complete polynomial vector fieldsin two complex variables

D. Cerveau and B. Scardua

0 Introduction

We consider a vector field X = a(x, y) ∂∂x

+ b(x, y) ∂∂y

on C2, where a(x, y) and b(x, y) are poly-

nomials in (x, y). In what follows we assume that X has isolated singularities, i.e., the singularset Sing(X) := a = b = 0 is finite. We denote by ϕt the local flow of X and we will say thatX is complete if for every point m ∈ C2, the holomorphic map t → ϕt(m) is defined on C. The

map (m, t) → ϕt(m) is therefore holomorphic from C2 to C3 and we have ∂ϕt(m)∂t

= X(ϕt(m)). Thevector field X induces a foliation FX on C2 with singularities at the points of Sing(X); if m ∈ C2

we denote by Lm the leaf of FX passing through m. If X is complete, and this is what we willsuppose in what follows, the leaf Lm can be of three types:

a. Lm = m, if m is a singular pointb. of type C, this is the case if and only if t → ϕt(m) is injectivec. of type C∗, this is the case if m is non singular and t → ϕt(m) is not injective (and therefore

periodic). In this last case there exists τ ∈ C − 0 such that ϕt+τ (m) = ϕt(m).These are the only possible cases, for t → ϕt(m) cannot be doubly-periodic because if so then

its image would be a (compact) elliptic curve in the affine space C2, what is not possible. A vectorfield X can exhibit all three types of leaves, for instance a linear vector field X = x ∂

∂x+ λy ∂

∂y,

λ ∈ C − Q:• the origin 0 ∈ C2 is the unique singular point• the leaves L(0,1) and L(1,0) are of type C∗

• the other leaves are of type C.In case λ ∈ Q − 0 all the leaves, except for L0 = 0, are of type C∗. In order to consider theleaves of type C∗ we proceed as follows. Consider on C2 ×C the closed analytic set Σ′ := (m, t) ∈C2 × C; ϕt(m) = m. Clearly Σ′ contains C2 × 0 and we may put Σ := Σ′ − (C2 × 0). Hence,the closure Σ of Σ is analytic and if (m, t) ∈ Σ with m non singular, then Lm is of type C∗. In caseϕt is globally periodic, i.e., there exists τ = 0 such that ϕτ = IdC2 , the vector field X induces aholomorphic action of the multiplicative group C∗, and we have an induced action of the compactgroup S1 what allows to evoke arguments of average.

The actions of C∗ on C2 have been described by M. Suzuki in a remarkable paper [22], inparticular a vector field X having periodic flow exhibits a meromorphic first integral f non constant.The leaves Lm are therefore all of analytic closure; more precisely Lm coincides with the irreduciblecomponent of f−1(f(m)) passing through m. Assume now that ϕt is not globally periodic; for anyτ = 0, the set of fixed points Fix(ϕτ ) := m ∈ C2; ϕτ (m) = m is not the entire C2. Then Fix(ϕτ )is a proper analytic subset of C2, and hence it is a union of curves and isolated points (necessarily

1

singularities of X). This remark shows that if Lm is of type C∗ then Lm is contained in someFix(ϕτ ); in particular Lm is a closed analytic curve, which is an irreducible component of Fix(ϕτ ).

We shall say that X is of type C∗ if Σ is of dimension 2, and of type C if dim Σ ≤ 1. In a certainsense if X is of type C∗ (respectively of type C), the major part of leaves are of type C∗ (respectivelyC); complete vector fields of type C∗ have been studied by Suzuki in [22] and [23]: actually, a flowon a Stein surface whose majority of leaves is of type C∗ admits a meromorphic fist integral. Incertain cases we can add some more precise information on the nature of this meromorphic firstintegral by the use of techniques we will employ to study the vector fields of type C, which are themain subject of this work.

The techniques we use rely essentially on three remarkable results; the first is the classificationby Lin and Zaidenberg [14] of simply connected algebraic curves (not necessarily connected) on theplane C2. The second, due to MacQuillan and exposed by M. Brunella in [6], describes the algebraicfoliations of surfaces admitting an invariant “entire curve”. We will employ repeatedly an affinegeneralization of this result [3] due to Brunella, that describes the foliations on C2 admitting a leafwith properly embedded isolated planar ends. The third is the description by Rebelo and Ghys-Rebelo of the singularities of semi-complete vector fields, local variant of the notion of completevector field [19],[8], [20].

Some of our results may be stated easily. The first (cf. §1) is a global elementary version of [19]:

Theorem 1. If m0 is an isolated singular point of a complete polynomial vector field X on C2, thenthe first jet of X at m0 is not zero.

Actually, we can describe the nature of the 1-jet j1m0

X: it is non nilpotent and of rank maximum2.

Theorem 2. If X is a complete polynomial vector field on C2, with isolated singularities, then# Sing(X) ≤ 1, i.e., X has at most one singular point.

The intermediate result that follows will be basic in order to state a list of normal forms, up topolynomial automorphisms, of vector fields having a unique singular point.

Theorem 3. If X is complete polynomial with isolated singularities and having exactly one sin-gularity, then X admits an invariant algebraic curve. If X is of type C then this curve is, up topolynomial automorphism, an affine line.

A complete vector field having at least three invariant algebraic curves has a rational firstintegral of type xpyq, p, q ∈ Z (up to polynomial automorphism). It is therefore natural to examinethe complete vector fields having exactly two algebraic invariant curves. This is done by means ofthe Borel type results concerning entire curves [13] and the works of Kizuka about “transcendent”diffeomorphisms of C2 leaving invariant an algebraic curve. We distinguish the following two caseswhere the curves intersect each other or not. We obtain the following classification results:

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Theorem 4. Let X be complete polynomial of type C with isolated singularities on C2. Assume that0 ∈ C2 is singular and that the 1-jet of X at 0 is reduced. Then, up to polynomial automorphism,X belongs to the following list:

1. λ1x∂∂x

+ λ2y∂∂y

, λ2/λ1 /∈ Q

2. λx ∂∂x

+ (a(x)y + b(x)) ∂∂y

, λ ∈ C∗, a and b polynomials a(0) = 0, b(0) = 0.

3. λ1x∂∂x

+ λ2y∂∂y

+ P (xpyq) (qx ∂∂x

− py ∂∂y

), λ1, λ2 ∈ C∗, P polynomial in one variable,

p, q ∈ N, 〈p, q〉 = 1.

We recall that j10 X is reduced if it is diagonalizable with eigenvalues quotient non rational

positive. All the vector fields in the above list are complete; evidently there are some conditions onλ, P , a and b in order to have X of type C and with isolated singularities.

If the 1-jet is not reduced but diagonalizable it is of type λ(qx ∂∂x

+py ∂∂y

), p, q ∈ N, λ ∈ C∗ and

if for instance p, q ≥ 2 then the vector field X is locally linearizable at 0 (Poincare LinearizationTheorem), and therefore the flow ϕt of X is periodic and X of type C∗. We obtain the followingresult (§4).

Theorem 5. Let X be polynomial and complete of type C∗ with isolated singularities and vanishingat 0. Then the 1-jet of X at 0 is of type λ

(qx ∂

∂x+ py ∂

∂y

), λ ∈ C∗, p, q ∈ Z, 〈p, q〉 = 1.

1. If p and q have the same sign there exists a global biholomorphism ψ : C2 → C2 that linearizesX: ψ∗X = λ

(qx ∂

∂x+ py ∂

∂y

).

2. If p and q have opposite signs, then there exists a holomorphic first integral f : C2 → CP (1) forX.

Chapters 5 and 6 are devoted to introduce techniques that may be useful in the exhaustiveclassification of complete vector fields. In particular, we prove

Theorem 6. Let X be polynomial, complete and without invariant algebraic curve on C2. Thereexists a holomorphic non constant first integral F : C2 → CP (1) for X.

If we look at theorem 4, theorem 6 and Suzuki ’s work we see that our complete vector fieldshas no dense orbite. We conjecture this is a general fact.

1 Proof of Theorem 1

This proof goes somehow like that of Rebelo [19], though the global behavior makes it a little bitmore precise. Let therefore X be a complete polynomial vector field, singular at 0. According toCamacho-Sad [7] there exists a local analytic curve γ passing through 0 and invariant by X. Let Lbe the leaf of F that contains γ − 0; if L is of type C then the adjunction of 0 to L produces amap from C ∪ ∞ ∼= CP (1) into L ∪ 0 and therefore L ∪ 0 is a compact holomorphic curveon C2. This is not possible. Assume now that L is of type C∗ and therefore the closure L is an

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analytic curve on C2, as we have already seen. Arguing as above, L is parameterized by C: thereexists Φ: C → C2 holomorphic and primitive such that Φ(C) = L and Φ(0) = 0. We may thereforelift the restriction X|L to a holomorphic vector field Z on the complex line C this one is completebecause its flow ψt satisfies: Φ(ψt(z)) = ϕt(Φ(z)) and also Z(0) = 0. Thus, Z is of the form λz ∂

∂z;

it comes from an elementary computation using the Taylor development of Φ of 0, that the 1-jet ofX is non zero at 0.

Remark 1. The hypothesis that X has isolated singularities is clearly necessary. For instance thevector field y2 ∂

∂xis complete but its 1-jet vanishes at 0.

Remark 2. The curve γ can be non smooth; for instance the curve (y2 − x3 = 0) is invariant bythe linear vector field of type C∗, X = 2x ∂

∂x+ 3y ∂

∂y· We will consider this situation later.

2 Proof of Theorem 2 and complements

This proof uses more or less the same ideas than the preceding one, nevertheless it requires morematerial. We begin by

Lemma 1. Let S1 and S2 be two algebraic curves on C2, disjoint, parameterized injectively by Cand invariant by X complete with isolated singularities. Then X is conjugate to the constant vectorfield ∂

∂x, by a polynomial diffeomorphism.

Proof: The algebraic (non connected) curve S1 ∪ S2 is simply-connected. According to Lin-Zaidenberg [14], up to polynomial automorphism, S1 ∪ S2 is a union of two parallel lines, sayS1 ∪ S2 = p = 0 ∪ p = 1 where p is the polynomial p(x, y) = y. Let m be such that p(m) = 0and the flow of X be denoted by ϕt . Since S1 and S2 are leaves of X the function t → p(ϕt(m))cannot assume the values 0 and 1, so that it is constant. Thus X is colinear to ∂

∂yand since X has

isolated singularities we obtain the result after conjugating X by a homothety. Now we prove Theorem 2. Assume by contradiction that X has two distinct singularities p1 and

p2 . We consider two local invariant curves γj pj , j = 1, 2 given by Camacho-Sad Theorem.Denote by Lj the leaf of FX containing γj − pj. We distinguish two cases according to the typeC∗ or C of X.

1. If X is of type C∗ then, according to Suzuki, X has a non constant meromorphic first integralF . As a consequence, the closure Lj is an irreducible component of some fiber of F and it istherefore a closed analytic curves on C2. Notice that, as in §1, the leaves Lj cannot be isomorphicto C; in that case Lj would be compact. Thus the possibility L1 = L2 is excluded: we would have“L1 ∪ p1 ∪ p2 C∗ ∪ 0 ∪ ∞” that would be compact. Finally the Lj ∪ pj are closed analytic

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curves Lj = Lj ∪ pj parameterized by C and disjoint.

2. If X is of type C we conclude that each Lj is of type C∗ and therefore is contained in a certaincurve Fix(ϕτj

). Hence, as in Case 1, the Lj ∪ pj = Lj are closed analytic curves, distinct andparameterized by C.

If the two analytic curves Lj ∪ pj are algebraic, Lemma 1 gives already a contradiction: thevector field X will be actually conjugate to ∂

∂xand therefore without singularities. We may therefore

assume that L1 ∪ p1 is transcendent. Since its closure is analytic, the leaf L1 C∗ has two ends:one is “the singular point p1 ” and the other defines a planar end, properly embedded, transcendentin the sense of Brunella [3]. In particular, always according to [3], since X has at least one singularpoint, there exists a polynomial P : C2 → C, that we will call Brunella’s polynomial, with connectedgeneric fiber (and of type C or C∗) such that the foliation FX is transverse to the fibers P−1(t)except, maybe, for a finite number of values ti ∈ C, for which at least one of the components ofP−1(ti) is invariant by X. We denote by TangFX , P the set of these components of fibers, thatcorrespond exactly to the set of points where FX and the foliation associate to P are tangent.Clearly, since X vanishes at pj , the curve TangFX , P is non trivial and contains the pj . Itfollows that there exists an irreducible algebraic curve Γj containing pj and invariant by X. Thesame arguments above show that Γj contains a leaf of FX of type C∗; and then Γj is parameterized(injectively) by C and Γ1 ∩ Γ2 = ∅. Again we obtain a contradiction via Lemma 1. The proof ofTheorem 2 is finished.

3 Polynomial complete vector fields - from local to global

We will, parting from properties of the first 1-jet at the singularity, deduce global properties of avector field. Conversely, the fact that X is complete restricts the type of the 1-jet. For instance wehave:

Proposition 1. Let X be a complete polynomial vector field on C2 with an isolated singularity at0. Then the first 1-jet of X is non nilpotent.

Proof: This is a straightforward consequence of the study carried out by Ghys-Rebelo [8]. If X is asin the statement and j1

0(X) is nilpotent, then X has a holomorphic first integral of one of the types:f = y2 + x3, f = y(y − x2) or f = y(y − x2)2. In the three cases the 1-relative cohomology of thefoliation given by the levels of f is non trivial of dimension ≥ 2. This allows to find two holomorphicdifferential 1-forms w1 and w2 , whose restrictions to the leaves of FX (restricted to a neighborhoodof 0) are homologically independent. Since we can choose w1 and w2 to be polynomial, the leavesof FX passing through this neighborhood cannot be of type C or C∗.

In the same spirit we have:

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Proposition 2. Let X be complete polynomial with an isolated singularity at 0. Then the 1-jet ofX at 0 has rank two.

Proof: The nilpotent case was already eliminated. Thus we may assume that the 1-jet is λy ∂∂y

;we know that X is formally conjugate to a vector field of the type

Xp,λ,µ := λy∂

∂y+

xp+1

1 − µxp

∂

∂x, p ≥ 1, λ, µ ∈ C.

Now we evoke the work of Rebelo [20] on semi-complete vector fields having non-null 1-jet; in thatsituation the foliation FX has two smooth and transverse invariant manifolds Γx and Γy , tangentto the coordinate axis (that we can assume to be adequately chosen). As before we consider theleaf L containing Γx − 0; this leaf is necessarily of type C∗, whose (analytic) closure and finallyΓx embeds in an invariant curve Γ parameterized by σ : C → Γ. The inverse image σ∗X is acomplete vector field on C, but an elementary computation shows that the 1-jet of σ∗X vanishes at0: σ∗X = (xp+1 + . . . ) ∂

∂x, what is not possible for a complete vector field on C.

Finally, the allowed 1-jets for a complete polynomial vector field on C2 with an isolated singu-larity at 0 are, up to linear automorphisms, the following:

1. λ1x∂∂x

+ λ2y∂∂y

, λ1, λ2 = 0 2. (λx + y) ∂∂x

+ λy ∂∂y

, λ ∈ C.

According to Briot-Bouquet we know that if X is a given germ of holomorphic vector field at 0 ∈ C2

whose 1-jet is of one the types 1 or 2 above, then X admits at least one smooth invariant curvepassing through 0. In Case 2. this curve is unique; in case 1 if the quotient λ1/λ2 is not a positiveinteger or an inverse of a positive integer, X has exactly two invariant curves, smooth passingthrough 0, that are transverse.

Corollary 1. Let X be complete and polynomial having an isolated singularity at 0. Then X hasan irreducible algebraic invariant curve Γ passing by 0. If Γ is smooth at 0, up to polynomialautomorphism, Γ is an affine line. Otherwise X is of type C∗ and the 1-jet of X is type λ

(qx ∂

∂x+

py ∂∂y

)where p, q ∈ Z and p, q ≥ 2 and 〈p, q〉 = 1. In particular, the flow of X admits a global

period.

Proof: We come back to the arguments in §2. If γ is an invariant curve then either γ is algebraicor γ is transcendent and produces via a Brunella’s polynomial, transverse to FX , an algebraicinvariant curve denoted Γ. In both cases we construct an irreducible algebraic invariant curve Γparameterized by σ : C → Γ. Necessarily σ−1(0) = 0, because σ∗X is complete : we avoid thecurves with double points. Hence, the germ Γ,0 of γ at 0 is irreducible. If γ, 0 is not smooth itis well-known that the 1-jet of X is of type λ

(qx ∂

∂x+ py ∂

∂y

), as in the statement. In this case

the field X is locally linearizable (Theorem of Poincare). In particular the flow of X admits aglobal period ϕη = Id and it is therefore of type C∗. If Γ, 0 is smooth then Γ is globally smooth

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(simply-connected) because X has no other singularities than 0. We conclude the corollary usingAbhyankar-Moh Theorem [1].

We note that we have obtain theorem 3.

As we have already remarked if the 1-jet of X is generic, then X has only two germs of invariantcurves at 0, that are smooth and transverse. If we consider the same arguments as in Corollary 1then at least one of these curves is contained in a “global” algebraic curve, isomorphic to the linex = 0. It may occur that both local invariant curves are actually (global) algebraic. In this case,according to Lin-Zaidenberg [14], these two curves are algebraically isomorphic to xy = 0. In thenext proposition we examine in details this situation:

Proposition 3. Let X be a complete polynomial vector field having 0 as isolated singular point.Assume that both axes x = 0 and y = 0 are invariant. Then, either FX has the first integralxpyq for some p, q ∈ Z and in this case X is linear; or FX is defined by a closed one form Ω =λ dx

x+ µ dy

y+ dH, where λ, µ ∈ C and H ∈ C[x, y].

Proof: We write X = x(a + A(x, y)) ∂∂x

+ y(b + B(x, y)) ∂∂y

with a, b ∈ C, A and B polynomials

vanishing at 0. Let m be a point off the axes xy = 0. We can write: ϕt(m) =(eα(t), eβ(t)

)where

α and β are two entire functions. Write A(x, y) =∑

i,j∈N

Aij xiyj and B(x, y) =∑

i,j∈N

Bij xi, yj and

assume for instance A ≡ 0 (clearly, if A = B = 0, then X is linear and satisfies the statement).The flows equation ∂ϕt

∂t(m) = X(ϕt(m)) gives:

α′(t) = a +∑i,j

Aij eiα(t)+jβ(t)

β′(t) = b +∑i,j

Bij eiα(t)+jβ(t)(*)

Suppose there exists an index (k, ) such that we have kAk + Bk = 0. The condition (*)implies the relation below:

[ka + jb− kα′(t)− β′(t)] · e−(kα(t)+β(t)) +∑

(i,j) =(k,)

(kAij + Bij)e(i−k)α(t)+(j−)β(t) = −(kAk + Bk).

This is an expression of type g0 eh0 + g1 eh1 + ... + gs ehs = 1 where the gj , hj are entire functions.Since gj is constant for j ≥ 1 and g0 = h′

0 + cte we are under the hypothesis of Borel’s Theorem(Lang [13] pages 191,192,193) which asserts the existence of indexes m = n such that hm−hn = cte.Thus we get a relation of type: pα(t) + qβ(t) = cte for certain p, q ∈ Z. This means that the leafLm is contained in a level of the rational function xpyq. Since the set Z is countable and the setof points m is not, we find two integers, still denoted p and q, such that xpyq is a meromorphic

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first integral for X. Necessarily X = f(qx ∂

∂x− py ∂

∂y

)with f polynomial and, since X has isolated

singularities, f is constant.Suppose now that for every (i, j) we have iAij + jBij = 0, and consider the differential form

w = y(b + B)dx − x(a + A)dy. This 1-form (polynomial) defines FX and we have:

w

xy= (b +

∑i,j

Bij xiyj)dx

x− (a +

∑i,j

Aij xiyj)dy

y·

We have

d(w

xy) = −((∂A/∂x)

/y + (∂B/∂y)

/x) =

∑i,j

(iAij + jBij)xi−1yj−1 dx ∧ dy = 0.

Thus wxy

is closed rational with simple poles (along the axes) and therefore wxy

is of type λ dxx

+µ dyy

+dH where λ, µ ∈ C and H is a polynomial.

Remark 3. The proof does not use exactly the fact that X is complete but, does use the fact thatthe flow ϕt(m) is defined on the entire C for uncountably many leaves of X. The proof appliestherefore every time X is rational with poles along xy = 0 and complete on C2 − xy = 0.Notice also that the condition of isolated singularities is not used except to pass from a first integralxpyq to a linear model for X.

Remark 4. The constants λ and µ appearing in the statement are non zero, otherwise w = xy dHwould vanish on a curve and so would X.

Now we give the normal forms for the vector fields X satisfying Proposition 3, but not necessarilywith isolated singularities.

Proposition 4. Let X be complete such that FX is defined by a 1-form Ω = λ dxx

+ µ dyy

+ dH,

H polynomial. There exist p, q ∈ N and a one variable polynomial τ such that H(x, y) = τ(xpyq).Moreover if X has isolated singularities then X is of one of the following types:

1. linear X = λ′x ∂∂x

+ µ′y ∂∂y

(λ′λ + µ′µ = 0)

2. X = λ′x ∂∂x

+ µ′y ∂∂y

+ τ(xpyq)(qx ∂

∂x− py ∂

∂y

), p, q ∈ N and τ is a one variable polynomial.

Furthermore, all these vector fields are complete.

Proof: We begin by verifying that a vector field of type 2 is complete. In order to integrate the

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system:

(1) x(t) = λ′ x(t) + q x(t)τ(xp(t)yq(t)), x(0) = x

(2) y(t) = µ′ y(t) = py(t)τ(xp(t)yq(t)), y(0) = y.

we put u(t) = x(t)p y(t)q; we obtain

u(t) = u(t)

(p

x(t)

x(t)+ q

y(t)

y(t)

)= u(t)(pλ′ + qµ′)

and u(t) = xpyq exp(pλ′ + qp′)t.The integration of (1) and (2) is therefore equivalent to the integration of

x(t) = x(t)(λ′ + qτ(xpyq exp[pλ′ + qµ′)t])

y(t) = y(t)(µ′ − pτ(xpyq exp[(pλ′ + qµ′)t])

which is immediate. We obtain ϕt(x, y) = (x(t), y(t)) where

x(t) = x expλ′t + qτ(xpyq exp[(pλ′ + qµ′)t]y(t) = y expµ′t − p′τ(xpyq exp[(pλ′ + qµ′)t])

and τ is a polynomial in one variable.We shall show now that H is of the form τ(xpyq). Notice that the multivalued function xλyµ ·eH(x,y)

is a first integral of X. Let m be a point outside the coordinate axes; we write ϕt(m) = (eα(t), eβ(t))where α and β are entire functions. If H(x, y) =

∑i,j∈N

hijxiyj we obtain:

λα(t) + µβ(t) +∑

(i,j)/hij =0

hij eiα(t)+µα(t) = c, c ∈ C (*)

If λ and µ are identically zero then Borel’s Theorem gives a relation of type pα(t) + qβ(t) = cte,p, q ∈ Z; by an argument involving countable and uncountable sets we obtain a first integral of typexpyq for FX . Since H is polynomial we can choose p and q positive and relatively prime, hence inthis case H splits as H(x, y) = τ(xpyq). If X has isolated singularities we have

X = λ′(

qx∂

∂x− py

∂

∂y

), λ′ ∈ C∗.

Assume now λ = 0 and H is not of the type τ(xpyq) for any τ , p and q. There are therefore twoindexes (k, ) and (m,n) such that hk · hm,n = 0 and d = kn − m = 0. Let u and v be entirefunctions defined by

du = kα + β

dv = mα + nβ

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Hence expression (*) becomes: λu + µv + hk edu + hmn edv +∑

qij =0 qij eiu+jv = cte., where λ, µand qij may be explicitly calculated in terms of λ, µ and hij . Once again evoking Borel’s Theoremand the fact that exp(f) grows faster than f (cf. [13]) we obtain a relation pu(t) + qv(t) = cte, forcertain integers p, q, relatively prime; as usual by means of a countable sets argument we proceedto find a first integral of type xpyq for FX . We conclude therefore that (λ, µ) = δ(p, q), δ ∈ C and(qx ∂

∂x− py ∂

∂y) · H = 0. If we choose p, q positive and relatively prime we obtain a factorization

H = τ(xpyq); what is an absurd.

Remark 5. As in Remark 3 we use in the proof of Proposition 4 only the fact uncountably manyleaves of X are parameterized by C.

We finish the study of complete vector fields having an isolated singularity at 0 and having twotransverse local invariant curves by the

Proposition 5. Let X be a complete vector field having an isolated singularity at 0 and admittingtwo local invariant curves, smooth and transverse at 0. If X is of type C then X is algebraicallyconjugate to one of the following models

λx∂

∂x+ µy

∂

∂y+ τ(xpyq)

(qx

∂

∂x− py

∂

∂y

), p, q ∈ N

λx∂

∂x+ (a(x) + b(x)y)

∂

∂y,

where λ, µ ∈ C, τ , a and b are polynomials in one variable.

Remark 6. Suppose X is complete and its 1-jet is λ1x∂∂x

+λ2y∂∂y

with λ1/λ2 /∈ Q. Then X admitstwo local invariant curves as in the statement and X is necessarily of type C. Indeed, if X is of typeC∗ then the leaves of FX have analytic closure what contradicts the local topological properties ofthe leaves of X at 0. We also notice that among the preceding models some of them are of type C∗.

Proof of Proposition 5: Let γ1 and γ2 be two local invariant analytic curves, L1 and L2 theleaves of FX containing γ1 − 0 and γ2 − 0 respectively. If L1 and L2 are algebraic there existsan automorphism mapping L1 and L2 onto the axes xy = 0; we evoke now Propositions 3 and4; otherwise one of these leaves, say L1 is transcendent and we produce a Brunella’s polynomial Pnormalized by P (0) = 0; we will here use more deeply the results of Brunella. In case the genericfiber of P is C then we have, up to a polynomial automorphism, P (x, y) = x and X(x, y) of theform X = a(x) ∂

∂x+(b(x)+c(x)y) ∂

∂y· Such a vector field is complete and has an isolated singularity

if, and only if, (up to translation) a(x) = λx, λ ∈ C∗.Suppose now that the polynomial P has generic fiber C∗. According to the terminology of

Brunella L1 is “at the infinity” with respect to P , what means that P |L1 : L1 → C is a finitecovering at infinity (see [3]).

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In fact, since L1 is closed and analytic, necessarily P |L1 : L1 → C is a finite morphism. Thefoliation FX is transverse to the fibration P outside a finite set of fibers. The holonomy of thisfoliation is a subgroup of the group of automorphisms of the generic fiber P−1(ε) C∗ that exhibitsa finite orbit because L1 ∩ P−1(ε) is a finite set of points. Since Aut(C∗) C∗ Z/2Z z →λz; λ ∈ C∗, z → 1

z, necessarily the holonomy group is a finite group (periodic rotations). Hence,

each leaf L of FX intersects the generic fiber P−1(ε) at a finite number of points. Now, if we supposeX is of type C, then for m generic t → ϕt(m) is injective and the map t → P (ϕt(m)) assumes onlya finite number of times the value ε (except for a finite set of ε). It follows that t → P (ϕt(m))is a polynomial of degree dm and always via an argument of countable sets we conclude that

P (ϕt(m)) =∑

j≤N

aj(m)tj where the aj are holomorphic on m. Since P (ϕt(m)) =∞∑0

tn

n!X(n)(P )(m),

where X(n)(P ) is the n-th derivative of P according to X, there is an integer n such that X(n)(P ) ≡ 0and X(n+1)(P ) = 0. If X(n)(P ) is non constant, then X(n)(P ) is a first integral for X, but thisimplies that L1 is algebraic and therefore non transcendent. Now it suffices to remark that X(n)(P )cannot be a non zero constant because X vanishes at 0; this gives a contradiction.

Comment: Above we have fully used the fact that X is complete.

The previous propositions imply Theorem 4.

4 Complements on vector fields of type C∗C∗C∗

Let X be polynomial and complete having an isolated singularity at 0 ∈ C2 and 1-jet λ(qx ∂∂x

+py ∂∂y

)

with λ ∈ C∗, p, q positive integers 〈p, q〉 = 1. According to Poincare Linearization Theorem, X islocally holomorphically linearizable in the following cases:(i) p ≥ 2 and q ≥ 2(ii) p = q = 1(iii) p = 1, q > 1 and X has two invariant analytic curves passing through 0, what is equivalentto X admit a local meromorphic first integral at 0.

In what follows in this chapter we suppose that one of the above conditions is satisfied. Inparticular, there exists a local holomorphic diffeomorphism F at 0 such that

(∗) ϕt F (x, y) = F(xeqλt, yepλt

)Indeed, by a standard argument F extends injectively to the entire C2. Its image F (C2) can, apriori, be different from C2: it is therefore a Fatou-Bierberbach domain, i.e., F (C2) is biholomorphicto C2 but different from C2. If all the local invariant curves at 0 globalize into algebraic curvesthen, as it is will be stated clearly in Chapter §5, X admits a rational first integral that, up to apolynomial automorphism, can be written as xp/yq; in this case X is therefore linearizable by an

11

algebraic automorphism. If this is not the case, we can find a Brunella’s polynomial P transverseto FX ; we argue more or less as in the proof of Proposition 5. If the generic fiber of P is C thenX is algebraically equivalent to a vector field α(x) ∂

∂x+ (β(x)y + γ(x)) ∂

∂y, α, β, γ polynomials

and clearly it is in fact of type λ(qx ∂

∂x+ (a(x)y + b(x)) ∂

∂y

), a(0) = p, b(0) = 0. If the generic

fiber of P is C∗ then, since FX is not equivalent to a trivial foliation F ∂∂x

(because X has a singular

point), the same arguments as in the proof of Proposition 5 say that each generic leaf L of FX

cuts the generic fiber P−1(ε) in a finite number of points. According to (*), ϕt is periodic, sayof period 2iπ for sake of simplicity (i.e., we assume λ = 1). Hence ϕt factorizes in an action ofC∗, i.e., ϕt(m) = ψet(m) where ψ : C1 × C∗ → C2, (m, s) → ψs(m) is holomorphic. Genericallyon m the map s → ψs(m) is injective, so that s → P (ψs(m)) is a map with finite fibers from C∗

to C. Since a holomorphic map with finite fibers from C∗ to C is the quotient of a polynomialQ(s) by a power of s we obtain P (ϕt(m)) =

∑Nj≡−N aj(m)ejt, where the aj are holomorphic on

C2. Notice that P (m) =N∑

j=−N

aj(m) and also X(P )(m) = ∂∂t

P ϕt(m)|t=0 =∑N

j=−N jaj(m) hence

X(k)P (m) = ∂k

∂tkP ϕt(m)|t=0 =

∑Nj=−N jkaj(m). We observe that the polynomials X(j)(P ) may

be expressed as a Van der Monde matrix in terms of the aj ; and thus the aj are also polynomials.From the flow (additive) property we get:

P (ϕt+s(m)) =N∑

j=−N

aj(m)ejtejs =N∑

j=−N

aj(ϕt(m))ejt

what gives aj(ϕs(m)) = ejsaj(m). Since ϕt is locally linearizable, if m is close enough to 0,ϕt(m) → 0 as t → −∞ for real values. We conclude that a0 constant and if aj is non identicallynull, j = 0, then j is positive:

P (ϕt(m)) = a0 +N∑

j=1

aj(m)ejt, a0 ∈ C.

Let us consider the holomorphic function aj = aj F . We have

aj

(xeqt, yept

)= aj(ϕt F (x, y)) = ejtaj F (x, y) = ejt aj(x, y).

A straightforward computation shows then that the aj are quasi-homogeneous polynomials; in

particular the holomorphic function P = P F =∑

aj is a polynomial. The holomorphic in-

jective map between algebraic curves F |(P=ε) : (P = ε) → (P = ε) admits an injective extension

(and a fortiori bijective) between smooth compactifications (P = ε) and (P = ε). It follows that

F |(P=ε) : (P = ε) → (P = ε) avoids at most a finite number of points. Since the image of F

is invariant by X (and therefore also its complementary), the image of F is the complementary

12

of a finite number of leaves of FX that are closed analytic curves. This is absurd by topologicalreasons; indeed F (C2) C2 is simply-connected and if (f = 0) is the equation of a leaf L containedin C2 − F (C2) then df

fhas a non null integral along a small loop γ, contained in F (C2), around

(f = 0). In particular γ is non trivial in π1(F (C2); ∗ ) what is absurd, therefore F is surjective.

As a consequence we have:

Proposition 6. Let X be a complete polynomial vector field having an isolated singularity at 0.

Assume that j10X = λ

(qx ∂

∂x+ py ∂

∂y

), λ ∈ C∗, p, q ∈ N, 〈p, q〉 = 1. If X is locally linearizable

at 0; then X is linearizable by a holomorphic diffeomorphism of C2.

Proof: According to the above discussion it is enough to remark that a vector field of type

λ(qx ∂

∂x+ (a(x)y + b(x)) ∂

∂y

), a(0) = p, b(0) = 0, what is locally linearizable, is globally lineariz-

able by a biholomorphism as above: this is a simple exercise on Riccati equations. We have the following

Corollary 2. Let X be polynomial complete having an isolated singularity at 0. Suppose thatj10X = λ

(qx ∂

∂x+ py ∂

∂y

), λ ∈ C∗, p, q ∈ N, 〈p, q〉 = 1. If X is of type C∗ then X is globally

linearizable by a biholomorphism of C∗.

Proof: If X is of type C∗, its leaves have analytic closure; in particular X is locally linearizable at0. Remark 7. Clearly, condition of type C∗ is automatic if (p, q) = (1, 1) or p ≥ 2 and q ≥ 2, theonly case where it may not be verified is therefore (p, q) = (1, n), n ∈ N − 1.Remark 8. If a germ of holomorphic vector field with non nilpotent linear part has a non smoothlocal invariant curve, then j1

0X λ(qx ∂

∂x+ py ∂

∂y

), λ ∈ C∗, p, q ∈ Z, 〈p, q〉 = 1 , p ≥ 2, q ≥ 2.

If X has two local invariant curves that are smooth and tangent at 0 then p = 1, q ≥ 2, and in thiscase X is locally linearizable.

The above statements give the proof of Theorem5 except for (2). This point is a simple remark:if X is of type C∗ it has a meromorphic first integral f whose only base point must be the origin;but the form of the 1-jet of X implies that f or 1

fis holomorphic at 0.

5 Complete vector fields and algebraic invariant curves -

Polynomial flows

We have already remarked that the presence of invariant algebraic curves allows to obtain importantinformation; for instance in Lemma 1 and Propositions 3 and 5. Corollary 1 and Proposition 6 (cf.

13

Remark 8) give information on the case we have a non smooth invariant curve. We will give someingredients in order to carry out the study of complete vector fields admitting such invariant curves.

We first notice that if X is complete and admits at least 3 distinct invariant algebraic curves,then X has infinitely many of such curves, more precisely all the leaves of FX have algebraic closure.This is a direct consequence of Borel-Nishino Theorem [13] which states that the image of an entiremap from C to C2, avoiding three distinct algebraic curves, is contained in an algebraic curve;indeed if S1, S2, S3 are invariant algebraic curves for X and if m ∈ C2 − S1 ∪ S2 ∪ S3 applyingBorel-Nishino Theorem to the flow map t → ϕt(m), we obtain:

Proposition 7. Let X be a complete polynomial vector field on C2 having isolated singularitieswhose leaves have algebraic closure. Then, up to polynomial automorphism, X admits a first integralof type xpyq, p, q ∈ Z not all zero, and therefore X is either of type ∂

∂xor of type λ

(qx ∂

∂x−py ∂

∂y

),

λ ∈ C∗, up to polynomial automorphism.

Comment. There are other complete vector fields admitting a rational first integral, but with nonisolated singularities. For instance xpyq

(qx ∂

∂x− py ∂

∂y

).

Proof (Proposition 7): The proof is a consequence of a result by H. Saito and M. Suzuki (see[23]: if R is a rational function on C2 whose fibers are of type C or C∗ then, up to polynomialautomorphisms of C2 and left composition by a Moebius map, R is of the form:(i) x(ii) xpyq, 〈p, q〉 = 1, p, q ∈ Z(iii) xp(xy +P (x))q, 〈p, q〉 = 1, p, q ∈ Z, ≥ 1; P polynomial of degree ≤ − 1, P (0) = 0.To each one of the cases (1), (2), (3) we associate the vector fields X1 = ∂

∂x, X2 = qx ∂

∂x− py ∂

∂y

and X3 = qx+1 ∂∂x

−(p + q)xy + pP (x) + qxP ′(x) ∂∂y

which are with isolated singularities. If X

is like in the statement then, according to Darboux-Jouanolou Theorem [11], X admits a rationalfirst integral. Thus, up to polynomial automorphism, X is C-colinear with some Xj because of theisolated singularities hypothesis. We observe that for ≥ 1 the vector field X3 is not complete andthis ends the proof of Proposition 7.

Now we shall give the ingredients of another proof, this one not so direct, but which permitto introduce results that will be useful later. We shall say that a complete vector field on C2 haspolynomial flow if for each t ∈ C the diffeomorphism ϕt : C2 → C2 is polynomial. The descriptionof polynomial flows is well known ([2],[24] for instance); up to polynomial automorphism ϕt is oftype

(i) (x, y ebt), b ∈ C

(ii) (x + t, y ebt), b ∈ C

(iii) (x, y + tp(x)), p(x) polynomial in x

14

(iv) (xeat, yebt), a, b ∈ C∗

(v) (xet, eakt(y + txk)), a ∈ C∗, k ∈ N∗.

Notice that the flows (i) and (iii) correspond to the vector fields that are not with isolatedsingularities: by ∂

∂yand p(x) ∂

∂x, respectively. The flow (ii) corresponds to the vector field without

singularities ∂∂x

+ by ∂∂y

· The flow (v) is the flow of “Poincare-Dulac”. We point out that there arecomplete polynomial vector fields whose flows are not algebraic; actually this is the case of mustexamples in Proposition 5.

The second ingredient, that may be useful in the complete classification we are concerned with,is a remarkable result due to Kizuka [12] that we shall now present. Let φ : C2 → C2 be anautomorphism, holomorphic and non polynomial. Suppose φ leaves invariant an algebraic curveS, not necessarily irreducible. Then S is, up to polynomial automorphism, of one of the followingtypes

(1) ϕ(x) + yψ(x) = 0, where ϕ and ψ are one variable polynomials.

(2) a union of irreducible components of fibers of xpyq, where p and q are positive integers with〈p, q〉 = 1.

(3) a union of irreducible components of fibers of a polynomial Q of type Q(x, y) = xp(xy+P (x))q,where , p, q are integers, 〈p, q〉 = 1 and P is a one variable polynomial of degree at most − 1, P (0) = 0.

We shall say therefore that S is of type (1), (2) or (3) in the obvious sense. It is now easy to guessthe proof of Proposition 7. If the flow ϕt of X is algebraic we regard the list of normal forms above;otherwise we apply Kizuka’s Theorem and proceed more or less like in the preceding proof.

5.2. We shall see in Chapter 6 the interest for the study of complete vector fields X admittingexactly two (irreducible) algebraic invariant curves S1 and S2 . We shall present here a study usingstrongly the result of Kizuka. Like always our vector fields X will be polynomial, complete andhaving isolated singularities. If our curves Sj intersect or are non smooth, then X has a singularpoint and the study is essentially done specially for the vector fields of type C, in the precedingchapters.

We will therefore suppose S1 and S2 smooth, disjoint invariant by the complete field X. Let usmake some useful remarks.

1. Since S1 and S2 contain orbits of X, S1 and S2 are of type C or C∗.

2. If Sj is of type C∗ then X has no singularity on Sj .

3. We may suppose X is non singular on C2−(S1∪S2); otherwise X would have a third algebraicinvariant curve and hence a rational first integral.

15

4. In particular, if S1 and S2 are of type C∗, we may suppose X is non-singular.

5. If S1 and S2 are of type C and S1 ∩ S2 = ∅ then, by Lemma 1, X is conjugate to ∂∂x

·6. We may assume S1 ∪ S2 belongs to the list of Kizuka, because the list of algebraic flows is

known.

We shall study specially the case S1 C and S2 C∗. Notice that case (1) in Kizuka’s list(ϕ(x)+ yψ(x) = 0) produces no factor isomorphic to C except if there exists a factorization of typeϕ(x) + yψ(x) = xm(ϕ(x) + yψ(x)). In this case S1 = x = 0 and S2 = ϕ(x) + yψ(x) = 0. SinceS1 and S2 are disjoint (by hypothesis) it follows that ϕ(0) = 0 and ψ(0) = 0; moreover S2 C∗

implies that ψ has some (unique) zero. Finally, we may consider (up to algebraic automorphism)S1 = x = 0 and S2 = xy + P (x) = 0, with ≥ 1, degree of P ≤ − 1 and P (0) = 0.

Examining Kizuka’s list we conclude, via the preceding remarks, that it remains to study thethree following cases:

(I)S1 = x = 0, S2 = xy + P (x) = 0, degree P ≤ − 1, P (0) = 0

(II) S1 = x = 0 S2 = xpyq = 1

(III)

S1 = x = 0, S2 = xp(xy + P (x))q = 1, degree P ≤ − 1, P (0) = 0

5.2.1. We shall begin by studying Case (I) where X leaves invariant the curves S1 = x = 0 andS2 = xy+P (x) = 0, X with isolated singularities. The birational map Π(x, y) = (x, xy+P (x))is biholomorphic on C∗×C; hence the direct image Π(FX) defines a foliation on C∗×C that extendsalgebraically to C2. We choose X ′ a vector field with isolated singularities that defines this foliation.We remark that Π(S2) = y = 0 is invariant by FX′ ; according to Remark 3 we may assume thatX and therefore X ′ has no singularity in C2 − x = 0. We have two cases to consider, accordingto the fact that the line x = 0 is invariant by FX′ or not.

(a) If x = 0 is not invariant by FX′ then X is of type C∗. Indeed, if (0, y0) is a generic point

of x = 0, the local leaf L′(0,y0) of FX′ by (0, y0) is contained in a certain leaf L of FX′|C∗×C . In

particular, the global leaf L′(0,y0) of FX′ contains Π(L) ∪ (0, y0). Since L′

(0,y0) is non compact,

Π(L) L is necessarily of type C∗. The vector field X ′ has a meromorphic first integral F (nonconstant) and therefore F = F ′Π is a first integral for X. We can summarize this study as follows.If X, that we may assume without rational first integral, has a singular point, that is a base pointfor F we apply Proposition 6. If X is without singularities or has a singularity which is not an

16

indeterminacy point of F , we prove, using FX′ , the existence of a Brunella’s polynomial for X ′ andtherefore for X.

(b) Suppose now x = 0 is invariant by FX′ . We can apply Remarks 3 and 5 to the “completerational vector field” Π∗X. The foliation FX′ is given by a rational 1-form of the following type:

Ω′ = λdx

x+ µ

dy

y+ dτ(xpyq).

The vector field (dual) X ′ defined by X ′ = µx ∂∂x

− λy ∂∂y

+ xpyqτ ′(xpyq)(qx ∂

∂x− py ∂

∂y

)is tangent

to FX′ and complete as we have seen in Proposition 4. It may occur that X ′ is not with isolatedsingularities, for instance if λ = 0 and p > 0. Notice that if Sing X ′ contains a curve different ofx = 0 and y = 0 then, always by an argument of Borel-Nishino Theorem, the foliation FX′ ,

and therefore FX , exhibits a non constant rational first integral. We can then assume X ′ is nonsingular in C2 − (x = 0 ∪ y = 0) = W . Let L′ be a generic leaf of FX′ , this is either an

embedded C or C∗ in W and equipped with two complete holomorphic vector fields Π∗X|L′ and

X ′|L′ , both without singularities.

These two vector fields are therefore C-colinear. Thus we may write Π∗X = K ′ · X ′ where K ′ isa rational function which is constant on the leaves of FX′ . Either K ′ is non constant and X has arational non constant first integral K = K ′ Π (and cf. Proposition 8) or K ′ ∈ C is constant, sayK ′ = 1. A straightforward calculation shows that

X = x(µ + qC)∂

∂x−

(xy + P (x))(λ + pC)

x+

(xy + xP ′(x))(µ + qC)

x

∂

∂y

with C = xp(xy + P (x))q · τ ′(xp(xy + P (x))q). Since X must be holomorphic it follows that x

divides (xy +P (x))(λ+pC)+(xy +xP ′(x)) · (µ+pC). Since ≥ 1 and P (0) = 0 it follows thatnecessarily λ = 0; if µ = 0, X ′ has a first integral xpyq and X has a first integral xp(xy + P (x))q.We have seen in Proposition 7 that such a first integral is not associated to a complete vector field,therefore µ = 0 and X ′ has the first integral yµ eτ(xpyq). In particular, the line x = 0 is notinvariant by FX′ as we have initially supposed. Therefore case (b) does not occur and we obtain:

Proposition 8. Let X be a complete polynomial vector field with isolated singularities and havingthe algebraic invariant curves S1 and S2 both smooth, disjoint of type (I). Then X is of type C∗.

5.2.2 We study now the possibility (II) where S1 = x = 0 and S2 = xpyq = 1. notice that ifq = 1 we are in the preceding situation (P = −1). We have

Lemma 2. Let p, q be positive integers with 〈p, q〉 = 1, q ≥ 2. Let σ : C2 − S1 ∪ S2 be an entiremap. Then the image of σ is contained in a level of xpyq.

17

Proof: Let f(x, y) = xp

(1−xpyq); f vanishes on S1 and is equal to ∞ on S2 . This function induces

a fibration of the open set W = C2 − S1 ∪ S2 ; its fibers xp

1−xpyq = ε, ε = 0, ∞, are connected

for 〈p, q〉 = 1 and if s ∈ C∗ the homothety (x, y) → (sqx, s−py) exchanges the fibers f−1(1) andf−1(spq). Notice that f−1(1) writes in the form xp = 1

1−yq and is therefore a covering with p leaves

of C − ξ1, . . . , ξq, where the ξj are the q-th roots of the unity. Let us consider the surjective map

F : W = C∗×f−1(s) → W defined by F (s, (x0, y0)) = (sqx0, s−py0), (x0, y0) ∈ f−1(1). We observe

that the image by F of the horizontal fibration of W onto C∗ is precisely the fibration of W by thelevels of xpyq. On the other hand the image of the vertical fibration in f−1(1) is the fibration of Wby the levels of f . We verify explicitly that if σ : t → (x(t), y(t)) is an entire map of C into W then

σ lifts into σ : C → W such that F σ = σ. If proj2 : W → f−1(1) is the second projection thenproj2 σ : C → f−1(1) is necessarily constant because q ≥ 2. This proves the lemma.

Corollary 3. Let X be polynomial complete with isolated singularities, leaving invariant S1 ∪S2 oftype (II). Then X is conjugate to a linear vector field λ(qx ∂

∂x− py ∂

∂y), p, q ∈ N.

Remark: Like always, the hypothesis of isolated singularities cannot be deleted. For instance thevector fields τ(xpyq)(qx ∂

∂x− py ∂

∂y) leave S1 ∪ S2 invariant and are non linear.

5.2.3. We arrive in case (III) where S1 = x = 0 and S2 = xp(xy + P (x))q = 1, P (0) = 0,degree P ≤ − 1. In an evident way there is an analogous of Lemma 2:

Lemma 3. Let σ : C → C2 − S1 ∪ S2 be an entire map with 〈p, q〉 = 1 and q ≥ 2. Then the imageof σ is contained in a level of xp(xy + P (x))q.

Proof: If Π(x, y) = (x, xy + P (x)) the entire map Π σ satisfies Lemma 2. Since, as we have already pointed out, there is no complete vector field with isolated singularities

associated to xp(xy + P (x))q, the above situation does not occur. It remains the case q = 1, thatis, S2 = xp(xy + P (x)) = 1 which may be studied more or less with the preceding techniques.Once again we consider Π(x, y) = (x, xy + P (x)) which is a biholomorphism of C2 − S1 ; let F ′ bethe direct image of FX by Π. If x = 0 is not invariant by the algebraic extension of F ′ to C2 thenwe may perform an analysis analogous to that in 5.2.1, and again X is of type C∗. We can alsoobtain new normal forms by means of an explicit calculation that we leave to the reader (pull-backby Π of the normal forms obtained in 5.2.1).

5.3. We shall examine now the case where X, polynomial complete with isolated singularities,leaves invariant S1 and S2 both smooth, disjoint and of type C∗. As we have already remarked, wemay assume X is non singular. The possible models of Kizuka are the following:

(I∗)

S1 = xpyq = αS2 = xpyq = β, 〈p, q〉 = 1, α = β, αβ = 0. (1)

18

(II∗)

S1 = xp(xy + P (x))q = αS2 = xp(xy + P (x))q = β 〈p, q〉 = 1, α = β, αβ = 0, P(0) = 0. (2)

(III∗)

S1 = xy + P (x) = 0S2 = xp(xy + P (x))q = β, 〈p, q〉 = 1, β = 0, P(0) = 0. (3)

Cases (I∗) and (II∗), and case (III∗) with p = 0, and q = 1, lead us, via Picard’s Theorem,to the existence of rational first integral for X. In the general case for (III∗) (i.e., p = 0), ifσ : C → C2 − S1 ∪ S2 is an entire map and Π(x, y) = (x, xy + P (x)), then Π σ has values inC2 − (y = 0 ∪ xpyq = β). If p ≥ 2 we can apply Lemma 2 to Π σ and construct again arational first integral. The only remaining case is the following

S1 = xpy + P (x) = 0S2 = x(xy + P (x))q = 1, p ≥ 1, P (0) = 0, degree P ≤ − 1.

We conjecture results identical to the preceding results. In particular it is our belief that a completevector field X leaving S1 and S2 invariant is always of type C∗.

6 Results of Brunella, Mendes, MacQuillan and applica-

tions ([3],[4],[5], [15],[16], [17],[18])

In the text [5] on the work of MacQuillan and Mendes, M. Brunella states the following result:

Theorem 7 (MacQuillan). Let F be a reduced foliation on an algebraic surface M ; assume thereexists an entire curve f : C → M non constant and tangent to F (f ∗F = 0) whose, image is Zariskidense. Then we have one of the three following cases:

1. F is transverse to the generic fibers of a rational fibration Π: M → B.

2. F is transverse to the generic fibers of an elliptic fibration E : M → C.

3. There exist a ramified covering p : N → M and a birational transformation F : Z − −− > Nsuch that the foliation F ∗p∗F of Z is defined by a global vector field with isolated singularities.

The above statement deserves some remarks; the term reduced foliation means reduced in the senseof Seidenberg’s Theorem for reduction of singularities ([3], [21]). If F is a foliation on a surface M

we may, by a sequence of blow ups at the singular points, obtain a birational map Π : M → M such

19

that the foliation Π∗F = F on M is reduced (this is Seidenberg’s Theorem). If F has an invariant

entire curve f : C → M , f ∗F = 0, then f lifts into f : C → M , entire curve invariant by F .In cases 1. and 2. the fibrations can exhibit singular fibers and each irreducible component of

a fiber not completely transverse to F is necessarily invariant by F .We remark that curves B and C necessarily are either CP (1) or elliptic curves; this comes from

the existence of a holomorphic map C → B or C. Moreover if M is rational it follows that B (orC) is a CP (1).

Let X be a complete polynomial vector field on C2; FX extends algebraically to CP (2) and weshall still denote this extension by FX . Clearly, unless X admits a non constant meromorphic firstintegral (Proposition 7), we may evoke the preceding theorem after the reduction of singularities ofFX . Evidently the techniques we have introduced, in particular the use of Brunella’s polynomials,may be seen as particular cases of Theorem7 above; but they sometimes are more efficient becausethey are concerned with the affine classification.

6.1. We shall place ourselves in the case of a complete vector field X on C2 such that the eventualsingularity of X in C2 is reduced. Nevertheless, in general, the extension of FX to CP (2) has non re-duced singularities. After performing the reduction of singularities of FX “at the infinity” we obtain

a reduced foliation FX on a rational surface CP (2); the pair (CP (2), FX) is a “compactification”

of the pair (C2,FX), i.e., FX |C2 = FX , where C2 ⊂ CP (2) naturally.

The complementary of C2 in CP (2) is a normal crossing divisor D = D1∪· · ·∪Dn where each Dj isisomorphic to CP (1). We notice that if X is of type C and does not have a (non constant) rational

first integral, then each Dj is invariant by FX .

6.2 Suppose that FX leads us to Case 3. of Theorem7. By following the construction of [5] we

remark that, with the above notations, the ramifications of p : N → M = CP (2) are on the divisorD. Thus, there exists a Zariski open set Σ ⊂ N such that p|Σ : Σ → C2 is a covering map, andtherefore a holomorphic diffeomorphism. Finally, in this situation, there exists a birational mapF : Z → CP (2), biholomorphic on C2 ⊂ CP (2), such that F ∗FX is defined by a global vector fieldY with isolated singularities on Z.

This can be interpreted as follows: up to polynomial automorphism, FX extends to a compact-ification of C2 over which it is defined by a global vector field Y with isolated singularities. SinceX is also (by hypothesis) with isolated singularities, we have X = h · Y |C2 where h is necessarilyconstant. Finally we obtain

Proposition 9. Let X be a complete polynomial vector field with isolated and reduced singularitieson C2, satisfying 3. of Theorem 7. Then the flow of X is algebraic.

Proof: According to the above argumentation we may embed X into X on a compactification

CP (2) of C2. By construction the flow ϕt of X coincides on C2 with the flow ϕt of X on C2, in

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particular ϕt(C2) = C2, ∀ t ∈ C. If P is polynomial in two variables then P ϕt is rational on

CP (2) and holomorphic on C2, and therefore it is polynomial. Thus ϕt is polynomial for all t.

6.3. Suppose now FX is transverse to a rational or elliptic fibration f : CP (2) → CP (1). Wedenote by Λ = t0, t1, . . . , ts ⊂ CP (1) the set of values t such that f−1(t) is not transverse to the

foliation FX . We have the following remarks:

(1) If t∗ ∈ CP (1)−Λ, then the fibration f is locally trivial over a neighborhood of t∗; the trivialization

is given by the transverse foliation FX .

(2) If m is a singular point of FX then f(m) ∈ Λ; the same holds if m is a singular point of thefiber f−1(f(m)).

(3) If tj ∈ Λ then at least one irreducible component of f−1(tj) is invariant by FX , more precisely,if m is a singular point of a fiber, then at least one local branch of f−1(f(m)) at m is invariant by

FX .

(4) The foliation without singularities FX |CP (2)−f−1(Λ)is transverse, in the sense of theory of folia-

tions, to the fibration f |CP (2)−f−1(Λ)

: CP (2)− f−1(Λ) → CP (2)−Λ. In particular, we can consider

the global holonomy of this foliation; it is a representation Hol : π1(CP (1)−Λ; t∗) → Aut(f−1(t∗)),where t∗ is a base point in CP (1) − Λ. The group G = Hol(π1(CP (1) − Λ; t∗)) ⊂ Aut(f−1(t∗))is therefore generate by the Hol(γi) where the γi are loops of index δi

j (Kronecker symbol) around

the tj . Notice that if f−1(tj) has a non invariant (by FX) irreducible component then Hol(γj) isperiodic; this irreducible component may be or not at finite distance.

Suppose, as it is the case if X is of type C, that the divisor D = D0 ∪ · · · ∪ Dn is invariant byFX , where D is the divisor resulting from the reduction of singularities of FX to which we haveadded the strict transform D0 of the line at infinity CP (2)\C2 CP (1). Notice that:

(i) D is not completely invariant by the fibration f .

(ii) To each non invariant component Dj of D by the fibration f we can associate a finite orbitf−1(t∗) ∩ Dj of the holonomy group G.

In particular if the fibration f is elliptic then G is a finite group (of order ≤ 6); in case the fibrationis rational we evoke the classification of elementary groups (cf. Beardon for instance). In precisewords, up to conjugacy, a subgroup of Aut(CP (1)) that leaves invariant a finite set is of one of

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following types:

1. a subgroup of the affine group z → az + b, a = 02. a subgroup of the group generated by the maps (z → λz) and the involution (z → 1

z)

3. a group of periodic rotations (z → e2πik

p z); and in this case G is finite.

We obtain this way a qualitative description of the foliation FX in the complementary of f−1(Λ)and therefore in C2 − f−1(Λ).

a. In Case 3, as in the elliptic case, the vector field X admits a non constant meromorphic firstintegral F defined in C2 − f−1(Λ); this first integral extends (clearly) to the branches of f−1(Λ)

that are not invariant by FX . The function F is not defined along the branches of f−1(Λ)∩C2 thatare invariant by FX . As habitually we may suppose that there are at most two such branches, andat this point we feel that the techniques of the preceding chapter can be useful.

b. In the generic Case 1. the group G has a unique fixed point. Suppose, as it occurs if X is oftype C, that all the components Dj of the divisor D are invariant by FX . Then the fibration f hasa unique dicritical divisor (i.e., a unique non invariant component Dj of D) say Dj0 and necessarilyf |Dj0

: Dj0 → CP(1) is injective. It follows from Abhyankar-Moh Theorem that the fibration f |C2 isup to polynomial automorphism, either of the form x = cte if Dj0 is at infinity, or of type x

y= cte

if Dj0 is at finite distance.In the first case, we have already seen, X is, up to polynomial automorphism, of type (i) (λx+µ) ∂

∂x+

(a(x)y + b(x)) ∂∂y

and in the second case of one of the following types: (i) ∂∂x

, (ii)λx ∂∂x

+ µy ∂∂y

,

or type (iii) λ[(x + yn) ∂∂x

+ my ∂∂y

], λ, µ ∈ C, n ∈ N.c. In case G has two fixed points, and exactly two fixed points, the group is up to conjugacy asubgroup of the group of homotheties. The two fixed points correspond to two components Dj0 andDj1 of D that are dicritical for f and over which f is injective. The reduction of singularities offunctions of type xpyq, p = q ∈ Z give such examples. Here we have the possibility of classifyingall these fibrations and deduce the classification of the vector fiels X, or the possibility of evokingclassical arguments from the Theory of Foliations. In the complementary of f−1(Λ) the foliation

FX is given by a closed meromorphic 1-form Ω that may be extended along the branches of f−1(Λ)

that are not invariant by FX ; it remains as above the invariant branches at finite distance whichwe suppose to be at most one and to which we will try to apply the techniques of Kizuka’s Theorem

type. In all known examples the meromorphic 1-form Ω extends rationally to CP (2) and thereforeto C2; there are good reasons to think that this is actually a general phenomena. In an ancientpre-print we have given, using the same techniques, the list of complete polynomial vector fieldswhose associate foliation is also given by a closed rational 1-form: the models found therein areamong those we have found above.

d. In case G is a subgroup of the group(

1z, λz

)only one branch Dj0 of the divisor D is dicritical for

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f and f |Dj0is generically 2:1. We do not know whether we can find complete vector fields with such

configuration. Anyway, the foliation FX admits a Liouvillan first integral on the complementary of

f−1(Λ).

6.4. Suppose now the complete vector field X has no algebraic invariant curve on C2; in particularX has no singularity and all the leaves of FX are Zariski dense. Examining the list given in 5.1we conclude that the flow of X is not algebraic. Thus, after the reduction of the singularities ofthe foliation FX extended to CP (2), the resulting foliation is transverse to a rational or ellipticfibration f . According to the above discussion, and with the same notations, we are in one of thefollowing situations:

1. X is polynomially conjugate to ∂∂x

+ (a(x)y + b(x)) ∂∂y

; we notice that, generically for a and b,such a vector field has no algebraic invariant curve.

2. FX has a non constant meromorphic first integral F defined in the complementary of f−1(Λ).

Since X has no algebraic invariant curve, the affine part f−1(Λ) ∩ C2 is not invariant by X and

hence F |C2−f−1(Λ) extends meromorphically to a meromorphic first integral F defined on the entire

C2; in fact F is holomorphic from C2 to CP (1).

3. FX is defined by a closed meromorphic 1-form Ω on the complementary of f−1(Λ) that extends

to C2 as a 1-form Ω as in 2. Indeed, by construction, the closed 1-form Ω has no finite distancepoles and thus Ω is holomorphic. Therefore X has a holomorphic first integral.

4. In the case the holonomy is a subgroup of (1z, λz), if ω is a polynomial 1-form with isolated

singularities defining FX on C2 (ω(X) = 0) we can find a closed meromorphic 1-form ω1 on thecomplementary of f−1(Λ) such that dω = ω1 ∧ ω and that extends to the entire C2. We obtaintherefore a closed holomorphic 1-form ω1 = dg

g= ω1|C2 , holomorphic on C2, such that dω = ω1 ∧ ω

and g holomorphic and non vanishing on C2. In particular, g−1ω is closed and holomorphic so thatω = gdF for some holomorphic first integral F for X on the entire C2. We have proved the followingtheorem 6 of the introduction :

Theorem 8. Let X be complete and polynomial without algebraic invariant curve on C2. Thereexists F : C2 → CP (1) a holomorphic non constant first integral for X. In particular the orbits ofthe flow of X are closed analytic curves.

Note that this theorem implies that all orbits are proper. In particular, we can apply the resultsof M. Suzuki to obtain analytical models for X (see [23]).

Before finishing, we point out that our study does not contain the classification of completevector fields without singularities and with exactly one algebraic invariant curve, as well as thevector fields having 1-jet of type λ(x ∂

∂x+ (x + y) ∂

∂y) and λ(x ∂

∂x+ ny ∂

∂y), λ ∈ C∗, n ∈ N, that

correspond to the normal forms of Poincare. These last cases that exhibit an invariant line, up to

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automorphism, may be related by blow-up techniques to non singular vector fields with an invariantline. It would be interesting to study these cases in order to finish, grosso modo, the classification.

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[12] T. Kizuka; Analytic automorphisms and algebraic automorphisms of C2; Tohoku Math. Journ.31, (1979), 553-565.

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[15] M. McQuillan; Diophantine approximations and foliations. Inst. Hautes Etudes Sci. Publ. Math.No. 87, (1998), 121–174.

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[17] L.G. Mendes; On algebraic and entire curves invariant by singular holomorphic foliations; Pre-print Univ. Dijon, 1999.

[18] L.G. Mendes; Kodaira dimension of holomorphic singular foliations. Boletim da SociedadeBrasileira de Matematica. Rio de Janeiro, v.31, n.2, p.127 - 143, 2000.

[19] J. Rebelo; Singularites des flots holomorphes; 46 2 (1996); 411-428. Ann. Inst. Fourier 1996.

[20] J. Rebelo; Realisation de feuilletages holomorphes pour des champs semi-complet en dimension2. Ann. Fac. Sci. Toulouse, Mat. (6), 9-(2000), no. 4, 735-763.

[21] A. Seidenberg; Reduction of singularities of the differential equation Ady = Bdx; Amer. J. ofMath. 90 (1968),248-269.

[22] M. Suzuki; Sur les operations holomorphes de C et de C∗ sur un espace de Stein; SeminaireNorguet, Springer Lect. Notes, 670 (1977), 80-88.

[23] M. Suzuki; Sur les operations holomorphes du groupe additif complexe sur l’espace de deuxvariables complexes; Ann. Sci. Ec. Norm. Sup. 4 e serie, t.10, 1977, p. 517 a 546.

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Dominique CERVEAUUniversite de Rennes I - IRMARCampus de Beaulieu - 35042 - Rennes CedexFRANCE

Bruno SCARDUAInstituto de MatematicaUniversidade Federal do Rio de JaneiroCaixa Postal 68530CEP. 21945-970 Rio de Janeiro - RJBRASIL

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