Total rigidity of polynomial foliations on the complex projective plane

ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2007, Vol. 259, pp. 60–72. c© Pleiades Publishing, Ltd., 2007.Original Russian Text c© Yu. S. Ilyashenko, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 64–76.

Total Rigidity of Polynomial Foliationson the Complex Projective Plane

Yu. S. Ilyashenko a,b,c,d

Received November 2006

To Vladimir Igorevich Arnold with admiration and love

Abstract—Polynomial foliations of the complex plane are topologically rigid. Roughly speak-ing, this means that the topological equivalence of two foliations implies their affine equivalence.There exist various nonequivalent formalizations of the notion of topological rigidity. Genericpolynomial foliations of fixed degree have the so-called property of absolute rigidity, which isthe weakest form of topological rigidity. This property was discovered by the author morethan 30 years ago. The genericity conditions imposed at that time were very restrictive. Sincethen, this topic has been studied by Shcherbakov, Gomez-Mont, Nakai, Lins Neto–Sad–Scardua,Loray–Rebelo, and others. They relaxed the genericity conditions and increased the dimension.The main conjecture in this field states that a generic polynomial foliation of the complex planeis topologically equivalent to only finitely many foliations. The main result of this paper isweaker than this conjecture but also makes it possible to compare topological types of distantfoliations.

DOI: 10.1134/S0081543807040050

1. TOTAL RIGIDITY THEOREMS

1.1. Topological rigidity of polynomial foliations. The foliations of the complex pro-jective plane by analytic curves are topologically rigid. Roughly speaking, this means that thetopological equivalence of two foliations implies their analytic and even affine equivalence. Withoutadditional stipulations, this assertion is false. For example, as a rule, a continuous deformation ofthe coefficients of a generic polynomial in two variables yields a local family of foliations that aretopologically equivalent but not analytically equivalent. There exist various nonequivalent ways toformalize the notion of topological rigidity. As a result, various notions with different names arise.In this paper, we suggest new versions of the notion of topological rigidity and state conjecturesand theorems about them.

1.2. Total rigidity: Conjectures. The foliation by analytic curves determined by a poly-nomial vector field P has the form

x = P(x), x = (x, y) ∈ C2. (1)

Denote by An the set of all foliations of the complex projective plane specified by equation (1)with a polynomial of degree at most n in a fixed affine neighborhood. The projective change

(x, y) �→ (z,w) =(

1x

,y

x

)(2)

a Faculty of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119991 Russia.b Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.c Independent University of Moscow, Bol’shoi Vlas’evskii per. 11, Moscow, 119002 Russia.d Cornell University, 217 Eastern Heights Drive, Ithaca, NY, 14850 USA.

E-mail address: [email protected]

60

TOTAL RIGIDITY OF POLYNOMIAL FOLIATIONS 61

transforms equation (1) intodz

dw= zR(z,w). (3)

Here, R is a rational function of degree n + 1 that can be expressed in terms of P. In the newcoordinates, the infinite line has the form z = 0. The zeros of the polynomial R(0, w) on this lineare the infinite singular points of equation (1). The number of such points is at most n + 1, and itequals n + 1 for a generic equation. The equations of class An with n + 1 infinite singular pointsform, by definition, a subclass A′

n. The ratios of eigenvalues of the linearizations of equation (3) atinfinite singular points are called the characteristic numbers of these singular points; the eigenvaluein the denominator corresponds to the tangent vector to the infinite line. These numbers are alsocalled the characteristic numbers of equation (1) at infinity and denoted by λj. It follows from theexplicit formulas for the function R and the residue theorem that

∑λj = 1.

Definition 1. A foliation of class An is said to be totally rigid if it is topologically equivalentto only finitely many foliations of this class up to affine equivalence.

In other words, a foliation from An is totally rigid if there exist only finitely many affine equiv-alence classes of foliations topologically equivalent to this foliation.

Definition 2. A singular point of an analytic foliation of the complex plane is called a complexcenter if it has a local analytic first integral with nondegenerate quadratic part.

Conjecture 1. Suppose that a foliation F ∈ A′n, n > 1,

(i) has no positive characteristic numbers at infinity ;(ii) has a nonsolvable monodromy group at infinity ;(iii) has n2 singular points in a distinguished affine complex plane, and none of these singular

points is a complex center.

Then the foliation F is totally rigid.For quadratic vector fields, the list of equations with a complex center was compiled by Dulac

and Zoladek.For the same fields, the list of equations with nonpositive characteristic values at infinity and a

nonsolvable monodromy group at infinity was compiled by Pyartli. Both these lists are given belowin Section 6.

Conjecture 2. Suppose that a complex quadratic vector field of class A′2

(i) has no positive characteristic numbers at infinity ;(ii) is not contained in Pyartli’s list ;(iii) is not contained in the Dulac–Zoladek list.

Then this field is totally rigid.The main results of this paper are weaker versions of these conjectures. We formulate and prove

them below.

1.3. Absolute rigidity. The phenomenon of topological rigidity was discovered in [2] andcalled absolute rigidity.

Definition 3. A foliation F of class An is said to be absolutely rigid if there exist a neigh-borhood of the foliation F in the class An and a neighborhood of the identity homeomorphism inthe space of all self-homeomorphisms of the complex projective plane such that any foliation in theformer neighborhood that is conjugate to F by a homeomorphism from the latter neighborhood isaffine equivalent to F .

Theorem [2]. Any generic foliation of class An is absolutely rigid.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 259 2007

62 Yu.S. ILYASHENKO

The genericity conditions in this theorem eliminate a dense subset of equations of class An.Shcherbakov [8, 9], Lins Neto with coauthors [4], and Nakai [6] substantially reduced this exceptionalset. Let us formulate Shcherbakov’s theorem. In addition to rigidity (this is the only propertydiscussed below), it includes the density of leaves and the existence of countably many complexcycles.

Theorem [9]. The space An with n ≥ 2 contains a real algebraic subset Σn and a nowheredense real analytic subset Σ′

n of real codimension at least 2 such that each foliation F in the setAn \ (Σn ∪ Σ′

n) has the following properties :

(1) each leaf of F is dense in C2;

(2) F is absolutely rigid ;(3) the foliation F has a countable set of homologically independent complex limit cycles

for n ≥ 3.

In the theorems presented below, we remove both closeness conditions in Definition 3; namely,the conjugating homeomorphism need not be close to the identity and the topologically equivalentfoliation need not be close to a given foliation.

1.4. Complexity of homeomorphisms. We endow the projective space of the parameters ofdegree n polynomial foliations with an arbitrary metric, e.g., the Fubini–Study metric. We identifyeach point in the parameter space with the corresponding foliation and use for points and foliationsthe same notation F ,G, . . . .

Consider the bundle of infinite solutions over the space A′n: the fiber over a point F ∈ A′

n is theinfinite solution LF of the equation F . Take an equation F0 for which the infinite points are the(n + 1)th roots of 1. Consider a set of generators γ1(F0), . . . , γn(F0) of the group π1(LF0). Eachloop γj consists of a line segment joining 0 to a point close to the root εj , a circle centered at εj

and going around εj in the positive direction, and the same segment with the opposite orientation.Consider the universal covering A of A′

n with the base point F0, projection π, and the metric dlifted from A′

n. Let Π be the bundle over A whose fiber over the point F is the fundamental groupπ1(LF ). Consider the continuous sections sj : An → Π for which sj(F0) = γj(F0). Finally, considerthe fundamental domain A0 defined by

A0 ={F ∈ A

∣∣ d(F0, F) ≤ d(F ′0, F) for each F ′

0 ∈ π−1(F0)}.

Thus, for each point F ∈ A0, we have constructed a basis, i.e., a system of generators, s(F) ={s1(F), . . . , sn(F)} in the group π1(LF ).

Let F and G be two points in A0, and let h : LF → LG be a homeomorphism. Consider twobases s(G) and h(s(F)) in the group π1(LG). Each generator h(sj(F)) can be expressed in terms ofthe generators s(G) as a word over an alphabet of n + 1 letters. Let us denote the minimum lengthof such a word by Cj(h). The complexity of the homeomorphism h is defined as

C(h) = maxj

Cj(h).

Consider a homeomorphism H : CP2 → CP2 that leaves the infinite line invariant and maps LFto LG . We define the complexity of the triple H,F ,G as

C(H,F ,G) = C(h), where h = H|LF .

Let A′′n denote the set of those foliations in A′

n that have precisely n2 singular points in adistinguished affine neighborhood.



Definition 4. A foliation F ∈ A′′n is C-rigid if the set of classes of affine equivalent foliations

F ′ ∈ A′′n conjugate to F by a homeomorphism H of complexity at most C,

C(H,F ,F ′) ≤ C,

forms a discrete subset in the space A′′n/Aff . A set of classes of affine equivalence is discrete in the

space A′′n/Aff if each compact set in the space A′′

n intersects only finitely many classes from this set.

1.5. Theorems on C -rigidity.Theorem 1. Suppose that a foliation F ∈ A′

n, n > 1,

(i) has no real characteristic numbers at infinity ;(ii) has a nonsolvable monodromy group at infinity ;(iii) has n2 singular points in a distinguished affine complex plane, and none of these singular

points is a complex center.

Then the foliation F is C-rigid for any C > 0.Theorem 2. Suppose that a complex quadratic vector field of class A′′

2

(i) has no real characteristic numbers at infinity ;(ii) is not contained in Pyartli’s list ;(iii) is not contained in the Dulac–Zoladek list.

Then the corresponding foliation is C-rigid for any C > 0.These theorems, together with Theorem 3 stated below, are the main results of this paper. The

proof of Theorem 3 is based on ideas of S.Yu. Yakovenko. These ideas were used in the proof of aweaker assertion contained in the book [3].

2. ISOMONODROMIC FOLIATIONS

Definition 5. Two foliations are said to be isomonodromic if they belong to the set A′n and

their monodromy groups at infinity are analytically equivalent.The latter condition means that there exists a correspondence between generators of these groups

such that the generators of one group are conjugate to the corresponding generators of the othergroup by the same biholomorphic mapping:

h ◦ fj = gj ◦ h, j = 1, . . . , n.

Below we define a subset of the set of isomonodromic foliations in which the dependence of thefundamental group of the infinite solution on the parameter is continuous in a certain sense.

Definition 6. Let F ∈ A′n. By I(F) we denote the set of all foliations F ′ ∈ A′

n that areisomonodromic to F ; to be more presice,

• for any F ′ ∈ I(F), there exists a neighborhood U ⊂ A′n in which a set of generators γj(G),

j = 1, . . . , n, of the fundamental group of the corresponding infinite leaf is defined that dependscontinuously on the point G ∈ U ;

• the monodromy transformations Δ(γj(F ′),F ′) and Δ(γj(G),G) for G ∈ U ∩ I(F) are conju-gated by the germ of a biholomorphic mapping h(F ′,G) that depends on F ′ and G but doesnot depend on j:

Δ(γj(F ′),F ′) ◦ h(F ′,G) = h(F ′,G) ◦ Δ(γj(G),G).

Let I0(F) denote the connected component of I(F) containing F .


64 Yu.S. ILYASHENKO

Theorem 3. If a foliation F satisfies the conditions of Theorem 1, then the set I0(F) coincideswith the class of affine equivalence of the foliation F .

Lemma 1. Let F be a foliation satisfying conditions (i) and (ii) in Theorem 1, and let F ′

and G be the same as in Definition 6. Then the germ h(F ′,G) is uniquely determined and dependson F ′ and G analytically.

Proof. This lemma coincides with a lemma from [2] up to insignificant details. A similarassertion is proved below in Section 5.

3. FOLIATED DEFORMATIONS

In this and the next section, we prove Theorem 3.

3.1. Definition and existence theorem.Definition 7. A one-parameter family of foliations {Ft ∈ A′

n | t ∈ (C, 0)} is called a foliateddeformation if there exists a holomorphic foliation T in X = (C, 0) × CP2 of codimension 1 thatinduces the deformation Ft in the following sense: For each p ∈ {0} × CP2 and each t ∈ (C, 0),there exists a leaf ϕt of the foliation Ft such that

Φp ∩ ({t} × CP2) = ϕt.

Here Φp is the leaf of T containing p.Lemma 2. Let F be a foliation satisfying conditions (i) and (ii) in Theorem 1. Then, for any

F ′ ∈ I(F) and any holomorphic mapping ψ : (C, 0) → I(F), 0 �→ F ′, the family {Ft = ψ(t) | t ∈(C, 0)} is a foliated deformation.

Lemma 2 is proved in the next four subsections.

3.2. Saturation lemma. Let h(F ′,G) be the same as in Definition 6. We set

ht = (F ′, ψ(t)), (4)

and let Γ be a transversal to the infinite solution such that all the mappings ht are defined on Γ.Lemma 3. There exists a neighborhood U of the infinite line in CP2 such that

(1) for any t ∈ (C, 0), the closure of the saturation of the transversal {t} × Γ with the leaves ofthe foliation Ft contains U ;

(2) for any p ∈ Γ, a connected component of the intersection ϕp,t ∩ U intersects Γ along an orbitof the monodromy group at infinity of the foliation Ft; here ϕp,t is the leaf of Ft containing p.

Proof. The infinite singular points Ojt of the foliation Ft are nonresonant of Poincare type,because their characteristic numbers are nonreal. Therefore, every such point has a neighborhood Vj

in which the equation is analytically equivalent to a linear equation. In normalizing coordinates,the corresponding foliation has the form zj = cw

λj

j , where wj is a local coordinate on the infinitesolution such that wj(Oj) = 0 and λj is the characteristic number of the point Oj . We denote theseparatrix Vj ∩ (wj = 0) of the foliation Ft by Cjt and the union of all such separatrices over tand j by C.

Let Ω be the complement of the infinite solution to the union of open disks Dj ⊂ Vj ∩ (zj = 0)centered at Oj . There exists a neighborhood U0 of Ω saturated with the leaves ϕp,t ∩ U0, p ∈ Γ. Ifthe neighborhood U0 is sufficiently small, then condition (2) of Lemma 3 holds in this neighborhoodby the continuous dependence theorem. Let Γjt ⊂ U0 ∩ Vj be a transversal intersecting Lt at apoint qjt ∈ Dj . The saturation of any such transversal with the connected components of the leavesϕq,t ∩ Vj , q ∈ Γjt, coincides with Vj \ Cjt. This follows from the formula for the leaves zj = cw

λj

j

and the fact that λ is nonreal.



Let U =(⋃n+1

1 Vj

)∪ U0. Then the saturation described in the lemma contains U \ C, and its

closure contains U . This completes the proof of Lemma 3.

3.3. Construction of the leaves of a “large foliation.” Let U be the same neighborhoodas in Lemma 3. We set

Y = (C, 0) × U and Z = (C, 0) × Γ.

The family of holomorphisms (4) determines a foliation τ of some open set Z0 ⊂ Z containing(C, 0) × {0}. The leaves of the foliation τ are defined as the curves swept out by points under thehomotopy ht:

τp = {(t, ht(p)) | t ∈ (C, 0)}, p ∈ Γ.

The leaf of a “large foliation” passing through a point p ∈ Γ is defined as the saturation of the leaf τp

with the leaves of the foliations Ft:Φp =

⋃q∈τp

ϕq ∩ U.

Here

q = (t, ht(p)) and ϕq = {t} × ϕt,ht(p). (5)

Recall that ϕt,s is the leaf of Ft passing through s. The leaf Φp is well defined: the leaves ϕq donot change when replacing p with a different point p′ ∈ ϕp ∩ Γ. Indeed, by Lemma 3, p′ = Δ0(p),where Δ0 is a monodromy transformation at infinity of the foliation F0. Hence,

Δt = ht ◦ Δ0 ◦ h−1t

is a monodromy transformation of the foliation Ft. Therefore,

ht(p′) = ht ◦ Δ0(p) = Δt ◦ ht(p) ∈ ϕq.

The leaves Φp, p ∈ Γ, are well-defined holomorphic 2-manifolds. This follows from the theoremon the analytic dependence of leaves on initial conditions and parameters. The manifolds Φp arepairwise disjoint, because so are the leaves of the foliations τ and Ft.

The leaves Φp, p ∈ Γ, thus defined fill Y \ C. To complete the construction of the leaves of alarge foliation T , we add the leaves swept out by the separatrices of infinite singular points:

Cj =⋃

t∈(C,0)

Cjt.

The leaves Cj are holomorphic 2-manifolds disjoint from each other and from the leaves Φp.

3.4. Analyticity of holonomy. To prove that the partition into leaves constructed aboveis a holomorphic foliation, we must show that the holonomy transformations of the topologicalfoliation T are biholomorphic.

For the leaves Φp, this follows directly from the definition. Indeed, it suffices to prove therequired property for any curve on Φp starting at p ∈ {0} × Γ. There is a homotopy on thisleaf that preserves the endpoints of such a curve and transforms the curve into the union of acurve τp with endpoint q = (t, ht(p)) and a curve on the leaf ϕq (see (5)). The correspondingholonomy transformation is the product of the holonomy transformations of τ and Ft. Since bothtransformations are holomorphic, their product is also holomorphic.


66 Yu.S. ILYASHENKO

It remains to prove that the holonomy transformations of the leaves Cj are holomorphic. Thisfollows from the fact that the foliations Ft near infinite singular points are analytically equivalentto linear foliations.

In more detail, we can show, as in the proof of Lemma 3, that each infinite singular point Ojt

of the foliation Ft has a neighborhood Vjt in which this foliation is analytically equivalent to alinear one; the point Ojt depends on t holomorphically. Let z = zjt and w = wjt be coordinates inwhich the foliation Ft linearizes. In the neighborhood Vjt, its leaves have the form z = cwλ withλ = λj . Since the foliations are isomonodromic, λ does not depend on t. Let Γjt be a transversalof the form w = c with the coordinate z on it. For this transversal, the monodromy transformationcorresponding to going around the point Ojt on an infinite solution is linear, i.e., has the formz �→ νz with ν = e2πiλ. Since Imλ �= 0, we have |ν| �= 1. For each t ∈ (C, 0), on the infiniteleaf, consider the curve γjt of the form {(s, 0, c) | s ∈ [0, t]} in the coordinates (t, z, w). We denotethe holonomy transformation of the infinite leaf along this curve by Δt = Δγjt : Γj0 → Γjt. Sincethe family Ft is isomonodromic, the transformation Δ commutes with the multiplication by ν.Therefore, it is linear: Δt(z) = μ(t)z.

Let us construct a special family of transversals Γjt to the leaf Cj. In the coordinates (t, z, w),we have

Cj = {w = 0} and Γjt = {(t, c, w) |w ∈ (C, 0)}.

Let γjt be a curve on Cj of the form {(s, c, 0) | s ∈ [0, t]} in the coordinates (t, z, w).The correspondence Γjt → Γjt along the leaves of Ft is defined for z �= 0 and has the form

z �→ w = c1z−λ. It conjugates the holonomy transformations along the curves γjt and γjt.

Taking into account the equality γjt(z) = νz, we see that

γjt(w) = γjt

(c1z

−λ)

= c1(νz)−λ = c1ν−λw

for z �= 0. Thus, the mapping γjt is linear for w �= 0 and, therefore, for all w.This shows that the holonomy transformations of the leaf Cj are analytic and completes the

proof of Lemma 3.

3.5. From foliations to distributions and vice versa. On the domain Y , the foliation Tis defined locally by holomorphic 1-forms ω = a dz + b dw + c dt. Two forms that define the samefoliation differ by a holomorphic factor. Note that the foliation T is transversal to the planardomains {t} × U at any point of the set Y . Therefore, the coefficients a and b of any form ω thatdefines the foliation do not vanish simultaneously; otherwise, the plane ω = 0 would be tangent tothe plane t = const. This allows us to choose a form ω globally. Namely, suppose that zR = P /Qin equation (3) that defines the foliation Ft, with P and Q coprime polynomials in z and w thatdepend analytically on t. Then the vector field X := (−Q, P , 0) annihilates any form ω definingthe foliation T . At each point of the set Y , one of the functions −Q/a and P /b is holomorphic.Hence, the function R = c(−Q/a) = c(P /b) is defined globally (it does not change when the form ωis multiplied by a nonzero factor). Thus, the foliation T in the intersection of Y with the chart(z,w, t) is defined by the form

ω = −Q dz + P dw + R dt. (6)

Suppose that P = (P,Q) in equation (1). Then, in the intersection of Y with the chart (x, y, t), thefoliation T is defined by the form

ω = −Qdx + P dy + R dt. (7)

The functions P and Q are polynomials in x and y that depend analytically on t. For t = const,the function R is holomorphic on the complement of a compact set to the plane. By the Hartogs–Poincare theorem on the removability of compact singularities, the function R extends to the entire



C2 × (C, 0)! Thus, having started with a foliation defined in a neighborhood of infinity, we con-

structed the corresponding form ω on the domain Y and then extended it to the entire space X,thereby extending the foliation T to X. Indeed, the integrability condition ω ∧ dω = 0 holds in aneighborhood of infinity and, hence, everywhere in X (by the principle of analytic continuation).

The field (P,Q, 0) defining equation (1) is tangent to the foliation T and annihilates the form ωin Y and, therefore, everywhere in X. Thus, the family F and the foliation T satisfy the requirementsof Definition 7. This proves Lemma 2.

4. GENERATOR

In this section, we construct an analytic vector field V such that the shift along this fieldconjugates the foliations F0 and Ft and is an affine transformation. Moreover, the field V istangent to the leaves of T .

Consider the function R in (7). The projective change (2) transforms the form (7) into theform (6), which is holomorphic in a neighborhood of infinity. Then it is easy to show that R is apolynomial of degree at most n + 1; this was done in [2].

Let Σ denote the set of singular points of the foliations Ft, t ∈ (C, 0).The first step in the construction of the field V is as follows.Proposition 1. The function R vanishes at the singular points of the foliation Ft.Proof. Since the foliation ω = 0 is integrable, it follows that ω ∧ dω = 0. Suppose that

P (a) = Q(a) = 0. Thenω ∧ dω(a) = R(Qω + Pz)(a).

The assumption R(a) �= 0 has two consequences. First,

(Pz + Qω)(a) = 0.

This means that the divergence of the field (P,Q) for a fixed t at the point a is equal to zero.Therefore, the eigenvalues of (P,Q) at a are opposite to each other; this is a resonance of type −1 : 1.

Second, the point a is nonsingular for the integrable distribution ω = 0. In a neighborhood of a,one can define integral surfaces of the distribution and a local analytic first integral of the largefoliation T . Restricting the latter to the plane t = const, we obtain an analytic first integral of thefoliation Ft.

Now, let us employ the fact that the foliation Ft has a −1 : 1 resonance at the point a. In otherwords, the point a is a complex center for the linear part. By the Poincare–Lyapunov theorem, if thispoint has an analytic integral, then it has an analytic first integral with nondegenerate quadraticform (is a “complex center”). By the hypothesis of Theorem 3, this is impossible for t = 0.

The condition of not being a complex center determines an open set. Therefore, R = 0 on theset Σ of singular points of the foliations Ft for all small t. Therefore, according to the principle ofanalytic continuation, we have R = 0 on the entire set Σ. This proves Proposition 1.

Proposition 2. The isomonodromic deformation {Ft | t ∈ (C, 0)} has a generator given by avector field

V = (a, b, 1)

whose components a and b are first-degree polynomials in x and y that are analytic in t.Proof. Let Pt, Qt, and Rt denote the restrictions of the polynomials P , Q, and R to the plane

{t} × C2. Proposition 2 follows immediately from the Max Noether theorem. Indeed, under the

conditions of Theorem 1, the polynomials Pt and Qt generate a maximal ideal at each singular pointof the foliation Ft. Therefore, by the Max Noether theorem, the polynomial R, which vanishes atthe common zeros of Pt and Qt, belongs to the ideal generated by Pt and Qt in the polynomial ring.


68 Yu.S. ILYASHENKO

Moreover,Rt = APt + BQt,

where A and B are first-degree polynomials in x and y that are analytic in t. Setting

V = (B,−A, 1),

we obtain the required field.Now, let us show that under the conditions of Lemma 2, the family {Ft | t ∈ (C, 0)} consists of

affine equivalent foliations. This follows immediately from Proposition 3 (see below), because thefield V is a generator of affine mappings Ht that conjugate F0 and Ft. Indeed, let

Ht = gtV ,

where gtV is the time t shift along the field V . The mapping Ht is affine and takes the plane {0}×C

2

to the plane {t} × C2. It conjugates the foliations F0 and Ft, because the field V is tangent to the

large foliation and the deformation {Ft | t ∈ D} is foliated.This proves an analog of Theorem 3 for one-dimensional families.In the general case, we join any two points on I0(F) by finitely many one-dimensional families.

This proves Theorem 3.

5. C-RIGIDITY IN THE CLASS An

In this section, we derive Theorem 1 from Theorem 3.Suppose that Theorem 1 is false. By Definition 4, this means that there exists a foliation F

satisfying the conditions of the theorem and a sequence of foliations Gm that are topologically equiv-alent to F , accumulate at some foliation G0 ∈ A′′

n, and are pairwise affine inequivalent. Moreover,the complexity of the homeomorphisms Hm conjugating F and Gm is uniformly bounded. Belowwe derive a contradiction from this assumption.

By definition, the condition of bounded complexity of the homeomorphisms Hm means thefollowing: Let U ⊂ A′

n be a neighborhood of G0 in which generators γj(G) of the fundamental groupof the infinite leaf LG of G are defined that depend continuously on the foliation G ∈ U . Let γj(F) besimilar generators of π1(LF ). We set hm = Hm|LF and γj(Gm) = hm(γj(F)). Then the generatorsγj(Gm) can be expressed in terms of γj(Gm) by means of words of length not exceeding some C.There are only finitely many such words. Therefore, for infinitely many m and all j = 1, . . . , n, theelements γj(Gm) are expressed in terms of γj(Gm) by means of the same word that does not dependon m. We denote this word by Wj and the set of all such m by M . The system of generatorsWj(γ1(G), . . . , γn(G)) depends continuously on G and coincides with γj(Gm) for all m ∈ M . Thus,we have extended the set of generators γj(Gm), m ∈ M , to a continuous family γj(G), G ∈ U .Consider the monodromy transformations fjG := Δ(γj(G),G). They are defined for all G ∈ Uand depend analytically on G. On the other hand, for each m ∈ M , there exists a biholomorphicgerm hm : (C, 0) → (C, 0) conjugating the monodromy at infinity of the foliations F and Gm; tobe more precise, it conjugates the germs of fjF and fjGm for all j. For any k,m ∈ M , the germhkm := hk ◦ h−1

m conjugates the monodromy groups of the foliations Gm and Gk.Proposition 3. There exists a connected analytic set V ⊂ U that contains G0 and all Gk,

k ∈ M, starting with some K and has the following property : the complement W := V \ I(Gk) iseither the empty set or a proper analytic subset of V .

Proof. The respective multipliers of the monodromy transformations at infinity of all foli-ations Gk, k ≥ 1, are equal, because the monodromy groups of these foliations are analyticallyconjugate. Passing to the limit, we see that the foliation G0 has the same multipliers at infinity.



Hence, there is a hyperbolic germ among the generators fjG0 of the monodromy group of G0. De-note it by f1G0 . We can reduce the neighborhood U , if necessary, so that there exists a family ofhyperbolic germs f1G in U that depends analytically on G. Let hG be a Schroder chart that dependsanalytically on G and conjugates the germ f1G with a linear germ.

Choose K so that Gk ∈ U for all k ≥ K, k ∈ M . We consider only these k hereafter. Thecommutator subgroup of the monodromy group of the foliation GK contains a parabolic germ Pthat does not commute with f1GK

. Hence, there is a family of parabolic monodromy transformationsΔ(G) of the foliations G ∈ U with Δ(GK) = P . In the chart z = hG , the germ Δ(G) has the formz + C(G)zm+1 + . . . , with C(GK) �= 0. If a germ hG conjugates the monodromy groups G(Gk) andG(G), then

(i) hG = c(G)hG for c(G) ∈ C∗, because hG conjugates the germs f1G and f1Gk

that are linearhyperbolic in the charts hG and hGK

, respectively;

(ii) (c(G))m = C(G)/C(Gk), because hG conjugates P and Δ(G):

P ◦ hG = hG ◦ Δ(G).

Thus, all generators fjG of the monodromy group of G that is analytically conjugate to G(GK)satisfy the equations

fjGK◦ (c(G)z) = c(G)fjG , (c(G))m = C(G)/c(GK).

These equations specify an analytic subset V of the neighborhood U , and this subset containsall Gk with k ≥ K. Since V is closed, it also contains G0. If C(G) �= 0 in U , then all mappings hGare biholomorphic and V ⊂ I(GK). If C(G) = 0 on an analytic set W (which does not contain Gk),then the mapping hG , G ∈ W , vanishes identically and is therefore noninvertible. The intersectionV ∩W is a proper (possibly, empty) analytic subset of V . The complement V \W belongs to I(GK).

Now, we can complete the proof of Theorem 1. Let V and W be the same as in Proposition 3.The complement V \W has only a finite number of connected components; if V is a manifold, thenthis number is equal to 1. At least one connected component contains infinitely many points Gk,k ∈ M . It is enough for our purposes that there exist two points Gk and Gl belonging to the sameconnected component V0 of V \ W . We will prove that V0 belongs to the affine equivalence classAff(Gk) of the foliation GK ; however, the foliations Gk and Gl are affine inequivalent by assumption,so we will arrive at a contradiction. To prove that V0 ⊂ Aff(Gk), we employ Theorem 3. To thisend, we must show that Gk satisfies all conditions of Theorem 1.

This is so because the foliation Gk is topologically equivalent to F and the foliation F satisfies theconditions of Theorem 1. Indeed, the monodromy groups of the foliations F and Gk are analyticallyconjugate. Therefore, conditions (i) and (ii) in the theorem hold for these foliations simultaneously.

Suppose that the foliation Gk does not satisfy condition (iii), i.e., has a complex center (thefact that Gk has n2 singular points in C

2 follows immediately from the topological equivalence of Fand Gk). Then one of the singular points of the foliation F is topologically equivalent to a complexcenter. By Naishul’s theorem [5], this singularity is analytically equivalent to a center, i.e., is itselfa complex center. Therefore, F does not satisfy condition (iii) in Theorem 1. This contradictioncompletes the proof of Theorem 1.

6. TOTAL RIGIDITY FOR QUADRATIC VECTOR FIELDS

In this section, we derive Theorem 2 from Theorem 1.

6.1. The Dulac–Zoladek list. Condition (iii) in Theorem 1 prohibits the foliation F fromhaving complex centers. For quadratic vector fields, a complete list of complex centers is well known.


70 Yu.S. ILYASHENKO

In the real case such a list was compiled by Dulac [1] at the beginning of the past century, and inthe complex case it was constructed by Zoladek in 1994 (see [10]).

Theorem (Dulac–Zoladek). A quadratic vector field has a center if and only if it is affineequivalent to a field of the form

z = iz + Az2 + Bzw + Cw2,

w = −iw + C ′z2 + B′zw + A′w2, A, . . . , C ′ ∈ C,

satisfying the conditionsAB − A′B′ = 0,

[(2A + B′)(A − 2B′)CB′ − (2A′ + B)(A′ − 2B)C ′B] = 0,

(BB′ − CC ′)[(2A + B′)B′2C − (2A′ + B)B2C ′] = 0.

These conditions constitute Dulac’s list.

6.2. Pyartli’s list. Recently, A.S. Pyartli has succeeded in enumerating all quadratic vectorfields with a nonsolvable monodromy group at infinity under the additional condition that thecharacteristic numbers at infinity are not positive integer multiples of 1

6 and 14 (see Theorems 4

and 5 below for more details). Under the conditions of Theorem 2, this requirement holds due toassumption (i).

6.3. Proof of Theorem 2. If a quadratic vector field does not belong to Pyartli’s list, thenthe monodromy group is nonsolvable.

As mentioned above, if a field does not belong to Dulac’s list, then it has no complex centers.Therefore, conditions (i) and (iii) in Theorem 2 imply conditions (i) and (iii) in Theorem 1. Now,Theorem 2 follows from Theorem 1.

6.4. Pyartli’s list (continued). Making affine changes in a distinguished affine chart and thenthe projective change (2), we can reduce a quadratic foliation to the form

dz

dw= z

s(w)(1 + αz) + βz + ηz2

r(w)(1 + ασz) + p(w)z2, α ∈ C; (8)

here r(w) = w2 − 1, s(w) = λ1(w − 1) + λ2(w + 1), σ = λ1 + λ2, p(w) = κ1(w − 1) + κ2(w + 1),η = κ1 + κ2, and α, β, κ1, and κ2 are complex parameters. The linear change z �→ cz transformsthe set (α, β, κ1, κ2) into (cα, cβ, c2κ1, c

2κ2). Therefore, each of these four parameters can be madeequal to 1 (if it does not vanish).

In the set of parameters λ1, λ2, consider the exceptional set L defined as follows: (λ1, λ2) ∈ R2,

λ1 ≥ λ2 ≥ 1 − σ, and (λ1, λ2) ∈ L, where

L0 ={

(λ1, λ2)∣∣∣ λ1, λ2 ∈ R, λ1 ≥ λ2 ≥ 1

2

},

L1 ={

(λ1, λ2)∣∣∣ λ1 ∈ 1

2Z \ Z, λ2 ∈

(13

Z ∪ 15

Z

)\ Z

},

L2 ={

(λ1, λ2)∣∣∣ λ1 ∈ 1

3Z \ Z, λ2 ∈

(14

Z ∪ 15

Z

)\ Z

},

L3 ={

(λ1, λ2)∣∣∣ λ1 ∈ 1

4Z \ 1

2Z, λ2 ∈ 1

6Z \ 1

2Z

},

L4 = L1 ∪ L2 ∪ L3, L5 = {(λ1, λ2) | (λ2, λ1) ∈ L4}, L = (L4 ∪ L5) ∩ L0.



Theorem 4 [7]. Suppose that (λ1, λ2) /∈ L in equation (3). Then the equation has a com-mutative monodromy group at infinity if and only if it is affine equivalent to one of the followingequations:

(1)dz

dw= z

s(w)(1 + αz)r(w)(1 + ασz)

with α ∈ C;

(2)dz

dw= z

s(w) + αwz

r(w)(1 + αz) + ε0z2with α, ε0 ∈ C for λ1 = λ2;

(3) any equation of the form (8) for λ1 ∈ Z, λ1 > 1;

(4)dz

dw= z

s(w) + (αw + β)z + εz2

r(w)(1 + αz) + δ(w + 1) + (εw + ε0)z2with α, β, δ, ε, ε0 ∈ C for λ1 = 1;

(5)dz

dw= z

s(w)(1 + αz) + εz2

r(w)(1 + ασz) + (εw + ε0)z2with α, ε, ε0 ∈ C for λ1 = λ2 = λ3 =

13;

(6)dz

dw= z

s(w) + εz2

r(w) + (εw + ε0)z2with ε, ε0 ∈ C for λ1 =

12

+ k, k = 1, 2, . . . , or ε = ε0 ∈ C for

λ1 =12;

(7)dz

dw= z

(αw + β)zr(w)(1 + αz)

with α, β ∈ C for λ1 = λ2 = 0.

Theorem 5 [7]. Suppose that λ ∈ Π, λ1, λ2 /∈ 14Z ∪ 1

6Z, λ3 �= 13 , and λ3 �= 1

4 . Then, in theset of quadratic vector fields with characteristic numbers λ1, λ2, and λ3 = 1− (λ1 + λ2) and with anoncommutative solvable monodromy group at infinity, there is at least one and at most seven orbitsof the action of the affine group.

Because of technical difficulties, we do not write out this list as explicitly as the list of vector fieldswith a commutative monodromy group. However, Theorem 5 gives an idea of how nonrestrictivecondition (ii) is for quadratic vector fields.

ACKNOWLEDGMENTS

The author is grateful to I. Gorbovitskis, N. Dimitrov, and V. Moldavskii for fruitful discus-sions. The author also thanks S.Yu. Yakovenko, without whom this paper could not be written.Unfortunately, Yakovenko did not agree to consider himself a coauthor of this paper.

This work was supported in part by NSF (project no. 0400495) and the Russian Foundation forBasic Research (project no. 05-01-02801).

REFERENCES1. H. Dulac, “Determination et integration d’une certaine classe d’equations diffeerentielle ayant pour point singulier

un centre,” Bull. Sci. Math., Ser. 2, 32, 230–252 (1908).2. Yu. S. Il’yashenko, “The Topology of Phase Portraits of Analytic Differential Equations in the Complex Projective

Plane,” Tr. Semin. im. I.G. Petrovskogo 4, 83–136 (1978) [Sel. Math. Sov. 5, 141–199 (1986)].3. Yu. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations (Am. Math. Soc., Providence, RI,

2007).4. A. Lins Neto, P. Sad, and B. Scardua, “On Topological Rigidity of Projective Foliations,” Bull. Soc. Math. France

126 (3), 381–406 (1998).


72 Yu.S. ILYASHENKO

5. V. A. Naishul’, “Complex Centers,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh. 30 (5), 66–69 (1975) [Moscow Univ.Math. Bull. 30 (5/6), 52–54 (1975)].

6. I. Nakai, “Separatrices for Nonsolvable Dynamics on C, 0,” Ann. Inst. Fourier 44 (2), 569–599 (1994).7. A. S. Pyartli, “Quadratic Vector Fields in CP2 with Solvable Monodromy Group at Infinity,” Tr. Mat. Inst.

im. V.A. Steklova, Ross. Akad. Nauk 254, 130–161 (2006) [Proc. Steklov Inst. Math. 254, 121–151 (2006)].8. A. A. Shcherbakov, “Topological and Analytic Conjugacy of Noncommutative Groups of Germs of Conformal

Mappings,” Tr. Semin. im. I.G. Petrovskogo 10, 170–196 (1984) [J. Sov. Math. 35, 2827–2850 (1986)].9. A. A. Shcherbakov, “Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential

Equations on C2,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 254, 111–129 (2006) [Proc. Steklov Inst.

Math. 254, 103–120 (2006)].10. H. Zoladek, The Monodromy Group (Birkhauser, Bazel, 2006).

Translated by O. Sipacheva


Documents

Total rigidity of polynomial foliations on the complex projective plane