Total rigidity of generic quadratic vector fields
aa r X i v : . [ m a t h . D S ] O c t Total rigidity of generic quadratic vector fields
Yu.Ilyashenko ∗† and V.Moldavskis ‡ To the memory of Vladimir Arnold, a teacher, a leader, a poet in math-ematics
Abstract
We consider a class of foliations on the complex projective planethat are determined by a quadratic vector field in a fixed affine neigh-borhood. Such foliations, as a rule, have an invariant line at infinity.Two foliations with singularities on C P are topologically equivalentprovided that there exists a homeomorphism of the projective planeonto itself that preserves orientation both on the leaves and in C P and brings the leaves of the first foliation to that of the second one. Weprove that a generic foliation of this class may be topologically equiv-alent to but a finite number of foliations of the same class, moduloaffine equivalence. This property is called total rigidity . Recent resultof Lins Neto implies that the finite number above does not exceed 240.This is the first of the two closely related papers. It deals withthe rigidity properties of quadratic foliations, whilst the second onestudies the foliations of higher degree. Total rigidity is a property opposite to structural stability. It is a formal-ization of the following paradigm.
For generic polynomial foliations in C P d , ∗ The first author was supported by part by the grants NSF 0700973, RFBR 10-01-00739- , RFFI-CNRS 050102801 † Cornell University, US; Moscow State and Independent Universities, Steklov Math.Institute, Moscow ‡ Cornell University, US opological equivalence implies affine equivalence. This is a heuristic principlerather than a theorem. In the above form it is proved only for linear vectorfields of strict Siegel type whose normal form has a non-trivial Jordan cell[11].A polynomial vector field in C d may be extended to a holomorphic linefield in C P d ; generically, the extended field has but a finite number of singularpoints. We will refer to it as a polynomial foliation in C P d , and denote by A n,d the class of all such foliations. Definition 1
A polynomial foliation of class A n,d is totally rigid providedthat there exists but a finite number of foliations of this class (up to affineequivalence) that are topologically equivalent to the foliation considered. In what follows, d = 2, so we write A n instead of A n, . Theorem 1
A generic foliation of class A , i.e. quadratic foliation on C P ,is totally rigid. The first (computer assisted) proof of this theorem was obtained by thesecond author [8]. Here we present a purely analytic proof of a slightlystronger result.
Theorem 2
Generic quadratic vector field is topologically equivalent to nomore than quadratic vector fields, modulo affine equivalence.
An earlier estimate given in [8] is 14 · . In subsection 2.2, we state evena stronger result, Theorem 6. It is proved together with Theorem 2. Thesketch of the proof follows.A generic foliation of class A n has n + 1 singular points at infinity, O , . . . , O n +1 , and n singular points in C : Q n +2 , . . . , O N N = n + n +1. Fora singular point O of a foliation let λ, µ be the eigenvalues of the linearizationof the corresponding vector field at O . These eigenvalues are defined up toa common factor. Let λµ + µλ = ν ( O ). The number λµ is usually called thecharacteristic number of O , and the number ν ( O ) is the Baum-Bott index. Theorem 3
The characteristic numbers (equivalently, the Baum-Bott in-dexes) are topological invariants for generic polynomial foliations of degree n ≥ in the projective plane. v be a polynomial vector field of degree n , M ( v ) = ( ν ( O ) , . . . , ν ( O N )) ∈ C N be the tuple of the Baum-Bott indexes for its singular points; recall that N = n + n + 1.Affine equivalent vector fields as well as those that differ by a constant fac-tor, have the same Baum-Bott indexes at the corresponding singular points.Hence, the map v ( ν ( O ) , . . . , ν ( O N )) may be descended to a map µ : ˜ A n = A n / Aff C × C ∗ → C N . This map is called the moduli map , see [8], and its image is denoted by M n . In [6], the moduli map is called the Baum-Bott map. We preserve theterm introduced in [8], in order to emphasize the relation of the map µ withthe moduli of topological classification of foliations. Theorem 4
The moduli map is algebraic. For generic quadratic vector field v , the derivative of the moduli map at v has the full rank. This theorem was proved in [8] by a computer assisted calculation. At thesame time, it follows from the result proved analytically in [3]. In Section 3we present a shorter analytic proof, yet based on the idea suggested in [3].Lins Neto [6] considered a moduli map of a larger space, namely, of a spaceon foliations on C P that have 2 tangent points with generic lines, and notnecessarily have the invariant line at infinity.The class ˜ A is a subset of this space. Lins Neto proved that the degreeof the moduli map on this larger space is exactly 240. Hence, the degree ofthe restriction of this map to ˜ A is no greater than 240.Therefore, Theorems 3 and 4 imply Theorem 1 and Theorem 2.Indeed, the moduli map is algebraic and of full rank at generic points.Hence, the critical locus of the moduli map is a proper algebraic submanifold.It may contain an algebraic subset Σ that is blown down by the moduli map.This means that for any F ∈ Σ , µ − ( µ ( F (0)) is an algebraic subset of positivedimension, which is squeezed to a point by µ . Any point F from ˜ A \ Σ hasno more than 240 points in the set µ − ( µ ( F (0)). If, in addition, F satisfiesthe genericity assumptions of Subsection 2.1, then F is totally rigid andtopologically equivalent to no more than 240 pairwise affine non-equivalentfoliations. 3 Topological invariance of Baum-Bott in-dexes for generic foliations
In this and the next section we prove an improved version of Theorem 3,Theorem 6 below, and provide the genericity assumptions.
Consider a foliation
F ∈ A n such that- it has exactly n + 1 singular points at infinity and n singular points inthe fixed affine neighborhood C ;- the monodromy group at infinity is non-solvable;- all the leaves are dense.Let us check than these are genericity assumptions indeed.The genericity of assumption on the singular points follows from the Be-zout Theorem. The fact that generic polynomial vector fields of degree n ≥ Theorem 5 (Pyartli, [12])
For any λ = ( λ , λ , λ ) , λ + λ + λ = 1 de-note by B λ the set of all quadratic vector fields with the characteristic numbers λ , λ , λ of the singular points at infinity. Let all the λ j be non-real. Thenany set B λ contains no more than seven classes of affine equivalence whosepoints correspond to foliations with solvable monodromy group at infinity. For future use, denote the representatives of these 7 classes of affine equiv-alence by A j ( λ ) , j = 1 , . . . , λ j may be real, but the following restrictions hold. IfRe λ ≥ Re λ ≥ Re λ , then λ , λ / ∈ Z ∪ Z , λ = , λ = . But thisimprovement of Pyartli theorem does not improve our result, because densityproperty requires that the numbers λ j are non-real.Density of leaves requires non-solvable monodromy at infinity plus hype-bolicity of singular points at infinity [13], [9], [14]. The genericity of the firstproperty was already discussed. The second one determines real Zariski openset. 4 .2 Improvement of the main result We can now include genericity assumptions stated above in Theorem 2. Thisgives us the following result.
Theorem 6
For any tuple λ = λ , λ , λ , λ + λ + λ = 1 , Im λ j = 0 , inany set B λ of quadratic vector fields with the tuple of eigenvalues of singularpoints at infinity equal to λ , there exists a Zarisski open subset T λ ⊂ B λ such that any foliation F ∈ T λ is topologically equivalent to no more than foliations of class A , modulo affine equivalence. The difference B λ \ T λ equals to ( ∪ A j ( λ )) ∪ Σ λ , where Σ λ is the set of all points in B λ that belongto a subset, which is blown down by the moduli map. This theorem follows from Theorem 4 proved in Section 3, and Theorem 7,improved version of Theorem 3. The latter theorem is proved in this section.As noticed in [6], the moduli map blows down the family of quadraticDarboux foliations with the first integral of the form ( xy + x + y )( x − ky ) α = c .In this family α is fixed, and k ∈ C is a parameter. Theorem 7
Suppose that foliation
F ∈ A n satisfies genericity assump-tions of subsection 2.1. Then its Baum-Bott indexes are topological in-variants in the following sense. Let G ∈ A n be topologically equivalent to F . The conjugacy induces a bijection h : sing F → sing G . Then for all j = 1 , . . . , N, ν ( O j , F ) = ν ( h ( O j ) , G ) . Proof Step . Conjugacy of monodromy maps. Let F and G beconjugated by a homeomorphism H . Then H topologically conjugates theirmonodromy groups at infinity, denoted by G F and G G , see [5] Proposition28.2. In more details, for any set f , . . . , f n of generators of G F there existsa set of generators g , . . . , g n of G G and a germ of a homeomorphism h :( C , → ( C ,
0) such that f j ◦ h = h ◦ g j , j = 1 , . . . , n. (1) Theorem 8 ([13], [9])
If two finitely generated non-solvable groups ofgerms ( C , → ( C , are topologically conjugated by an orientating pre-serving homeomorphism, then this homeomorphism is in fact holomorphic. tep . Induced maps of cross-sections and transversal holomor-phy. Arguments of step 2 and 3 are very close to those of [5], Lemma 28 , F and G are topologically equivalent. Thenfor any point p and any two germs of cross-sections: Γ at p , Γ ′ at p ′ = H ( p ),Γ and Γ ′ being transversal to the leaves of the foliations F and G respectively,there exists an induced germ of the homeomorphism h : (Γ , p ) (Γ ′ , p ′ ) (2)defined in the following way. Consider two flow boxes: of the foliation F near p and of the foliation G near p ′ . The local leaves in these flow boxes arein one to one correspondence with subdomains of Γ and Γ ′ respectively. Thehomeomorphism H sends the leaves of the first flow box to those of anotherone. This induces the homeomorphism (2).Note that germ h that conjugates the monodromy groups in (1) is inducedin a sense of the previous paragraph.The induced germs respect the holonomy. namely, let γ be a nontrivialloop with the endpoint p on a leaf of F , and ∆ γ, F : (Γ , p ) → (Γ , p ) be the germof its holonomy transformation. Let γ ′ = Hγ , and ∆ γ ′ , G : (Γ ′ , p ′ ) → (Γ ′ , p ′ )be the corresponding holonomy. Let h be the induced germ from (2). Then h ◦ ∆ γ, F = ∆ γ ′ , G ◦ h. (3) Definition 2
A homeomorphism H that conjugates two foliations is called transversally holomorphic , if all the induced germs (2) are holomorphic. Itis called transversally holomorphic at p , if the germ h in (2) is holomorphic. Step . Extending transversal holomorphy. Theorem 8 impliesthat the homeomorphism H is transversally holomorphic near non-singularpoints of the infinity leaf. Density of leaves allows us to extend the transversalholomorphy to all the non-singular points of F . We will use the followingobvious remark. If a homeomorphism is transversally holomorphic at allpoints of one cross-section of a flow box, then it is transversally holomorphicat all points of this flow box.
Now let us extend the transversal holomorphy of H to all the non-singularpoints of F . Take any such point q and a curve γ on the leaf passing through q that connects q with some point p ∈ Γ, where Γ is a cross-section to6he infinite leaf of F , on which the relation (1) holds. Recall that h in(1) is a conformal map induced by H . We may assume that γ is non-selfintersecting. In the opposite case we delete from γ all the loops producedby self intersections. Then there exists a neighborhood of γ on the leaf of F which is biholomorphic equivalent to a disk. This neighborhood may beincluded in a flow box, for which one of the cross-sections belongs to Γ. Thisallows us to conclude that H is transversally holomorphic at q . Hence, H istransversally holomorphic everywhere. Step . Topological invariance of Baum-Bott indexes. For anynondegenerate singular point O of a complex planar foliation, there existsa holomorphic separatrix S through O . For the case when the character-istic number is non-positive, this follows from the complex version of theHadamard-Perron theorem, [5], theorem 7 .
1. For the case when this num-ber is positive, this follows from the Poincar´e-Dulac theorem, ibid., Theorem5 .
5. Consider a nontrivial small loop γ O on S around O and a correspondingholonomy transformation ∆ O . It is well known that ∆ ′ O (0) = exp 2 πi λµ where λµ is a characteristic number of O, µ corresponds to the tangent vector to S at O . Let O ′ = H ( O ) , λ ′ µ ′ be the corresponding characteristic number, and∆ O ′ be a holonomy map of G corresponding to the loop γ ′ = H ( γ ). Thehomeomorphism H is transversally holomorphic. Hence, the maps ∆ O and∆ O ′ are complex conjugated. This implies the coincidence of their derivativesat zero. Hence, the characteristic numbers of O and O ′ coincide modulo Z .This is almost the desired statement. Proposition 1 ([4], [10])
Suppose that two planar foliations in a neighbor-hood of a non-degenerate singular point are topologically equivalent, and thecorresponding holonomy maps are analytically conjugate. Then the charac-teristic numbers of these singular points coincide.
Together with the previous arguments, this proposition implies the theo-rem. (cid:3)
Remark 1
Note that all hyperbolic singular points of planar foliations aretopologically equivalent.In particular, two singular points whose characteristic numbers differ bya nonzero integer are topologically equivalent. Proposition 1 claims that thisequivalence can not be transversally holomorphic. Moduli map for quadratic vector fields
In this section we will check that the dimensions of the factorized space ofquadratic vector fields and the moduli space are both equal 5. We will provethat there exists at least one point where the moduli map has the full rank.This will imply Theorem 4.
The space of all quadratic polynomial in the plane has dimension 6. Hence,the space of all planar quadratic vector fields equals 12. The affine groupaction on the phase space induced a transformation on the space of quadraticvector fields. Moreover, multiplication of a vector field by a nonzero numberpreserves the foliation. The affine group has dimension 6. Hence,dim C A / Aff C × C ∗ = 5 . This is the “effective” dimension of the space of quadratic vector fields. De-note by ˜ A the factor-space above:˜ A = A / Aff C × C ∗ . (4)Any class of vector fields from ˜ A with at least three singular points in C has a regular representative with singular points (0 , , (2 , , (0 , C . Yet it issubject to at least two relations. The Baum-Bott equality [1] for quadraticvector fields in C P implies: X ν ( O j ) = 2 . (5)On the other hand, for the singular points at infinity X λ j µ j ( O j ) = 1 . (6)Here µ j are eigenvalues that correspond to eigenvectors tangent to the lineat infinity.This equality is usually called Camacho-Sad [2].So, the dimension of the image of the moduli map is no more than 5.We will prove that at a special point the moduli map has rank 5 indeed. Asmentioned before, this will imply Theorem 4.8 .2 Counting the rank In this subsection we will prove Theorem 4.
Proof
The idea of the proof goes back to [3]. In fact, Theorem 4 was provedin [3], but not explicitly stated. Our proof is shorter and more explicit.Suppose that, contrary to the statement of Theorem 4, rank of the modulimaps drops everywhere. Then the fibers of this map are analytic sets ofdimension at least one. Take such a fiber passing through a special vectorfield v chosen below. By the curve selection lemma, there exists an analyticcurve γ = { v ε | ε ∈ ( C , } such that M ( v ε ) ≡ M ( v ) , v ε v . Note that theBaum-Bott indexes for all the fields v ε are the same.Take a quadratic vector field v having three invariant lines. It is affineequivalent to a field with invariant lines x = 0 , y = 0 , x + y = 1. Withoutloss of generality, we may assume that all the vector fields v ε have singularpoints (0 , , (0 , , (1 , Proposition 2
The assumption M ( v ε ) ≡ M ( v ) implies that the fields v ε have the same invariant lines as v for all ε . Proof
The invariance of the line l passing through two singular points ofa quadratic vector field v is established like follows. Take any nonsingularpoint p ∈ l . The line l is invariant for v iff v ( p ) ∈ T p l . Hence, the set ofquadratic vector fields with the singular points (0 , , (1 , , (0 ,
1) for whichthe line l = { y = 0 } is invariant has codimension one. Denote this set by IL,for invariant line .The line y = 0 contains a third singular point of the extended foliation v at infinity. On the other hand, let U ⊂ ˜ A be a small neighborhood of v , O = (0 , , O = (1 , O be an infinite singular point of the vectorfield v ∈ U , close to the infinite singular point of v that belongs to l . Thenthe set of vector fields v ∈ U such that λ µ ( O , v ) + λ µ ( O , v ) + λ µ ( O , v ) = 1 (7)has codimension 1. Here µ j are eigenvalues that correspond to eigenvectorstangent to l . Denote this set by CS, for Camacho and Sad. Now, invarianceof the line l for v implies (7). Hence, CS ⊃ IL. But both algebraic sets havecodimension one. It is easy to see that both sets CS and IL are irreduciblenear v . Hence, CS ∩ U = IL ∩ U . By assumption, γ ⊂ CS . Hence, γ ⊂ IL. Therefore, the line l is invariant for all the fields v ε . The same argument9hows that the other two lines x = 0 , x + y = 1 are invariant for v ε for all ε . (cid:3) It is well known that a quadratic vector field having three invariant lineshas a multivalued Darboux first integral. For v ε it is F ( x, y, ε ) = x a ( ε ) y b ( ε ) ( x + y − c ( ε ) . Dividing all the exponents by a non-vanishing one preserves the firstintegral. Hence, we may assume that c ( ε ) = 1. Then a ( ε ) and b ( ε ) are thecharacteristic numbers at the points (1 ,
0) and (0 , ε . Hence, the Darboux integral above does not depend on ε ,and the foliations determined by the vector fields v ε coincide, a contradiction. (cid:3) This implies Theorem 4. Together with Theorem 7, and the quoted resultof Lins Neto , see Subsection 1.2, this implies our main result, Theorem 2.
Theorem 3 holds for polynomial foliations of arbitrary degree. Yet the targetof the moduli map M n has the same dimension as ˜ A n = A n / Aff C × C ∗ for n = 2, and smaller dimension for larger n . Indeed,dim ˜ A n = ( n + 1)( n + 2) − , dim M n ≤ n + n − . The difference is dim ˜ A n − dim M n = 2 n − . (8)It is 0 for n = 2 only. For larger n the arguments above fail.Yet the moduli space may be extended. Indeed, topological equivalenceof generic foliations implies analytic equivalence of the corresponding mon-odromy groups. We used a very weak corollary of this equivalence, namely,the coincidence of the linear terms of the monodromy only. Equivalence re-lations on higher jets imply relations on the higher Taylor coefficients of themonodromy maps. It may be shown that the equivalence relations on thequadratic and cubic terms generate extra 2 n − µ .10 roblem 1 Is it correct that the extended moduli map has full rank at ageneric point?
The first author is grateful to the organizers of the “School in holomorphicfoliation and dynamical systems”, Mexico, August 2010, especially to LauraOrtiz and Ernesto Rosales; to Adolfo Guillot, who introduced him in therecent progress in the study of the Baum-Bott map; to all the participantsof the above mentioned School for the enthusiastic and creative atmosphere,and to the UNAM that provided an ideal environment for writing of thispaper. Both authors are grateful to the Cornell University, where the firsttheorem on total rigidity of quadratic foliations was proved by the secondauthor in the thesis [8] done under the advisory of the first one.
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