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: rsmol@mat.ufmg.br
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 ωx∈ C x∀ x ∈ U \ Sing(
), then ωx∈ 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, Ca is the tangent sheaf of a foliation a=
( Ca ). 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
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PROPOSITION 1. Let and
be reduced foliations of dimensions i < j on a complex manifold M . Then,
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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 ∈ Ak 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 ) ⊂ Tx 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
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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,
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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
oln (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
oln (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
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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
oln (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
oln (1, d).
PENCIL OF FOLIATIONS ON
Let us now consider
oln (n - 1, d ), the space of foliations of codimension one and degree d on![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/206a9f5aeb14cc441a254069b43e217cc53ebe83.jpg)
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(i) ω ∧ dω = 0 (integrability);
(ii) ir ω = = 0, where r = X0∂/∂X0 + ∙∙∙ Xn∂/∂Xn is the radial vector field (Euler condition);
(iii) codim Sing(ω) > 2,
where Sing(ω) ={A0 = A1 = ∙∙∙ = An = 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,
oln(n - 1, d ) is an open set of an algebraic subvariety![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/723388f67d48213742886ec70548a3922291decf.jpg)
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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 = 0 denotes the homogeneous polynomial equation for this set, we write ω1 ∧ ω2 = θ, for some 2-form θ whose coefficients are homogeneous polynomials and whose singular set has codimension two or greater. Since
ir(ω1 ∧ ω2) = irω1 ∧ ω2 - ω1 ∧ irω2 = 0
we have ir 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
oln (n - 1, d ), is called a pencil of foliations. Remark that two foliations in
oln (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 = 1θ , where θ is a polynomial 2-form that induces, having singular set of codimension at least two, and 1 is a non-zero homogeneous polynomial. Similarly, we have ω2 ∧ ω3 = 2θ, for some non-zero homogeneous polynomial 2. 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 oln (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 ∂
oln(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 Ai and Bi 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 ωt = ω1 + tω2 has a component of codimension one in its singular set. For such a t, take gt = 0 as an equation of this component, where gt is non-constant reduced homogeneous polynomial. Fix i, j, with 0 < i, j < n . We have that both Ai + t Bi and Aj + tBj vanish over { gt = 0}. If gt is a factor of neither Bi nor Bj, then we have that Ai Bi = t = Aj Bj over { gt = 0}, which means that AiBj - Aj Bi vanishes over { gt = 0}. The same will be true if gt is a factor of Bi (or Bj ), since, in this case, it will also be a factor of Ai (or Aj ). In any case, we have that gt is a factor of AiBj - Aj Bi . Finally, the hypothesis on the singular sets of ω1 and ω2 implies that, by varying t, there are infinitely many different polynomials gt. This gives that AiBj - Aj Bi = 0, that is Ai Bi = Aj Bj = 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 ∂ oln(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 ∂oln (n - 1, d ), where d + 1 is the degree of αi ωi, its generic element lies in
oln (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
oln (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 Lj ⊂ .
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 Lr - k +2 do not span
. It follows straight from the definition that
Let Ak (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 [
] ∈ Am - k (M) is independent of the flag, where Am - 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 : ∙∙∙ : Xn ) 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
![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/7fdeffddf5c3a6a4084624a0293ff3b2c62159b5.jpg)
![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/9b25d7f6dac8769d03bfcd6b30236dabd9031c0d.jpg)
α0A0 + α1A1 +∙∙∙+ α n An = 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 Lm -1 is a line in
, which we suppose to be generated by points of coordinates ( α0 : α1 : ∙∙∙ : αn ) and (β1 : β2 : ∙∙∙ : βn ). Thus,
is contained in the variety V2 given by the pair of equations
We assume that Lm-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 2G ∪ 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 < ∙∙∙ < ik < m and 1-∙∙∙ -
k be a flag of foliations on the m-dimensional projective manifold M ⊂
. Fix a flag of linear subspaces of
:
D : Ln ⊂ Ln -1 ⊂ ∙∙∙ ⊂ Lj ⊂ ∙∙∙ L1 ⊂ L0 =
For i = i1,. . ., ik, 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 < ∙∙∙ < ik . 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, Li- r + 2has 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 (
![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/849fd9451f855603b7989d9855286b9d08b6420b.jpg)
Thus,
dim (
![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/849fd9451f855603b7989d9855286b9d08b6420b.jpg)
which finishes the proof.
Let us now consider a flag of foliations
1![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/df4f06c485dd3ece214cbb00e97b2f5de23a7df1.jpg)
![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/df4f06c485dd3ece214cbb00e97b2f5de23a7df1.jpg)
![](https://minio.scielo.br/documentstore/1678-2690/NjdRvs5ScZ8XHrJnVRN7sYy/206a9f5aeb14cc441a254069b43e217cc53ebe83.jpg)
THEOREM 4. Let i < j be two integers of the list i1 < ∙∙∙ < ik . 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 Lj -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 Lj -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
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
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Publication Dates
-
Publication in this collection
29 July 2011 -
Date of issue
Sept 2011
History
-
Received
29 Mar 2010 -
Accepted
03 Mar 2011