Theorem, X ∂Tω∧∂Tω=X d(Tω∧∂Tω)=0 (Tω∧∂Tωis a current, being ∂Tωsmooth, and its differential is ∂Tω∧∂Tω because ∂∂Tω≡0). This forces ∂Tωto be identically zero, because ∂Tω∧∂Tω≥0. Remark. We have used the projectivity of Xonly to reduce the problem to dimC(X)=p+ 1. In other words: any logarithmic p-form on any compact complex manifold of dimension p+ 1 is closed. 598 M. Brunella, L. Gustavo Mendes In the context of manifolds with Pic(X)=Z, we shall use the following well-known fact: Lemma 2.2. Let Xbe a complex projective manifold with Pic(X)=Z and V⊂Xa normal crossing hypersurface. Let ωbe a global logarithmic p-form with poles along Vand Res(ω)≡ 0,1≤p≤n−1. Then Vis not smooth. Proof: By contradiction, assume that Vis smooth, so that η= Res(ω)≡ 0 is a holomorphic (p−1)-form on V. The line bundle OX(V) is ample, so that by Kodaira Vanishing Theorem Hp(X, OX(−V)) = 0, because p<n. Hence the restriction map Hp−1(X, OX)→Hp−1(V,OV)is surjective. The conjugate form ηisa(0,p −1)-form on Vwhich is ∂-closed, because ηis ∂-closed. Hence ηdefines a class in Hp−1(V,OV)(`a la Dolbeault), which arises from Hp−1(X, OX), whence it follows that there exists a ∂-closed (0,p−1)-form βon Xwhose restriction to Vis cohomologous to η, that is equal to η+∂γ for some (0,p−2)-form γon V. After extending γto Xand replacing βby β−∂γ, we may and will suppose that βcoincides with ηon V. Let now θbeaK¨ahler form on X. By considering ηas a current Tη of bidegree (p, 1) on X, we may evaluate it on the (n−p, n −1)-form θn−p∧β: Tη(θn−p∧β)=V η∧θn−p∧β=V η∧η∧θn−p. But θn−p∧βis ∂-closed and Tη=1 2πi∂Tωis ∂-exact, so that the integral is zero. Contradiction, because η∧η∧θn−p≥0 and η∧η∧θn−p≡ 0. It will be useful to reformulate Lemma 2.1 in a more abstract form, due to Bogomolov [Bo], [Re]. To this end, we recall the definition of Kodaira dimension of a holomorphic line bundle Lof a projective variety X, denoted κ(X, L)[Ii]. Consider the ring R(X, L):= ∞  m=0 H0(X, L⊗m) and the homogeneous field of fractions Q(X, L):={li lj|li,l j∈H0 (X,L ⊗m), m≥0}. Then we define κ(X, L) as the transcendence degree of Q(X, L), if R(X, L)=Cor κ(X, L):=−∞,ifR(X, L)=C. One has κ(X, L)≤ n= dimC(X), and κ(X, L)=nif Lis ample. Degree of Solutions to Pfaff Equations 599 Lemma 2.3. [Bo]Let Xbe a complex projective n-manifold, V⊂Xa normal crossing hypersurface and L∈Pic(X). If there exists a nontrivial global section σof Ωp X(log V)⊗L, then κ(X, L−1)≤p. Proof: Suppose by contradiction that κ(X, L−1)≥p+ 1, i.e. for some m≥1 there exist p+ 2 global sections l0,...,l p+1 ∈H0(X, L⊗−m) such that the meromorphic functions on Xgiven by fi:= li l0are algebraically independent. Let us suppose for a moment that m= 1. We can multiply σby each li, obtaining global sections ωi:= liσ∈H0(Ωp X(log V)). Since ωi=fiω0, the closedness of each ωi(Lemma 2.1) gives, for i=1,...,p+1, df i∧ω0≡0. The nontriviality of ω0implies df 1∧df 2∧...∧df p+1 ≡0, contradicting the algebraic independence. The case m>1 is reducible to the case m= 1 by passing to a suitable m-fold ramified covering. We refer to [Re] for details. 3. Bounding the degree of solutions Let σ∈H0(X, Ωp X⊗N) be a Pfaff equation and V⊂Xa normal crossing hypersurface, which is a solution to σ. We can look at σas a global holomorphic section of Ωp X⊗OX(−V)⊗ N⊗O X(V), that is, a meromorphic section ˆσof Ωp X⊗O X(−V)⊗N with simple poles (ˆσ)∞⊂V. Locally, i.e. after local trivialization of OX(−V)⊗N, we can see ˆσas a meromorphic p-form ωwith simple poles (ω)∞⊂Vand we assert: Lemma 3.1. The meromorphic p-form ωis logarithmic. Proof: We have to check that that, if {f=0}is a local reduced equation of V, then df ∧ωis holomorphic. From iVσ≡0 (Definition 1.2) it follows that df ∧σis identically zero along V, i.e. df ∧σ=f·θ, for some regular local section θof Ωp+1 X⊗N. Hence df ∧σ fis a regular section of Ωp+1 X⊗Nand df ∧ωis holomorphic, because (up to local trivialization) ω=σ f. Remark that the converse of Lemma 3.1 is also true: if ωis a logarithmic p-form with polar set given by {f=0}, then V={f=0}is a solution to the Pfaff equation defined by fω. That’s the reason for 600 M. Brunella, L. Gustavo Mendes which the use of logarithmic forms is particularly well adapted to the study of solutions to Pfaff equations. The meaning of Lemma 3.1 is that σis in fact a global holomorphic section of Ωp X(log V)⊗O X(−V)⊗N. Now, Bogomolov’s Lemma (Lemma 2.3) gives: κ(X, OX(V)⊗N−1)≤p<dimC(X),(3) which is the “bound on d(V)” we were looking for. Let us now specialize to the case Pic(X)=Z. Then OX(V)⊗N−1=lH, where l=d(V)−d(N)∈Zand His the positive generator of Pic(X)= Z. Since l>0 implies κ(X, lH) = dimC(X), we conclude from (3) that d(V)−d(N)≤0 as desired. Moreover, d(V)−d(N) = 0 means OX(V)⊗N−1=OXand therefore the Pfaff equation is globally defined by a (closed!) logarithmic p-form ωwith (ω)∞=Vand hence Res(ω)≡ 0. In this case, Lemma 2.2 says that Vis not smooth. In order to apply this result to foliations of CPn, we remark: Lemma 3.2. Let σ∈H0(CPn,Ωp CPn⊗N)be a Pfaff equation associated to a foliation Fof CPnwith codimCSing(F)≥2(and codimC(σ)0≥2). Then d(N)=d(F)+p+1. Proof: Take a generic linear subspace Π ≃CPpand consider the restriction of σto Π, denoted iΠσ, with hypersurface of zeros on Π denoted by (iΠσ)0. Observe that iΠσis a global regular section of KCPp⊗N|Π vanishing on (iΠσ)0; hence OΠ((iΠσ)0)=KCPp⊗N|Π, where KCPpis the canonical line bundle, whose degree is −(p+1). Since (iΠσ)0is the tangency set between Fand Π and d(F) is defined as the degree of (iΠσ)0in Π, we obtain d(F)=−(p+1)+d(N). Returning to the general inequality (3), we stress that it gives informations whatever Pic(X) is. Roughly speaking, it says that OX(V) is “partially less positive” than N. First of all, let us observe that if κ(X, OX(V)⊗N−1)=r≥0 then (as the proof of Lemma 2.3 shows) on a suitable ramified covering of Xthe Pfaff equation will be defined by a global (and closed) logarithmic p-form and will have ralgebraically independent first integrals. Next, when Xis a surface we have the following Degree of Solutions to Pfaff Equations 601 fact (probably, a similar statement holds in any dimension). Recall [Dm] that a divisor Dis nef if D·C≥0 for every irreducible curve C⊂X. Lemma 3.3. Let Mbe a line bundle on a projective surface Xwith κ(X, M)≤1. Then there exists a non-trivial nef divisor D, with real coefficients, such that M·D≤0. Proof: Let NSR(X)⊂H2(X, R) be the real Neron-Severi group of X, let Nnef ⊂NSR(X) be the nef cone (i.e., the closure of the ample cone), and let Npsef ⊂NSR(X) be the pseudoeffective cone (i.e., the closure of the effective cone). See for instance [Dm] for these notions. Then, by Kleiman criterion (in its dual form), a line bundle Mbelongs to the interior of Npsef if and only if M·D>0 for every D∈Nnef \{0}. On the other hand, to say that Mbelongs to the interior of Npsef is the same as to say that κ(X, M) = 2, by [Dm, Prop. 6.6]. Whence the result. In our case, applied to M=OX(V)⊗N−1this fact gives V·D≤N·D. On the other hand, one finds in [Br] the inequality V·V≤N·V (and Vis not necessarily nef). The relation between these inequalities is not clear to us: one is a “global” statement about the line bundle OX(V)⊗N−1over X, while the other is a “local” statement, i.e. about the restriction of OX(V)⊗N−1to V. We now discuss some possible extensions. Given an analytic hypersurface V⊂Xwhose singularities are worst than normal crossings, we can generalize the definition of logarithmic forms (Section 2) in two ways, which are not equivalent in general: 1) Given a meromorphic p-form ωwith simple poles along V={f=0}, we say that ωis weakly logarithmic if fdω (or equivalently df ∧ω)is holomorphic. 2) Given ωas in 1), we say that it is strongly logarithmic if, on a neighbourhood of any x∈V,ωbelongs to the OX-module generated by holomorphic forms and the forms df 1 f1,...,df k fk, where V= {f1·f2...f k=0}, with fireduced equations of local branches of V at x. The previous Lemmata 2.1, 2.3 are still valid for strongly logarithmic forms, as was observed in [No]. There are at least two ways to see this: