Vector fields with simply connected trajectories transverse to a polynomial

Advances in Mathematics 285 (2015) 1339–1357

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Vector fields with simply connected trajectories transverse to a polynomial ✩ Alvaro Bustinduy a,∗ , Luis Giraldo b a

Departamento de Ingeniería Industrial, Escuela Politécnica Superior, Universidad Antonio de Nebrija, Calle Pirineos 55, 28040 Madrid, Spain b Instituto de Matemática Interdisciplinar (IMI), Departamento de Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 31 July 2014 Received in revised form 1 July 2015 Accepted 4 August 2015 Available online 8 September 2015 Communicated by the Managing Editors of AIM

a b s t r a c t We classify the polynomial vector fields X on C2 which have simply-connected trajectories and satisfy dP (X) = 1 for P in C[x, y]. In particular, we obtain that such vector fields are complete and have all its trajectories proper and of type C. Finally, we apply this result to classify surjective C-derivations of C[x, y], and provide a proof of a conjecture of Cerveau [7]. © 2015 Published by Elsevier Inc.

MSC: primary 32M25 secondary 32L30, 32S65 Keywords: Foliation transverse to a fibration Foliation P -complete Simply-connected trajectories Surjective derivations

✩

Supported by Spanish MINECO project MTM2011-26674-C02-02.

* Corresponding author. E-mail addresses: [email protected] (A. Bustinduy), [email protected] (L. Giraldo). http://dx.doi.org/10.1016/j.aim.2015.08.003 0001-8708/© 2015 Published by Elsevier Inc.

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1. Introduction and results Vector fields and trajectories. (See [8].) A holomorphic vector field X on C2 is a section of the tangent bundle of C2 . If we fix coordinates x, y in C2 , X is expressed analytically as X=P

∂ ∂ +Q , ∂x ∂y

where P , Q are holomorphic functions in C(x, y). Given a holomorphic vector field X, there exists a differential equation: ϕz (t) = X(ϕz (t)),

ϕz (0) = z ∈ C2 ,

whose local solution ϕz defines the local flow of X in a neighborhood of (0, z) ∈ C × C2 . Fixed z ∈ C2 , ϕz can be extended by analytic continuation along paths from t = 0 in C to a maximal connected Riemann surface πz : Ωz → C, which is a Riemann domain over C. Note that if πz : Ωz → C is single sheeted, Ωz is identified with the domain πz (Ωz ) ⊂ C. We define: • The solution of X through z as ϕz : Ωz → C2 . • The (complex) trajectory Cz of X through z as ϕz (Ωz ). Any nontrivial trajectory Cz of X is a Riemann surface immersed in C2 . If Ωz = C (as domain in C), X is said to be complete on Cz . In this case, Cz is uniformized by C, and hence it is analytically isomorphic to (= of type) C or C∗ (maximum principle in C2 ). X is complete, if it is complete on Cz for all z. In particular, its flow ϕ : C × C2 → C2 , given by (t, z) → ϕ(t, z) = ϕz (t), defines a holomorphic action of the additive group of the complex numbers (C, +) on C2 (by analytic automorphisms). Reciprocally, any holomorphic action of (C, +) on C2 is the flow of a complete holomorphic ∂ vector field: X = ∂t ϕ(t, z)|t=0 . C, +) on C 2 . (See [17] and [15].) Let ϕ : C × C2 → C2 be a Holomorphic actions of (C holomorphic action of (C, +) on C2 . For any t ∈ C, let ϕt be the analytic automorphism of C2 defined by ϕt (z) = ϕ(t, z), where z ∈ C2 . Then, • ϕ is algebraic, if ϕ is a polynomial map. • ϕ is quasi-algebraic, if ϕt , for any t ∈ C, is a polynomial automorphism. • ϕ is proper, if the topological closure C z of Cz in C2 , for any z, is an analytic curve. Let us recall that a polynomial (resp. holomorphic) automorphism in C2 provides a polynomial (resp. holomorphic) change of coordinates. If X is a vector field and Ψ is an

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automorphism, after Ψ, we obtain Ψ∗ X. In coordinates, if Ψ(x, y) = (z, w), Ψ∗ X = dΨ(Ψ−1 (z, w))X(Ψ−1 (z, w)). Let us also recall that if ϕ is the flow of X, the flow ϕ¯ of Ψ∗ X satisfies ϕ¯t = Ψ ◦ ϕt ◦ Ψ−1 . It is said that ϕ is conjugated of ϕ¯ by a change of coordinates. Let us consider a holomorphic action ϕ without fixed points, or equivalently, a complete holomorphic vector field X without zeros and flow ϕ. Let us assume moreover that X is polynomial. Let us recall some classic results: If ϕ is algebraic, Rentchsler’s Theorem [15] implies that there is a polynomial change of coordinates Ψ such that Ψ ◦ ϕt ◦ Ψ−1 = (x + a(y)t, y), with a ∈ C[y]. In our case, we are using moreover that X has not zeros, then a(y) is a non-vanishing constant a. Only in this case, we can perform another change of coordinates (a−1 x, y) to obtain that the action (x + at, y) is conjugated to (x + t, y). Then we can assume that a = 1. Therefore, up to a polynomial automorphism, X=

∂ , ∂x

and all the trajectories of X are algebraic (then, proper) and of type C. If ϕ is quasi-algebraic, Suzuki Theorem [17, Théorème 2] implies that there is a polynomial change of coordinates Ψ such that Ψ ◦ ϕ ◦ Ψ−1 must be an action of the form α(t, x, y) = (x + at, y), with a ∈ C∗ , or of the form δ0 (t, x, y) = (x + t, yebt ). In this second case, up to a polynomial automorphism, X=

∂ ∂ + by , ∂x ∂y

where b ∈ C, and in particular, the trajectories of X are proper and of type C. If ϕ is proper and all the trajectories of X are of type C, Suzuki Theorem [17, Théorème 4] implies that there is a holomorphic change of coordinates Ψ such that Ψ ◦ ϕ ◦ Ψ−1 must be an action of the form α(t, x, y) = (x + a(y)t, y), with a a nowhere vanishing holomorphic function in C(y). Therefore, after a holomorphic change of coordinates: X=

∂ . ∂x

It follows from the analysis above that, if X is a complete polynomial vector field on C2 without zeros whose flow is ϕ, it holds:

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(I) Algebraic ϕ ⇒ Quasi-algebraic ϕ ⇒ Proper ϕ and trajectories of type C. (II) In any of the three cases of (I) for ϕ, after a holomorphic automorphism, there is a polynomial P = x such that dP (X) = 1. Motivating questions. Given a polynomial vector field X on C2 , there is a C-derivation DX of C[x, y] associated to X: DX : C[x, y] → C[x, y] f → X(f ). A slice s of DX is s ∈ C[x, y] such that DX (s) = 1. Questions about slices and derivations have been studied by many authors (see [9] and references therein), and are related to important problems such as the Cancellation Problem in affine spaces, as we can see in [9, Chapter 10]. Note also that the Jacobian Conjecture can be formulated as a problem in terms of derivations with a slice [9, Chapter 3]. In this paper, we will study the following questions: Let X be a polynomial vector field on C2 with simply-connected trajectories such that DX has a slice. Can X be determined modulo a polynomial automorphism? Is X complete? We will answer these questions (and apply the solution to prove a conjectured problem about surjective derivations asked by Cerveau.) Main results Theorem 1. Let X be a polynomial vector field in C2 . Let us suppose that (1) There exists P ∈ C[x, y] such that dP (X) = 1. (2) All the trajectories of X are simply connected. Then, up to a polynomial change of coordinates, X=

∂ ∂ + [b(x)y + c(x)] , ∂x ∂y

with b, c ∈ C[x]. In particular, X is complete and with all its trajectories proper and of type C, and after a holomorphic change of coordinates: X=

∂ . ∂x

As a consequence of Theorem 1, and using some results in [6], we prove the following Conjecture stated by Cerveau in [7]: If DX is surjective, then, up to a polynomial change of coordinates, X= where b ∈ C.

∂ ∂ + by , ∂x ∂y

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This result appeared as a theorem in [6], but the proof used a technical lemma ([6, Lemma 5.5]) whose proof is not correct, as can be seen in [7] and [10]. About Theorem 1 and its proof. Let G be the extension of P on CP2 and Λ be the pencil of curves in CP2 defined by G. After resolution of the base points of Λ one obtains a rational surface S, a proper map f : S → CP2 and a holomorphic fibration R on S whose fibers are projected by f to curves of Λ. Let FX be the foliation induced by X on CP2 . First, due to the simply-connectedness of the trajectories of X, we show that E = f −1 (CP1∞ ) is G-invariant, where G = f ∗ FX . It allows us to prove that G is transverse to the generic fiber F of R and to assume that G is reduced. As a first conclusion we obtain (after projecting G to the initial coordinates) that FX |C2 is P -complete [1]. The next point is to analyze the possible cases for F . For F of genus g = 1 (Turbulent) [4] or g ≥ 2 [14], the foliation has a rational first integral. For F of genus g = 0 (Riccati), using the local models described in [4, p. 439], the connectedness of E, and the invariancy of E by G we can prove that E intersects F in one or two points. Then, P is of type C or C∗ . Finally, we can deduce that P is of type C: for P of type C∗ there is a FX -invariant connected component of the singular fiber (a copy of C), which contradicts dP (X) = 1. There exists a difficulty which makes not possible a more direct proof of Theorem 1. The possibility, in principle, that all the fibers of R are transverse to the foliation. In order to exclude that, we need to prove two things. First, the existence of a unique horizontal component of E with respect to R, using that E is connected. Second, the existence of an algebraic trajectory Cz for X, using mainly that Kodaira dimension of G is zero. This is done in Lemmas 4 and 5. It will allow us to obtain, by a careful analysis of the leaf L of G such that L \ E is Cz and of the unique irreducible component of E that cuts L, a contradiction. Examples. We give two examples related to the hypotheses in the statement of Theorem 1. The first example is a family of vector fields that satisfy (1), but not (2). Example 1. Let us consider Y1 =

∂ ∂ + B(x, y) , ∂x ∂y

with B(x, y) ∈ C[x, y]. Let us write B = ad (x)y d + · · · + a1 (x)y + a0 (x). For P = x, then dP (Y1 ) = 1. If Y1 has no rational first integral (Lemma 1) and d ≥ 2 there are infinitely many trajectories of Y1 which are not simply-connected. First, let us extend the foliation defined by Y1 on C2 to a foliation H on CP1 × CP1 : take coordinates (x, v) = φ(x, y) = (x, 1/y), and φ∗ Y1 =

∂ ∂ − v 2 B(x, 1/v) . ∂x ∂v

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We define Y2 as the polynomial vector field without poles v d−2 · φ∗ Y1 . Take coordinates (u, y) = ϕ(x, y) = (1/x, y), then ϕ∗ Y1 = −u2

∂ ∂ + B(1/u, y) . ∂u ∂y

If d¯ is the maximal of the degrees of ai , we define Y3 as the polynomial vector field ¯ without poles ud · φ∗ Y1 . The foliation defined by Y1 , Y2 and Y3 is H. A simple inspection 1 shows that CP × C (= {u = 0}) is H-invariant and that C × CP1 (= {v = 0}) is not H-invariant. Take a generic point p ∈ {v = 0} and the leaf L of H through it. We can assume that L is not an algebraic curve (Darboux). Let D = L ∩ {v = 0}. Then L∗ = L \ D is a trajectory of Y1 . Suppose that L∗ is simply-connected and we shall derive a contradiction. Uniformization Theorem implies that L∗ is of type CP1 , C or D. On the other hand,  → L is the universal covering map of L, if β : L −1  β|L\β (D) → L∗  −1 (D) : L \ β

 is simply-connected then L  is is a covering map, and then it is a biholomorphism. As L 1 of type CP and β −1 (D) is one point. Thus L∗ is of type C and L \ L∗ = {p}. Therefore, as L is a compact leaf of H, the inclusion L → CP1 × CP1 is an embedding and L is a subvariety of CP1 × CP1 . Chow’s Theorem implies that L is an algebraic leaf, which is a contradiction. The second example is a family of vector fields that satisfy (2), but not (1). Example 2. Let us consider, for m, n ≥ 1, Y = n2 xm+1 y n−1 If ϕ(x, y) = (xen(x

m n

y )

, ye−m(x

m n

ϕ∗

∂ ∂ − (1 + mnxm y n ) . ∂x ∂y

y )



) = (z1 , z2 ), then  m n ∂ = −em(x y ) · Y. ∂z2 m n

We see that Y is not complete. However, Y multiplied by −em(x y ) is complete, and analytically equivalent (but not algebraically equivalent) to the horizontal vector field. Thus, Y has all its trajectories proper and of type C; and Y has no zero. Note that Y generates an algebraic foliation on C2 which is P -complete with P = m n x y of type C∗ . According to the proof of Theorem 1 there is not a polynomial Q such that dQ(Y ) = 1. m n m n Remark that d(ye−m(x y ) )(−em(x y ) · Y ) = 1. Then, the conclusion of Theorem 1 is not valid for X non-polynomial and P ∈ C(x, y).

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2. Preliminaries 2.1. Foliations generically transverse to a fibration [14] Let us consider a complex compact projective surface M . A fibration on M is a holomorphic surjective map g : M → B where B is a Riemann surface. A point p ∈ M is a critical point of g if dg = 0 at p. A critical value at B is the image by g of a critical point. Since B is compact, the set of critical values is a finite set Σ. The fibers g −1 (t), for t ∈ B \ Σ, are all smooth compact curves in M biholomorphic to the same Riemann surface. Any such fiber is called a generic fiber. The fibers g −1 (t), for t ∈ Σ, are called singular fibers. The fibration is connected if its generic fiber is connected. In this case, the singular fibers are also connected, by Stein factorization theorem, and they define compact curves in M with one or more irreducible components. Definition 1. A holomorphic foliation F with isolated singularities on M is generically transverse to a holomorphic fibration g : M → B if there is a generic fiber of g transverse to F. We refer to [14, pp. 43–44, 126] for the following well-known proposition: Proposition 1. If F is a holomorphic foliation generically transverse to a connected fibration g : M → B, then there exists a finite subset B0 ⊂ B such that for any t ∈ B \ B0 : (i) g −1 (t) is a generic fiber which is transverse to F (ii) There are an open neighborhood Ut ⊂ B of b and a biholomorphism from g −1 (Ut ) to Ut × g −1 (t) which transforms F|g−1 (Ut ) into the horizontal foliation and the fibration g|g−1 (Ut ) into the vertical fibration. Remark 1. The set B0 is defined by Σ ∪ {t ∈ B \ Σ | g −1 (t) is F − invariant}. A singular fiber could have several irreducible components, some of which can be invariant by F while the others are not. 2.2. Resolution of singularities of F [2] Let F be a foliation with isolated singularities on M . Let p ∈ Sing(F) and X be the vector field that generates F in a neighborhood U of p. Let us consider the eigenvalues λ and μ of the linear part (DX)(p) of X at p. The singularity p is reduced if at least one of these eigenvalues is not zero (say, μ = 0) and the quotient λ/μ ∈ C \ Q+ . In this case, either λ = 0 = μ or λ = 0. In the former case, p is non-degenerate and there are only two local curves through p invariant by F. These two curves, called separatrices, are

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transversal at p. In the latter case, p is a saddle-node and there is always a nonsingular local curve through p invariant by F, which is called the strong separatrix of p, and at most other local curve through p invariant by F. If it exists, it is always transversal to the strong separatrix, and it is called the weak separatrix. We refer to [2, p.13] for the proof of the following well-known theorem: Theorem (Seidenberg). For any pj ∈ Sing(F), there exists a finite sequence of blowingups Πj over pj such that the pull-back foliation has only reduced singularities on the exceptional divisor.  → M be the finite composition of any sequence of Let Π = ΠN ◦ · · · ◦ Π1 : M blowing-ups Πj over pj ∈ Sing(F) given by Seidenberg’s Theorem. The foliation F =  is the resolution of singularities of F. The exceptional divisor D of the Π∗ F on M  is reduced resolution of singularities of F is defined by Π−1 (Sing(F)). Any p ∈ sing(F) and it is contained in D. Any connected component of D is a finite union of embedded projective lines with normal crossings and without triple points, which is biholomorphic to Π−1 j (pj ). 3. Proof of Theorem 1 3.1. Rational first integrals Let FX be the natural extension to CP2 = C2 ∪ CP1∞ of the foliation defined by a polynomial vector field X on C2 . Lemma 1. If X has no zeros, its trajectories are simply-connected, and FX has a rational first integral, then up to a polynomial change of coordinates, X=

∂ . ∂x

Proof. Any leaf L of FX |C2 is of type C because the trajectory of X defined by L is simply-connected and L is contained in an algebraic curve [16]. On the other hand, L must be contained in a fiber of the rational first integral of FX , say H. As H|C2 has no indetermination points, and we can assume that it is primitive, up to a polynomial automorphism, H|C2 = y [17, p. 527]. 2 The proof of Theorem 1 follows from Lemma 1 if there are rational first integrals. From now on, we will assume that FX has no rational first integral. 3.2. Invariance of the total transform of CP1∞ by f Proposition 2. Let F be a foliation with isolated singularities on a complex compact projective surface M , and without rational first integral. Let C be a curve in M , and

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suppose that all the leaves of F restricted to M \ C are simply-connected. Let us consider a proper map h : M  → M defined by a finite composition of sequences of blowing-ups  over points in C, which may not be in Sing(F). Let us consider the foliation F = h∗ F  and the curve C  = h−1 (C). Then C  is F -invariant. 

Proof. Let C1 be an irreducible component of C  . Suppose that C1 is not F -invariant, we shall reach a contradiction.  Take a generic point p ∈ C1 and the leaf L of F through it. Let L be the leaf of F defined by h(L ). As h restricted to M  \ C  is a biholomorphism over M \ C, L \ C  is projected by h to a simply-connected leaf L \ C of F|M \C . Then L \ C  is simply-connected, and by Uniformization Theorem it is of type CP1 , C or D. On the other hand, take the universal covering map β : L → L of L and T = L ∩C  . As β|L \β −1 (T ) : L \ β −1 (T ) → L \ C  is a biholomorphism and L is simply-connected, the unique possibility is L of type CP1 and β −1 (T ) one point. In fact L \ C  is of type C and L ∩ C  = L ∩ C1 = {p}. Hence L is an algebraic leaf of F defined by a rational curve (Chow), and F has a rational first integral (Darboux), which contradicts our assumptions. 2 Corollary 1. Let f : S → CP2 a finite composition of sequences of blowing-ups over points in CP1∞ , which may be not in Sing(FX ). Let us consider the foliation G = f ∗ FX and E = f −1 (CP1∞ ). Then E is G-invariant. Proof. It follows from Proposition 2 applied to FX .

2

3.3. Resolving base points of Λ Let G be the extension of P to CP2 as rational function. Let Λ be the pencil of curves in CP2 given by {G−1 (t)}. Note that Λ has no fixed components and that its base points are the indetermination points Indet(G) = {p1 , . . . , pN } ⊂ CP1∞ of G. We refer to [13] for the following well-known consequence of resolving Indet(G). Lemma 2. There exists a birational map f = fN ◦ · · · ◦ f1 : S → CP2 , where each fj is a finite sequence of blowing-ups over pj , and a holomorphic fibration R : S → CP1 , such that the fibers R−1 (t) project by f to the curves of the original pencil Λ.

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Remark 2. (a) An exceptional curve of f is an irreducible component of the curve f −1 (Indet(G)). Note that any exceptional curve of f projects by R to a single point, and then it is contained in a fiber, or it maps onto CP1 and it cuts any fiber of R. In the latter case, the exceptional curve is said to be horizontal with respect to R. (b) E = f −1 (CP1∞ ) ⊂ S is a connected curve defined by a tree of embedded rational curves with normal crossings. The irreducible components of E are the exceptional curves of f and the strict transform of CP1∞ by f . (c) Set v h E = E1v ∪ · · · ∪ EN ∪ E1h ∪ · · · ∪ EN , 0 1

where Eiv are the irreducible components of E contained in fibers of R; and Eih , are the horizontal components of E (with respect to R). Note that the strict transform of CP1∞ by f is contained in a fiber of R. Let us denote it by E1v . Moreover, as we will prove in Proposition 3, there always exists at least one horizontal component. Then N1 ≥ 1. (d) f ∗ (Λ) = {f ∗ (G−1 (t))} is a pencil of curves in S with fixed components contained in f −1 (Indet(G)). (e) The pencil {R−1 (t)} is the strict transform of Λ by f . It is obtained after removing the fixed components of f ∗ (Λ). 3.4. FX |C2 is P -complete Proposition 3. Let f be as in Lemma 2. If G is the foliation f ∗ FX , then (i) G is generically transverse to R (over S), and (ii) Moreover, we can assume that G is reduced. Proof. Recall that, by Corollary 1, E = f −1 (CP1∞ ) is G-invariant. Let us prove (i). Let F be the generic fiber of R. Since f restricted to S \ E is a biholomorphism over C2 and dP (X) = 1, any fiber of R is transverse to G in S \ E. In particular, F is transverse to G in S \ E. On the other hand, E ∩ F = ∅. Otherwise f (F ) is contained in C2 , which is impossible by maximum principle. Take an irreducible component Eih of E which cuts F . Note that Eih = E1v , that Eih is G-invariant, and that Eih defines an exceptional curve of f which is horizontal with respect to R. Thus F and Eih are transversal at any point in E ∩ F . Hence F is transverse to G in S and we obtain (i). To prove (ii), let us consider the resolution of singularities G of G. Recall that G is the reduced foliation Π∗1 G where Π1 : S → S is the birational map obtained by the sequences of blowing-ups over points of Sing(G) (Seidenberg). Let D = Π1 −1 (Sing(G))  the holomorphic fibration such that {R  −1 (t)} is be the exceptional divisor of G and R −1 the strict transform of {R (t)} by Π1 .

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By construction, the leaves of G restricted to S \ E are simply-connected and  = Π−1 (E) (and then D) is G-invariant.  Sing(G) ⊂ E. Then, according to Proposition 2, E 1  If an exceptional curve of Π1 (that is, in D) is horizontal with respect to R, it must be  and then G-invariant.  an irreducible component of E, Then G is generically transversal  to R. 2 Proposition 4. FX |C2 is P -complete. Proof. Note that P is primitive. Otherwise, P (x, y) = h(P0 (x, y)), with h ∈ C[z] of degree > 1 and a primitive P0 (x, y) (Stein factorization). Hence dP (X) vanishes on {h (P0 (x, y)) = 0}, which contradicts dP (X) = 1. It follows from Propositions 1 and 3 that there is a finite set Q ⊂ C such that, for all t∈ / Q: (1) P −1 (t) is transverse to FX , and (2) There exist a neighborhood Ut ⊂ C of t and a biholomorphism from P −1 (Ut ) to Ut ×P −1 (t) which sends FX |P −1 (Ut ) to the horizontal foliation and the fibration P|P −1 (Ut ) into the vertical fibration. Therefore FX |C2 is P -complete [1]. 2 / Q is biholomorphic to the same Riemann surface: the generic Any fiber of P over t ∈ fiber F of R minus the finite set of points F ∩ E. 3.5. P is of type C Proposition 5. F is of genus 0, and G is a Riccati foliation. Proof. If F is of genus ≥ 2, it follows from [14, Theorem III.6.1] that G has a rational first integral, which is contrary to our assumptions. Then this case does not occur. Let us suppose that F is of genus one (G Turbulent). E ∩F = ∅ by maximum principle. Then there is at least one irreducible component of E, which is horizontal, say E1h , such that E1h ∩ F = ∅. Moreover, E1h is G-invariant by Corollary 1. The existence of E1h is enough to construct a rational first integral for G [4, p. 438]. Then, this case is not possible with our assumptions. 2 3.5.1. Local models for the fibers of R Since G is reduced, for any t ∈ CP1 , after contracting irreducible components of −1 R (t), one obtains one of the following models described in [4, p. 439] (see also [3, §7], [12, Section IV.4]): (a) Transverse fibre: the fibre is a rational curve and G  is transverse to it. (b) Dicritical fibre: the fibre is a rational curve of multiplicity k with two quotient singularities p and q of order k and G  is transverse to the reduced fibre outside p and q. Around p and q the foliation G  is the cyclic quotient of a regular foliation.

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(c) Nondegenerate fibre: the fibre is a rational curve tangent to G  with two nondegenerate singularities, or one saddle-node of multiplicity 2, with strong separatrix transverse to the fiber. (d) Semidegenerate fibre: the fibre is a rational curve tangent to G  with two saddlenodes of the same multiplicity m, with strong separatrices inside the fibre. (e) Nilpotent fibre: the fibre is a rational curve of multiplicity 2 with two quotient singularities p and q of order 2, and a saddle-node of multiplicity l, with strong separatrix inside the fibre. The foliation G  is tangent to the reduced fibre outside p and q. Around p and q the foliation G  is the cyclic quotient of a regular foliation. Then, there exists a contraction π : S → S  of irreducible components inside fibers of R, that produces a new surface S  , possibly with quotient singularities, and a foliation G  regular on Sing(S  ), such that any fiber of R follows one of the above models after π. Let us denote by Σ the exceptional divisor of π. Set  = R(Σ) = {t1 , . . . , tM } ⊂ CP1 . Σ 0  are fibers of R such that some of their components are contracted Recall that R−1 (Σ)  by π. Note that Σ ⊂ R−1 (Σ).  Set Σt = Ft ∩ Σ. Then, Take Ft = R−1 (t) with t ∈ Σ. Σ = Σt1 ∪ · · · ∪ ΣtM0 . Remark 3. Note that π contracts all the irreducible components of Ft except one of them, that we will denote by Ft0 . According to the models, Σt is G-invariant, Ft = Σt ∪ Ft0 , 0 and π(Ft ) = F  t ⊂ S  is a fiber of type (a), (b), (c), (d) or (e). Proposition 6. There exists t ∈ CP1 such that R−1 (t ) is of type (c), (d) or (e) after π. Proof. We will assume that R−1 (t) is of type (a) or (b) after π for every t ∈ CP1 , and derive a contradiction.  because E v is G-invariant. Note that Σ = ∅: R(E1v ) must be in Σ 1 Set Es = irreducible components of Fs contained in E. It is clear that there is only a finite set of values {s1 , . . . , sM0 } ⊂ CP1 for which Esi = ∅. Remark that, 1) Esi ⊂ S is a curve, which can have several connected components. The irreducible components of Esi are vertical with respect to R. Then, v Esi = Fsi ∩ (E1v ∪ · · · ∪ EN ), 0

and E can be also defined in the following way h E = Es1 ∪ · · · ∪ EsM  ∪ E1h ∪ · · · ∪ EN . 1 0

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2) Distinct fibers of R do not intersect (R is a fibration). Then, Esi ∩ Esj = ∅ if si = sj . 3) Esi is G-invariant (Corollary 1). To conclude the proof, we need a few lemmas. Lemma 3. It holds that – Esi ⊂ Σ, – Σti ⊂ E. Proof. Remark 3 implies that Fsi = Σsi ∪ Fs0i and that F  si ⊂ S  as (a) or (b), which is not G  -invariant. Then Fs0i is not an irreducible component of Esi according to 3), and Esi ⊂ Σsi . Let L be an irreducible component of Σti . We know that L is G-invariant (Remark 3). If L is not contained in E, L ∩ (S \ E) projects to C2 as an algebraic curve which is contained in a fiber of P and is invariant by X. Then dP (X) is not 1, which is impossible. 2 0

After Lemma 3, we conclude that M0 = M0 and sj = tj . Then, by 1): h E = Σt1 ∪ · · · ∪ ΣtM0 ∪ E1h ∪ · · · ∪ EN . 1

Lemma 4. N1 = 1. Proof. Let E  = π(E). It is clear that E  is a connected curve in S  . Moreover, according to Corollary 1, E  is G  -invariant. Let us suppose that N1 ≥ 2. We know that E  = E  1 ∪ · · · ∪ E  N1 , h

h

where E  i , with 1 ≤ i ≤ N1 , is the rational curve in S  obtained from Eih after π. Let h us fix E  1 . h h h h It holds that E  1 ∩ E  i = ∅ for i = 1: the existence of i0 = 1 such that E  1 ∩ E  i0 = ∅, together with the invariancy of E  by G  , gives h

E  1 ∩ E  i0 ⊂ Sing(G  ). h

Then Sing(G  ) = ∅, against the assumption that R−1 (t) is of type (a) or (b) after π, which implies that G  has not singularities. h Therefore E  1  E  defines a connected component of E  , and it is not possible because E  is connected. Then N1 = 1. 2

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Let B be CP1 with the natural structure of orbifold where the points over which the fiber is of type (b) are affected with multiplicity m. In this case the orbifold universal  is C or CP1 [11, p. 181]. covering of B, B, If KG is the canonical bundle of G it holds [3, §7]: deg(R∗ KG ) = −χorb (B). On the other hand, deg(R∗ KG ) ≥ 0. Otherwise G has a rational first integral, (is a rational fibration) against our assumptions.  is a disk D, which is not possible Moreover, if deg(R∗ KG ) > 0 then χorb (B) < 0 and B 1  because B is C or CP . Then deg(R∗ KG ) = 0, and the Kodaira dimension of G is zero. According to [12, §III and §IV] we can contract G-invariant rational curves on S via a contraction s to obtain a new surface S (maybe with cyclic quotient singularities), a  which is regular on Sing(S),  and a finite covering map r from a reduced foliation G on S, smooth compact projective surface S to S such that: 1) r ramifies only over singularities  is generated by a complete holomorphic vector field Z0 of S and 2) the foliation r∗ (G) on S with isolated zeroes [4, p. 443]. f

CP2

S s

S

r

S

It follows from [4, p. 443] that r can be lifted to S via a birational morphism g : W → S¯ and a ramified covering h : W → S such that s ◦ h = r ◦ g. h

S

W

s◦h

s

g r◦g

S

r

S

Let Y be the lift of Z0 on W via g. Then Y must be a rational vector field on W generating the foliation H given by g ∗ (r∗ (G )) = h∗ G. On the other hand, H is also  on W given by (π ◦ h)∗ X. generated by the rational vector field X We remark from the above construction: • The map g is a composition of blowing-ups over a finite set Θ ⊂ S of regular points of Z0 . • The divisor P of poles of Y is g −1 (Θ). In fact, P is the exceptional divisor of g. Note that P is invariant by H, and that Y is complete and holomorphic on W \ P . Moreover Y has only isolated zeroes in W \ P .

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• It holds that h(P ) ⊂ S is the exceptional divisor of s, which is G-invariant. Moreover, by definition, h only can be ramified on h(P ). Then u = h|W \P : W \ P → S \ h(P ) is a regular covering map. Lemma 5. X has an algebraic trajectory. Proof. Let us suppose that X has no algebraic trajectories. Then h(P ) ⊂ E. As S \ E ⊂ S \ h(P ), it is clear that there is a Zariski open set W  ⊂ W \ P such that u|W  is a regular covering map over S \ E. Thus, π ◦ u|W  is a biholomorphism from W  to C2 , and as a consequence of that, h and r are biholomorphisms. In particular, G is generated by  which is a smooth complex projective the complete holomorphic vector field r∗ Z0 on S, surface. Hence G is generated by the complete rational vector field Z = s∗ (r∗ Z0 ) = h∗ Y on S. The fact that Z is complete and holomorphic on S \ h(P ), and without zeros on S \ h(P ), implies that π∗ (Z|S\E ) is a complete polynomial vector field on C2 , without zeros, and such that it generates FX |C2 . Therefore, X = λ · (π∗ Z|S\E ), with λ ∈ C∗ , and X is complete. Moreover, the flow of X is composed by algebraic automorphisms of C2 : X is defined in terms of r∗ Z0 on S whose flow necessarily generates  algebraic automorphisms of S. According to [4, p. 444], up to a polynomial automorphism, X is one of the two following vector fields: (a) X = λx

∂ ∂ + μy , where λ, μ ∈ C∗ , λ/μ ∈ / Q. ∂x ∂y

(b) X = λx

∂ ∂ + [λny + xn ] , where λ ∈ C∗ , n ∈ N. ∂x ∂y

Booth (a) and (b) have at least one algebraic trajectory, which contradicts our assumption. It finishes the proof of Lemma 5. 2 Let us finish the proof of Proposition 6. Take the algebraic trajectory Cz of X. As Cz is simply-connected, Cz is of type C [16]. Let us consider the leaf L of G such that L \ E is Cz . Then L ∩ E = {p}. After Lemma 4, E = Σt1 ∪ · · · ∪ ΣtM0 ∪ E1h .

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We shall derive a contradiction in the two cases that may arise: Case 1: p ∈ E1h . We know that L ∩ E1h = {p}. As L and E1h are G-invariant, p ∈ Sing(G). The fact that G is generically transverse to R implies that the tangency locus between G and R is a union of several components of a finite amount of fibers of R, and Sing(G) must be contained in it. Hence p must yield on it. We conclude that there must exist a component C1 of a fiber R−1 (t) that passes through p and which is G-invariant. Then C1 must be L or E1h . If C1 = L, then Cz is an algebraic curve contained in a fiber of P , which is invariant by X, against dP (X) = 1. On the other side, C1 is not E1h because is contained in a fiber of R. Case 2: p ∈ E \ E1h . Let Σt ⊂ (Σt1 ∪ · · · ∪ ΣtM0 ) such that L ∩ Σt = {p}. Let C1 be the connected component of Σt such that L ∩ Σt = L ∩ C1 = {p}. It holds that E1h cuts any connected component of Σt : Let C2 be a connected component of Σt such that C2 ∩ E1h = ∅. As C2 ∩ Σti = ∅, if i = , by 2), one concludes that a strict subset C2 ⊂ E defines a connected component of E, which is not possible because E is connected. Let L = π(L). We know that L is a connected curve in S  because L is not contracted h to a point by π. According to E1h ∩ C1 = ∅ and L ∩ C1 = ∅, one obtains that E  1 and L 0 0 must intersect at one singular point of G  on F  t , against the assumption F  t of type (a) or (b). Therefore, Proposition 6 is proven. 2 To finish the proof of Theorem 1 we just need to show that Proposition 7. P is of type C. Proof. Let t ∈ CP1 such that R−1 (t ) is of type (c), (d) or (e). As Eih is G-invariant for h h any i, E  i cuts R−1 (t ) after π in a singular point p ∈ Sing(G  ). Then, E  i is a separatrix of G  through p. According to the models (c), (d), or (e), one concludes that N1 is 1 or 2, and that near t , R−1 (t) ∩ E is either one or two points for any t = t . Then F \ E is C or C∗ . According to [17, pp. 526–528], up to a polynomial automorphism, (1) P = x, if P is of type C, or (2) P = xm (x y + p(x))n , m, n ∈ N+ , (m, n) = 1,  ∈ N, p ∈ C[x] of degree <  with p(0) = 0 if  > 0 or p(x) ≡ 0 if  = 0, if P is of type C∗ . P of type C. A foliation FX |C2 which is x-complete is generated by a vector field of the form ∂ ∂ + [b(x)y + c(x)] , X = a(x) ∂x ∂y with a, b, c ∈ C[x] [1, p. 1230]. As dP (X) = 1, then a(x) = 1.

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P of type C∗ . Let us prove that this case does not really occur. The following lemma uses similar ideas to those on [5, Lemma 3.2]. We include its proof to make the exposition more self-contained. Lemma 6. The line x = 0 is invariant by X. Proof. Consider F2 = R−1 (0). At most one irreducible component of F2 can be noninvariant by G (just look at the blow-up of models of [4, p. 439]). Moreover, if such a non-invariant component exists, then it is everywhere transverse to the foliation. This concludes the case  = 0 in P since at least one irreducible component of {xy = 0} must be invariant. If  > 0, {P = 0} has two disjoint irreducible components, one ({x = 0}) isomorphic to C and other ({x y + p(x) = 0}) isomorphic to C∗ . Let us prove that the first is necessarily invariant. Assume the contrary, and we will obtain a contradiction. Let C be the irreducible component of F2 corresponding to {x = 0} and assume that it is transverse to G. There is only one p ∈ C ∩E. Moreover p is also the unique intersection point between C and the other components of F2 . Because E and F2 \ C are G-invariant, and the foliation is regular at p, there exists a common irreducible component L ⊂ E ∩F2 such that, on a neighborhood U of C, we have E∩U =L∩U

and F2 ∩ U = (L ∩ U ) ∪ C.

On the other hand, if one contracts components of F2 different from C we obtain C0 of type (a) of [4, p. 439] (not like (b), which contains two quotient singularities). The direct image E0 of E is then an invariant divisor which cuts C0 at a single point p0 . Hence it cuts a generic fibre also at a single point, which contradicts that P is of type C∗ . 2 Lemma 6 implies that dP (X) vanishes along {x = 0}, which contradicts dP (X) = 1. Then, P is not of type C∗ . It finishes the proof of Proposition 7. 2 4. Proof of Cerveau’s conjecture As we mentioned in the introduction, Theorem 1, together with some results in [6], gives the following Theorem 2. Let X be a polynomial vector field in C2 . Let us suppose that the derivation DX of C[x, y] is surjective. Then, up to a polynomial change of coordinates, X= where b ∈ C.

∂ ∂ + by , ∂x ∂y

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Proof. If DX is surjective, X satisfies (1) and (2) in the statement of Theorem 1: (1) because 1 ∈ Im(DX ), and (2) by [6, Proposition 1.6]. Therefore, according to Theorem 1, after a polynomial change of coordinates, X=

∂ ∂ + [b(x)y + c(x)] . ∂x ∂y

On the other hand, there is a nonconstant Q ∈ C[x, y] such that DX (Q) is divisible by Q (see [6] and [10, Theorem 2]). Then, any irreducible component {Q(x, y) = 0} defines an algebraic trajectory of X. The existence of one algebraic trajectory implies that c ≡ 0, up to a polynomial automorphism [6, Lemma 6.2]. Let us prove that b(x) is constant. Take g ∈ C[x, y] such that X(g) = y. Explicitly, ∂g ∂g + b(x)y = y. ∂x ∂y Thus, g has the form y˜ g (x, y) + g0 (y), where g˜ ∈ C[x, y], with g˜(0, 0) = 0, and g0 ∈ C[y]. Therefore, the above equation can be rewritten as   ∂˜ g ∂˜ g  + g˜(x, y) + y + g0 (y) b(x)y = y. y ∂x ∂y Then,   ∂˜ g ∂˜ g + g˜(x, y) + y + g0 (y) b(x) = 1. ∂x ∂y In particular, if y = 0 ∂˜ g (x, 0) + (˜ g (x, 0) + g0 (0)) b(x) = 1. ∂x ∂g ˜ Note that ∂x (x, 0) has degree less or equal to the degree of g˜(x, 0) minus 1. Then the relation above can be verified only for a constant b(x). 2

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