Flags of holomorphic foliations

Abstracts

A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties oft hese objects and, in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" />, n > 3, we establish some necessary conditions for a foliation, we find bounds of lower dimension to leave invariant foliations of codimension one. Finally, still in <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" /> involving the degrees of polar classes of foliations in a flag.

holomorphic foliations; polar varieties; invariant varieties

Uma bandeira de folheações holomorfas em uma variedade complexa M é um objeto que consiste de um número finito de folheações holomorfas singulares em M de dimensões crescentes tais que o feixe tangente de uma folheação fixa é subfeixe do feixe tangente de cada folheação de dimensão maior. Estudamos algumas propriedades básicas destes objetos e, em <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" />, n > 3, estabelecemos condições necessárias para que uma folheação de dimensão menor deixe invariante folheações de codimensão um. Finalmente, ainda em <img src="/img/revistas/aabc/2011nahead/aop2411pcn.jpg" align="absmiddle" />, encontramos quotas envolvendo graus das classes polares de folheações em uma bandeira.

folheações holomorfas; variedades polares; variedades invariantes

Flags of holomorphic foliations

Rogério S. Mol

Departamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, 30123-970 Belo Horizonte, MG, Brasil. E-mail: [e-mail]

ABSTRACT

A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties oft hese objects and, in , n > 3, we establish some necessary conditions for a foliation, we find bounds of lower dimension to leave invariant foliations of codimension one. Finally, still in involving the degrees of polar classes of foliations in a flag.

Key words: holomorphic foliations, polar varieties, invariant varieties.

RESUMO

Uma bandeira de folheações holomorfas em uma variedade complexa M é um objeto que consiste de um número finito de folheações holomorfas singulares em M de dimensões crescentes tais que o feixe tangente de uma folheação fixa é subfeixe do feixe tangente de cada folheação de dimensão maior. Estudamos algumas propriedades básicas destes objetos e, em , n > 3, estabelecemos condições necessárias para que uma folheação de dimensão menor deixe invariante folheações de codimensão um. Finalmente, ainda em , encontramos quotas envolvendo graus das classes polares de folheações em uma bandeira.

Palavras-chave: folheações holomorfas, variedades polares, variedades invariantes.

INTRODUCTION

Let M be a complex manifold of dimension M with tangent bundle TM . Let us denote by = (TM ) its tangent sheaf. A singular holomorphic foliation, or shortly foliation, is a coherent analytic subsheaf of e that is involutive, which means that its stalks are invariant by the Lie bracket:

The sheaf is called the tangent sheaf of the foliation. We will denote a foliation by or by () . when a reference to its tangent sheaf is needed.

The singular set of = () is the analytic set S = Sing() defined as the singular set of the sheaf / , which on its turn consists of the points where the stalks are not free modules over the structural sheaf . The dimension of is defined as the rank of the locally free part of . The locally free sheaf _|M\S is the sheaf of sections of a rank p vector bundle T, which is a subbundle of TM _|M\S . The involutiveness of implies that the distribution of p-dimensional subspaces of T M induced by T on M \ S is integrable, that is, there exists a regular holomorphic foliation on M \ S such that the tangent space to the leaf passing through each point x ∈ M \ S is, the fiber of T over x . This is the so-called Theorem of Frobenius.

We say that a foliation is reduced if is full . This means that, whenever U ⊂ M is an open subset and v is a holomorphic section of |U such that v _x∈ ∀ x ∈ U ∩ (M \ S ), then v x ∈ holds also for x in U ∩ S . We remark that, given an involutive sheaf , which induces a foliation with singular set S, there is a unique sheaf that is both full and involutive, and such that _{T |M \ S} = _{|M \S} . We can therefore restrictour attention to full involutive sheaves as a way to avoid artificial singularities (see Baum and Bott 1972,Suwa 1998).

We can describe foliations in a dual way by means of differential forms. Let Ω = (T^*M ) be the cotangent sheaf of the m-dimensional complex manifold M . Let C be an analytic coherent subsheaf of Ω of rank p , where 1 < p < n - 1, which satisfies the integrability condition:

The sheaf C defines a singular holomorphic foliation denoted by = (C ). The singular set of , denoted by Sing(), is equal to Sing( Ω /C). On M \ S (), the sheaf C is the sheaf of sections of a rank n - p vector subbundle of T ^*M . The local sections of this subbundle are holomorphic 1-forms whose kernels, at each point x, define a subspace of , which is the tangent space at x of a regular foliation of codimension p on M \ Sing().

We say that = (C ) is reduced if C is full, that is, whenever U ⊂ M is an open subset and ω is a holomorphic section of Ω _|U with the property that ω_x∈ C _x∀ x ∈ U \ Sing(), then ω_x∈ C _x for all points x in U . The sheaf C is called the conormal sheaf of .

Both definitions of foliation that we have just introduced are related as follows. Let be the tangent sheaf of a foliation = () of dimension p . Define

where i _v denotes the contraction by the germ of vector field v. We have that ^a is the conormal sheaf of a codimension M - p foliation ^a= ( ^a ). We clearly have Sing( ^a) ⊂ Sing(). Furthermore, ^ais a reduced foliation.

Similarly, given C the conormal sheaf of a codimension M - p foliation = F(C) on M , we define

Then, C^a is the tangent sheaf of a foliation ^a= ( C^a ). We have that Sing( ^a) ⊂ Sing() and that ^ais a reduced foliation.

If is the tangent sheaf of a foliation = (), then ^r= ( ^a )^ais the tangent sheaf of a reduced foliation ^r= ( ^r). As a consequence of the definitions, we have that is a subsheaf of ^r. Thus,

Sing() = Sing(/ ^r) ⊂ Sing(/ ) = Sing().

Furthermore, on M \ Sing(), the regular foliation induced by ^rcoincides with the one induced by . We also notice that reduced foliations are stable by this reduction process: if is full, then ^r= . In a similar way, a reduction process can be defined for a foliation defined by a conormal sheaf.

Let = () be a foliation with tangent sheaf . If is reduced, then codim Sing() > 2. The converse holds when is locally free. The equivalent is true for a foliation = (C ) defined by its conormal sheaf C : If is reduced, then codim Sing() > 2, and both facts are equivalent when C is locally free. A proof for these facts can be found in [Su1, Lemma 5. 1].

DEFINITION. The foliations, . . ., on the m-dimensional holomorphic manifold M form a flag of foliations if

(i) is reduced ∀ l = 1,. . ., k .

(ii) 1 < i ₁ < ∙∙∙ i _k m and dim = _{i l} ∀ l = 1,. . ., k .

(iii) is a subsheaf of ₊₁ ∀ l = 1,. . ., k - 1, where is the tangent sheaf of .

In the definition, we say that leaves invariant or that is invariant by whenever i _r < i _s . This terminology is due to the fact that, for x ∈ M \ (Sing() ∪ Sing()), the inclusion relation ⊂ holds, giving that the leaves of are contained in leaves of . We will use the notation - .

Let and be foliations of dimensions i < j on a complex manifold M such that

. The tangent sheaves of these foliations satisfy ⊂ where " ⊂ " means subsheaf. We produce conormal sheaves by taking annihilators: C_i = ()^a and C_j= ()^a. This gives C_j⊂ C_i. By taking annihilators again, since our sheaves are full, we have = C_i^a ⊂ C^a_j= . That is,

if and only if ⊂ C^a_j. In terms of local sections, this is equivalent to the following: whenever v is a local vector field tangent to and ω is a local integrable 1-form tangent to , then i _vω = 0. As a consequence, since the singular set of a foliation is a proper analytic set, we have

PROPOSITION 1. Let and be reduced foliations of dimensions i < j on a complex manifold M . Then,

and only if ⊂ holds for every x ∈ M \ (Sing () ∪ Sing ( )).

We now recall some facts about the structure of the singular set of a foliation (see Yoshizaki 1998 and Suwa 1998 as well). Let, as above, be a reduced foliation of dimension p , with tangent sheaf, on an m-dimensional complex manifold M . For each x ∈ M let

be the subspace of formed by the directions induced by . For each integer k with 0 < k < p , we define

.

Then, S ^{(k )} is an analytic variety in M and we have a filtration

,

where S ^(p)= M and S ^{(p -1)} = Sing() is the singular set of . It is proved in (Yoshizaki 1998) that, for each k = 0,. . ., p , there is a Whitney stratification of S ^{(k )} such that, for any α ∈ A_k and x ∈ M _α, the inclusion T (x ) ⊂ M _α holds. Moreover, induces a non-singular foliation of dimension k on M _α \ S ^{(k -1)} whose tangent space at x ∈ M _α is T (x ).

If V is an analytic subvariety of M with singular set Sing (V ), we say that V is invariant by if T (x ) ⊂ T_x V holds for each x ∈ V \ Sing (V ) . The above discussion says, in particular, that the analytic set Sing() is invariant by . We obtain:

THEOREM 1. Let M be a complex manifold of dimension n, and let and be foliations of dimensions i and j, where 1 < i < j < n, such that

. Then, Sing () is invariant by .

This has the following simple consequence:

COROLLARY 1. Let M be a complex manifold of dimension n, and let be a foliation of dimension one. If is a foliation of dimension i > 1 such that , then the isolated points of Sing () are contained in Sing ().

FLAGS OF FOLIATIONS ON

In this section we consider, on the projective space = of dimension n > 3, a foliation of dimension one and a foliation of codimension one. Let us suppose that leaves invariant, that is,

in our notation. If is the tangent sheaf of, then = (1 - d ), where d > 0. This number d is the degree of, which is the degree of the variety of tangencies between and a generic hyperplane H ⊂ . Now, if C is the cotangent sheaf of, then C = (-2 - ), where > 0 is the degree of and counts the number of tangencies, considering multiplicities, between and a generic line L ⊂ .

The study of genericity properties of the set of foliations in without invariant algebraic varieties is known as the Jouanolou problem . It was considered by many authors, such as J. P. Jouanolou, A. Lins Neto, M. Soares, X. Gomez-Mont, L. G. Mendes and M. Sebastiani, among others. We consider here the following result by S. C. Coutinho and J. V. Pereira (see Coutinho and Pereira 2006), Theorem 1. 1 and the remark after its proof): if ol_n (1, d) denotes the space of foliations on of dimension one and degree d , then, for d > 2, there is a very generic set (1, d) ⊂ ol n (1, d) such that if ∈ (1, d), then does not admit proper invariant algebraic subvarieties of non-zero dimension. Here very generic means that its complementary set is contained in a countable union of hypersurfaces. In the case of invariant algebraiccurves, (1, d) can be taken to be open and dense in ol_n (1, d), as a consequence of a result by A. Lins Neto and M. Soares (see Lins Neto and Soares 1996, Soares 1993).

Let now be a foliation of dimension one and degree d > 2 on , n > 3. Suppose that there is a foliation of codimension one on such that

. We recall that the singular set of a codimension one foliation on necessarily has at least one component of codimension two (see Jouanolou 1979). So, by Theorem 1, if Sing() has codimension greater than two, then the components of dimension n - 2 in Sing () are invariant by . This implies that lies outside the subset (1, d) ⊂ ol_n (1, d) above. We recall that the foliations in ol_n (1, d) with isolated singularities form a generic set. Thus, for n > 3, the set of foliations ∈ ol_n (1, d) such that codim Sing()> 2 contains a generic set. This allows us to conclude the following:

THEOREM 2. The set of foliations of dimension one and degree d > 2 on , n > 3, which do not leave invariant a foliation of codimension one, is very generic. When n = 3, this set contains a subset that is open and dense in ol_n (1, d).

We say that a foliation of dimension one on admits a rational first integral if there is a rational function Φ in such that the leaves of are contained in the level surfaces of Φ. In homogeneous coordinates in , by writing Φ = P /, where P and are homogeneous polynomials of the same degree, this means that the 1-form dP - Pd induces a codimension one foliation on that is invariant by . This gives:

COROLLARY 2. The set of foliations of dimension one and degree d > 2 on , n > 3, which do not admit rational first integral, is very generic. When n = 3, this set contains a subset that is open and dense in ol_n (1, d).

PENCIL OF FOLIATIONS ON

Let us now consider

ol_n (n - 1, d ), the space of foliations of codimension one and degree d on . Such foliations are given, in homogeneous coordinates X = (X₀ : X₁ :∙∙∙: X_n ) ∈ , by holomorphic 1-forms of the type, where each A_i is a homogeneous polynomial of degree d + 1, satisfying the following:

(i) ω ∧ dω = 0 (integrability);

(ii) i_r ω = = 0, where r = X₀∂/∂X₀ + ∙∙∙ X_n∂/∂X_n is the radial vector field (Euler condition);

(iii) codim Sing(ω) > 2,

where Sing(ω) ={A₀ = A₁ = ∙∙∙ = A_n = 0} is the singular set of ω. We consider the projectivization of the space of polynomial forms in with homogeneous coefficients of degree d + 1. Here

Then, in Zariski's topology,

ol_n(n - 1, d ) is an open set of an algebraic subvariety of . We remark that the elements in the border

are integrable 1-forms satisfying Euler condition, but having a singular set of codimension one.

Let 1 and 2 be two distinct foliations on induced, in homogeneous coordinates, by integrable 1-forms ω₁ and ω₂. The 2-form ω₁ ∧ ω₂ might be zero on a set of codimension one, which corresponds to the set of tangencies between 1 and 2. If ƒ = 0 denotes the homogeneous polynomial equation for this set, we write ω₁ ∧ ω₂ = ƒ θ, for some 2-form θ whose coefficients are homogeneous polynomials and whose singular set has codimension two or greater. Since

i_r(ω₁ ∧ ω₂) = i_rω₁ ∧ ω₂ - ω₁ ∧ i_rω₂ = 0

we have i_r θ = 0, so the field of (n - 1)-planes on defined by θ goes down to an integrable field of n - 2-planes on whose singular set has codimension two or greater. This defines a foliation of codimension two on , which leaves both 1 and 2 invariant. Following the terminology on (Ghys 1991), is called the axis of 1 and 2.

A line of the space , which is entirely contained in and whose generic element is in ol_n (n - 1, d ), is called a pencil of foliations. Remark that two foliations in ol_n (n - 1, d ) represented by 1-forms ω₁ and ω₂ define a pencil of foliations if and only if ω = ω₁ + tω₂ is integrable for all t ∈ . This means

0 = ω ∧ dω = (ω₁ + tω₂) ∧ ( dω₁ + td ω₂)

= t (ω₁ ∧ dω₂ + ω₂ ∧ dω₁),

which is equivalent to

One value of t ∈ \ {0} for which ω₁ + tω₂ is integrable is sufficient for assuring condition (1). So, if three foliations are on a line, then they define a pencil of foliations. Of course, given a pencil of foliations in , a foliation of codimension two is intrinsically associated to it as being the axis of anytwo foliations in the pencil. It leaves invariant all the foliations in the pencil.

For foliations of codimension one on there is a conjecture due to M. Brunella, which asserts that, if is such a foliation, then one of the alternatives holds:

(a) leaves an algebraic surface invariant;

(b) is invariant by a holomorphic foliation by algebraic curves.

In (b) we mean that the closure of each leaf of is an algebraic curve. In (Cerveau 2002), the following result is proved:

THEOREM 3. Let be a foliation of codimension one on, which is an element of a pencil of foliations. Then, satisfies (a) or (b) above.

It is worth remarking that, in Cerveau's proof, the foliation that appears in alternative (b) is the axis of the pencil and is given by two independent rational first integrals. We next prove the following simple lemma:

LEMMA 1. Let be a foliation of codimension two on , which leaves invariant three foliations of codimension one induced, in homogeneous coordinates, by integrable polynomial 1-forms ω₁, ω₂ and ω₃. Then, there are non-zero homogeneous polynomials α₁, α₂ and α₃, relatively prime two by two, such that

PROOF. We write ω₁ ∧ ω₃ = ƒ₁θ , where θ is a polynomial 2-form that induces, having singular set of codimension at least two, and ƒ₁ is a non-zero homogeneous polynomial. Similarly, we have ω₂ ∧ ω₃ = — ƒ₂θ, for some non-zero homogeneous polynomial ƒ₂. We thus have

This implies that there is a rational function Φ such that

By canceling denominators, we get homogeneous polynomials α, α₂ and α₃, which satisfy (2). Finally, a common factor for two of these polynomials would be a factor of the third and, so, could be canceled. We can thus suppose that α₁, α₂ and α₃ relatively prime two by two.

Before proceeding we make a simple remark: if ω is an integrable 1-form with homogeneous coefficients of the same degree d + 1 inducing a foliation in ol_n (n - 1, d ), and α is a homogeneous polynomial of degree k, then = αω is also integrable. Of course, if α is non-constant, then has a codimension one component in its singular set. It will be regarded as representing an element of Actually, it is an element in the border ∂ol_n(n - 1, d + k ), if k 0.

LEMMA 2. Let ω₁ and ω₂be 1-forms in with homogeneous polynomial coefficients of the same degree, defining different distributions of n -planes in the sense that ω₁ ∧ ω₂is not identically zero. Suppose also that the singular sets of ω₁ and ω₂do not have a common component of codimension one. Then, the generic element of the pencil of 1-forms

{t₁ω₁ + t₂ω₂; (t₁ : t₂) ∈ }

has singular set of codimension two or greater.

PROOF. Let us write

where A_i and B_i are homogeneous polynomial of the same degree. Suppose that the result is false. Then, for all values of t ∈ but a finite number, the 1-form ω_t = ω₁ + tω₂ has a component of codimension one in its singular set. For such a t, take g_t = 0 as an equation of this component, where g_t is non-constant reduced homogeneous polynomial. Fix i, j, with 0 < i, j < n . We have that both A_i + t B_i and A_j + tB_j vanish over { g_t = 0}. If g_t is a factor of neither B_i nor B_j, then we have that A_i B_i = t = A_j B_j over { g_t = 0}, which means that A_iB_j - A_j B_i vanishes over { g_t = 0}. The same will be true if g_t is a factor of B_i (or B_j ), since, in this case, it will also be a factor of A_i (or A_j ). In any case, we have that g_t is a factor of A_iB_j - A_j B_i . Finally, the hypothesis on the singular sets of ω₁ and ω₂ implies that, by varying t, there are infinitely many different polynomials g_t. This gives that A_iB_j - A_j B_i = 0, that is A_i B_i = A_j B_j = Φ,where Φ is a rational function of degree zero. Doing this to all values of i and j, we get ω₁= Φ ω₂, which is a contradiction with the fact that ω₁ ∧ ω₂ ≠ 0.

It is worth mentioning the following result, which is a corollary of the above lemma:

COROLLARY 3. Let ω₁ and ω₂be integrable 1-forms in with polynomial coefficients of the same degree d + 1, such that ω₁ ∧ ω₂ = 0. Suppose that the pencil of 1-forms

{t₁ω₁+ t₂ω₂; (t₁: t₂) ∈ }

lies entirely in ∂ ol_n(n - 1, d ). Then, the singular sets of the elements of this pencil have a common component of codimension one.

We have the following result:

PROPOSITION 2. Let be a foliation of codimension two on that leaves invariant three foliations of codimension one. Then, leaves invariant a whole pencil of foliations.

PROOF. Suppose that the codimension one foliations are induced in homogeneous coordinates by 1-forms ω₁, ω₂ and ω₃. In view of the previous lemma, there are homogeneous polynomials α₁, α₂ and α₃, such that

If ω₁ and ω₂ lies in a pencil of foliations, the result is done. Otherwise, we necessarily have that either α₁ or α₂ is non-constant. Expression (3) gives that the integrable 1-form α₃ ω₃ lies in the pencil generated by the integrable 1-forms α₁ω₁ and α₂ω₂. Thus, this whole pencil is composed by integrable 1-forms. Finally,even though the generators of this pencil may lie in ∂ol_n (n - 1, d ), where d + 1 is the degree of α_i ω_i, its generic element lies in ol_n (n - 1, d ). This is a consequence of Lemma 2 above. Therefore, α₁ω₁ and α₂ω₂ generate a pencil of foliations whose axis is .

The above proposition together with Theorem 3 give:

COROLLARY 4. Let be a foliation of dimension one on . Suppose that no hypersurface in is invariant by . Then, the number of foliations of codimension one invariant by is at most two.

PROPOSITION 3. Let be a foliation of codimension two on that leaves invariant a pencil of foliations in ol_n (n - 1, d ). Suppose that, outside this pencil, there is another foliation of codimension one and degree at least d that leaves invariant. Then, admits a rational first integral.

PROOF. Suppose that the pencil of foliations is generated by the 1-forms ω₁ and ω₂, and that is induced by the 1-form ω₃. Lemma 1 assures the existence of homogeneous polynomials α₁, α₂ and α₃, two by two without common factors, such that

Since does not lie in the pencil of foliations generated by ω₁ and ω₂, we have that α₁ and α₂ are non-constant. The integrability condition applied to α₃ω₃ reads

0 = α₃ω₃ ∧ d ( α₃ω₃) = ( α₁ω₁ + α₂ω₂) ∧ ( d α₁ ∧ ω₁ + d α₂ ∧ ω₂ + α₁dω₁ + α₂dω₂),

which gives

where we used that ω₁ ∧ dω₂ + ω₂ ∧ dω₁ = 0. The rational function α₁/ α₂, which is non-constant since α₁ and α₂ are non-constant and without common factor, is thus a rational first integral for .

POLAR CLASSES

We now consider an r-dimensional foliation defined on a projective manifold M ⊂ of dimension m . Let be the tangent sheaf of . For each x ∈ M \ Sing(), there is a unique r-dimensional plane ⊂ passing through x with direction ⊂ M.

Let us fix

a flag of codimension j linear subspaces L_j ⊂ .

for k = 1,. . ., r + 1, the k -th polar locus of with respect to is defined as

where the closure Cl is taken in M . We remark that a point x ∈ M \ Sing() belongs to if and only if the subspaces of corresponding to and to L_{r - k +2} do not span . It follows straight from the definition that

Let A_k (M ) denote the Chow group of M , where k st and s for the complex dimension. In (Mol 2006,Proposition 3. 3), it is proved that, for a generic choice of a flag and for k = 1,. . ., r + 1, the set is empty or is an analytic variety of pure codimension k whose class [] ∈ A_{m - k} (M) is independent of the flag, where A_{m - k} (M ) stands for the Chow group of M of complex dimension m - k . We then have a well-defined class that is called polar class of . The polar degrees of arethe degrees of these polar classes. We denote them by = deg [], k = 1,. . ., r + 1.

EXAMPLE 1. Let be a foliation of dimension one on . We have

This means that the hyperplane generated by L₂ and x is tangent to at x . The tangency locus between and a non-invariant hyperplane H ⊂ is a hypersurface in H of degree deg(). We then conclude that is a hypersurface in of degree deg() + 1, since L₂ ⊂ .

EXAMPLE 2. Let now be a foliation of codimension one on with Sing () of codimension at least two. If X = (X₀ : X₁ : ∙∙∙ : X_n ) is a system of homogeneous coordinates in , then is induced by a polynomial 1-form with homogeneous coefficients of degree deg () + 1, which is integrable and satisfies the Euler condition. We have

that is, the hyperplane

contains the point L_m . Writing in homogeneous coordinates L_m = ( α₀ : α₁ :. . . : α_n ), we have that has equation

α₀A₀ + α₁A₁ +∙∙∙+ α _n A_n = 0

and we see that is a hypersurface of degree deg () + 1.

EXAMPLE 3. Let us now examine, where is a foliation of codimension one on . We have

Suppose that is given in homogeneous coordinates in by the polynomial 1-form ω of the previous example. The space L_m -1 is a line in , which we suppose to be generated by points of coordinates ( α₀ : α₁ : ∙∙∙ : α_n ) and (β₁ : β₂ : ∙∙∙ : β_n ). Thus, is contained in the variety V₂ given by the pair of equations

We assume that L_m-1 is generic, so V₂ has pure codimension two and has degree (deg () + 1)². It contains two types of points. Outside Sing (), the points of V₂ correspond to those of . On the other h and, since Sing () ⊂ V₂, the remaining points of V₂ are contained in the component of codimension two of Sing (), which will be denoted by S₂. We then have V₂ = P ₂^G ∪ S₂, and the two sets of this union do not have a common component of codimension two. We conclude that

+ deg(S₂) = (deg () + 1).²

Let 1 < i₁ < ∙∙∙ < i_k < m and ₁-∙∙∙ -_k be a flag of foliations on the m-dimensional projective manifold M ⊂ . Fix a flag of linear subspaces of :

D : L_n ⊂ L_{n -1} ⊂ ∙∙∙ ⊂ L_j ⊂ ∙∙∙ L₁ ⊂ L₀ =

For i = i₁,. . ., i_k, let be the k -th polar locus of with respect to . We have the following result:

PROPOSITION 4. Let i < j be two integers of the list i₁ < ∙∙∙ < i_k . For integers r and s such that r = 1,. . ., i and r + s < j, it holds

PROOF. We start by remarking that the inclusion follows immediately from the definition of polar locus. Thus, all we have to prove is that . Let x ∈ M \ (Sing() ∪ Sing( )). We have

and

Since and are foliations in a flag, is a subspace of codimension j - i. Furthermore, L_{i- r + 2}has codimension

(i - r + 2) - ( j - (r + s) + 2) = i - j + s

in Lj _{-(r +s )+2}. Thus, if dim ( ∩ _j-(r+s)+2) > (r + s) - 1,then

dim (

∩ _{Lj -(r +s )+2}> (r + s) - 1) - (j - i).

Thus,

dim (

∩ _{Li - r + 2}) > ((r + s) - 1) - (j - i)) - (i - j + s) = r - 1

which finishes the proof.

Let us now consider a flag of foliations

₁∙∙∙

_kon , where 1 < i₁ < ∙∙∙ < i_k n .

THEOREM 4. Let i < j be two integers of the list i₁ < ∙∙∙ < i_k . For integers r and s such that 1 < r < i and r + s < j, with j - i ≠ s - r, it holds

PROOF. This is a consequence of Bezout's Theorem ([Fu]). All we have to do is to prove that and can be chosen to be transverse. These polar loci are induced by L _{i - r +2} and L_j -s +2, which are distinct linear spaces, since j - i = s - r . Thus, transversality occurs for generic choices of L i -r +2 and of L_j -s +2 as a consequence of Piene's Transversality Lemma (see Piene 1978, Mol 2006).

Taking into account the calculations made in Examples 1 and 2, Theorem 4 gives:

COROLLARY 5. Let and be foliations on , n > 3, where has dimension one and has codimension one. Suppose that

. Then, the following inequality holds:

where deg() and deg () are the degrees of and , respectively.

As seen in Example 3, + deg(S₂) = (deg () + 1)², where S₂ corresponds to the component of codimension two of Sing () . Putting this in (6) gives

COROLLARY 6. Let and be as in Corollary 5. Then,

deg(S₂) > (deg () + 1)(deg () - deg()),

where S₂ st and s for the component of codimension two in Sing () .

EXAMPLE 4. Take a foliation of degree d on 2defined in homogeneous coordinates by an 1-form . Let Φ : ³ → ² be a rational projection, for instance the one defined in homogeneous coordinates by

Φ(X₀ : X₁ : X₂ : X₃) = (X₀ : X₁ : X₂).

Then, ω = Φ ω defines a foliation of codimension one and of degree d on . The linear fibration given by the levels of Φ is a foliation of dimension one on whose degree is zero. It leaves invariant. Corollary 6 gives in this case deg(S₂) > ( d + 1) d = d ²+ d . However, Sing () = Φ^-1(Sing()) is a finite family of lines. Thus, Sing () = S₂. In the generic situation, has d ²+ d + 1 singularities (see Baum and Bott 1972), and we find deg(S₂) = d ²+ d + 1, which is larger than the bound obtained.

ACKNOWLEDGMENTS

The author thanks to Jorge V. Pereira for his suggestions. This work was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Fundação de Amparo à Pesquisado Estado de Minas Gerais (FAPEMIG) and Programa de Apoio a Núcleos de Excelência (PRONEX) /Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ).

Manuscript received on March 29, 2010; accepted for publication on March 3, 2011

• BAUM P AND BOTT R. 1972. Singularities of holomorphic foliations. J Differential Geometry 7: 279-342.
• CERVEAU D. 2002. Pinceaux linéaires de feuilletages sur CP (3). Publ Mat 46(2): 441-451.
• COUTINHO SC AND PEREIRA JV. 2006. On the density of algebraic foliations without algebraic invariant sets. J Reine Angew Math 594: 117-135.
• GHYS E. 1991. Flots transversalement affines et tissus feuilletés. Mém Soc Math France (N. S.) 46: 123-150.
• JOUANOLOU JP. 1979. Equations de Pfaff algébriques. Lecture Notes in Mathematics, 708, Springer-Verlag, Berlin.
• LINS NETO A AND SOARES M. 1996. Algebraic solutions of one-dimensional foliations. J Differential Geometry 43: 652-673.
• MOL R. 2006. Classes polaires assocíées aux distributions holomorphes de sous espaces tangents. Bulletin of the Brazilian Mathematical Society 37: 1-20.
• PIENE R. 1978. Polar classes of singular varieties. Ann Scient Ec Norm Sup 11: 247-276.
• SOARES M. 1993. On algebraic sets invariant by one-dimensional foliations on CP (3). Annales de l'Institut Fourier 43: 143-162.
• SUWA T. 1998. Indices of vector fields and residues of singular holomorphic foliations. Paris, Hermann.
• YOSHIZAKI J. 1998. On the structure of the singular set of a complex analytic foliation. J Math Soc Japan 50: 837-857.

Publication Dates

• Publication in this collection
  29 July 2011
• Date of issue
  Sept 2011

History

• Received
  29 Mar 2010
• Accepted
  03 Mar 2011