Abstract
We study the singular set of a codimension one holomorphic foliation on ℙ 3 . We find a local normal form for these foliations near a codimension two component of the singular set that is not of Kupka type. We also determine the number of non-Kupka points immersed in a codimension two component of the singular set of a codimension one foliation on ℙ 3.
holomorphic foliations; Kupka sets; non-Kupka points
1 - Introduction
A regular codimension one holomorphic foliation on a complex manifold , can be defined by a triple where
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is an open cover of .
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is a holomorphic submersion for each .
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A family of biholomorphisms such that
Since the set is a subbundle. Alsodefine a line bundle . The family of 1–forms glue to a global section. We have
where . We also obtain
Definition 1.1. Let be a compact complex manifold of dimension .A singular codimension one holomorphic foliation on , may be defined by one of the following ways:
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A pair , where is an analytic subset of , and is a regular codimension one holomorphic foliation on .
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A class of global sections , where such that
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the singular set has .
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in .
In this case, we denote by the foliation represented by .
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An exact sequence of sheaves
where is a reflexive sheaf of rank with torsion free quotient where is an ideal sheaf for some closed scheme .
These three definitions are equivalents.
Remark 1.2. Let be a section.
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The section may be defined by a family of 1-forms
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The section is a morphism of sheaves . The kernel of is the tangent sheaf. The image of is a twisted ideal sheaf . It is called the normal sheaf of .
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As in the non-singular case, the following equality of line bundles holds
where are the canonical sheaf of and respectively.
We denote by
The number is called the degree of the foliation represented by .
1-1 Statement of the results
In the sequel, is a compact complex manifold with . We will use any of the above definitions for foliation. The singular set will be denoted by . Observe that decomposes as
For a foliation on represented by the Kupka set (Kupka 1964r15 KUPKA I. 1964. Singularities of structurally stable Pfaffian forms. Proc Nat Acad of Sc USA 52: 1431-1432.; De Medeiros 1977r12 DE MEDEIROS A. 1977. Structural stability of integrable differential forms. Palis J and do Carmo M (Eds), Geometry and Topology, Springer LNM 597, p. 395-428.) is defined by
We recall that for points near the foliation is biholomorphic to a product of a dimension one foliation in a transversal section by a regular foliation of codimension two (Kupka 1964r15 KUPKA I. 1964. Singularities of structurally stable Pfaffian forms. Proc Nat Acad of Sc USA 52: 1431-1432.) and in particular we have .
In this note, we focus our attention on the set of non-Kupka points of . The first remark is
We analyze three cases, one in each section, the last two being the core of the work.
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, then .
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There is an irreducible component such that .
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For a foliation . Let be a connected component such that is a finite set of points.
The first case has been considered in Brunella (2009r2 BRUNELLA M. 2009. Sur les Feuilletages de l’espace projectif ayant une composante de Kupka. Enseig Math (2) 55(3-4): 227-234.); Calvo-Andrade (1999r4 CALVO-ANDRADE O. 1999. Foliations with a Kupka Component on Algebraic Manifolds. Bull of the Brazilian Math Soc 30(2): 183-197., 2016r5 CALVO-ANDRADE O. 2016. Foliations with a radial Kupka set on projective spaces. Bull of the Brazilian Math Soc, 13 p. doi 10.1007/s00574-016-0158-6
10.1007/s00574-016-0158-6...
); Calvo-Andrade andSoares (1994r6 CALVO-ANDRADE O, CERVEAU D, GIRALDO L AND LINS NETO A. 2004. Irreducible components of the space of foliations associated with the affine Lie algebra. Ergodic Theory and Dyn Sist 24: 987-1014.); Cerveau andLins Neto (1994r8 CERVEAU D AND MATTEI JF. 1982. Formes intégrables holomorphes singulières. Astérisque 97, Paris: Soc Math de France.). Let be a foliation with and connected, then has a meromorphic first integral. In the generic case, the leaves define a Lefschetz or a Branched Lefschetz Pencil. The non-Kupka points are isolated singularities . In this note, we present a new and short proof of this fact when the transversal type of is radial.
In the second section, we study the case of a non-Kupka irreducible component of . These phenomenon arise naturally in the intersection of irreducible components of . The following result is a local normal form for near the singular set and is a consequence of a result of Loray (2006r16 LORAY F. 2006. A preparation theorem for codimension-one foliations. Ann of Math (2) 163(2): 709-722.).
Theorem 1. Let , , be a germ of integrable 1–form such that , is a smooth point and on . If , then or either
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there exists a coordinate system such that
and is biholomorphic to the product of a dimension one foliation in a transversal section by a regular foliation of codimension two, or
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there exists a coordinate system such that
such that with , or
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has a non-constant holomorphic first integral in a neighborhood of .
The alternatives are not exclusives. The following example was suggest by the referee and show that the case (3) of Theorem 1 cannot be avoid.
Example 1.3. Let be a germ of a 1-form at defined by
where . We have
The singular set of is and , therefore the singular set has an embedding point and vanish along . We will show that has a holomorphic first integral in a neighborhood of . In fact, let and set defined by
Let be 1-form at , note that and moreover , this implies that is non-singular at and by Frobenius theorem has a holomorphic first integral on . Defining , we get is a holomorphic first integral for in a neighborhood of .
We apply Theorem 1 to a codimension one holomorphic foliation of the projective space with empty Kupka set.
About the third case, consider a foliation . Let be a connected component of . We count the number of non-Kupka points of in .
Theorem 2. Let be a foliation and a connected component of . Suppose that is a local complete intersection and is a finite set of points, then is a global section of and the associated divisor has degree
Note that the section vanishes exactly in the non-Kupka points of in then the above theorem determine the number (counted with multiplicity) of non-Kupka points of in .
2 - The singular set
Let be a codimension one holomorphic foliation then singular set of may be written as
The fact that implies that . To continue we focus in the components of singular set of of dimension at least three.
2.1 - Singular set of codimension at least three
We recall the following result due to B. Malgrange.
Theorem 2.1.(Malgrange 1976r17 MALGRANGE B. 1976. Frobenius avec singularités. Codimension 1. IHES Publ Math 46: 163-173.)Let be a germ at , of an integrable 1–form singular at , if , then there exist and such that
We have the following proposition.
Proposition 2.2. Let be a foliation and let an isolated singularity, then any germ of vector field tangent to the foliation vanishes at .
Proof. Let be a 1-form representing the foliation at . Let be a vector field tangent to the foliation, i.e., .If there exists a coordinate system with and , then
and , but this function does not have an isolated singularity.∎
Now, we begin our study of the irreducible components of codimension two of the singular set of . Note that, given a section , along the singular set, the equation implies Then
2.2 - The Kupka set
These singularities has bee extensively studied and the main properties have been established in (Kupka 1964r15 KUPKA I. 1964. Singularities of structurally stable Pfaffian forms. Proc Nat Acad of Sc USA 52: 1431-1432.; De Medeiros 1977r12 DE MEDEIROS A. 1977. Structural stability of integrable differential forms. Palis J and do Carmo M (Eds), Geometry and Topology, Springer LNM 597, p. 395-428.).
Definition 2.3. For . The Kupka set is
The following properties of Kupka sets, are well known (De Medeiros 1977r12 DE MEDEIROS A. 1977. Structural stability of integrable differential forms. Palis J and do Carmo M (Eds), Geometry and Topology, Springer LNM 597, p. 395-428.).
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is smooth of codimension two.
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has local product structure and the tangent sheaf is locally free near .
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is subcanonically embedded and
Let be a foliation with .By Calvo-Andrade and Soares (1994r6 CALVO-ANDRADE O, CERVEAU D, GIRALDO L AND LINS NETO A. 2004. Irreducible components of the space of foliations associated with the affine Lie algebra. Ergodic Theory and Dyn Sist 24: 987-1014.), there exists a pair , where is a rank two holomorphic vector bundle and , such that
and the total Chern class
In 2009, Marco Brunella proved that following result, which in a certain sense say that the local transversal type of the singular set of foliation determines its behavior globally. Here we present a new proof of this fact. The techniques used in the proof could be of independent interest.
Proposition 2.4. Let be a foliation with , (connected if ) and of radial transversal type. Then is a complete intersection and has a meromorphic first integral.
To prove Proposition 2.4, we require the following lemma. This result may be well known but for lack of a suitable reference we include the proof in an appendix.
Lemma 2.5. Let be a rank two holomorphic vector bundle over with and . Then , is holomorphically trivial.
Now, we prove Proposition 2.4.
Proof of Proposition 2.4. Let be the vector bundle with a section defining the Kupka set as scheme. The radial transversal type implies (Calvo-Andrade and Soares 1994r6 CALVO-ANDRADE O, CERVEAU D, GIRALDO L AND LINS NETO A. 2004. Irreducible components of the space of foliations associated with the affine Lie algebra. Ergodic Theory and Dyn Sist 24: 987-1014.)
The vector bundle, has and . Let be a linear embedding. By the preceding lemma we have
and by the Horrocks’ criterion (Okonek et al. 1980r18 OKONEK CH, SCHNEIDER M AND SPINDLER H. 1980. Vector Bundles on Complex Projective spaces. Progress in Math Vol 3, Boston: Birkhauser, vii + 389 p.),
is trivial and hence splits as and is a complete intersection. The existence of the meromorphic first integral follows from Theorem A of (Cerveau and Lins Neto 1994r8 CERVEAU D AND MATTEI JF. 1982. Formes intégrables holomorphes singulières. Astérisque 97, Paris: Soc Math de France.). ∎
If is such that and connected, the set of non-Kupka points of is
A generic rational map, that means, a Lefschetz or a Branched Lefschetz Pencil, has only isolated singularities away its base locus. The singular set of the foliation defined by the fibers of is . The Kupka set corresponds away from its base locus and are the singularities as a map. is empty if and only if the degree of the foliation is . The number of isolated singularities counted with multiplicities can be calculated by (Cukierman et al. 2006r11 CUKIERMAN F, SOARES M AND VAINSENCHER I. 2006. Singularities of Logarithmic foliations. Compositio Math 142: 131-142.). If is a germ of form that defines the foliation at , we have
We have that .
3 - Foliations with a non-Kupka component
It is well known that , but the converse is not true. Our first result describes the singular points with this property.
3.1 - A normal form
Now, we analyze the situation when there is an irreducible non-Kupka component of .
Proof of Theorem 1. By hypotheses, for any . Since
we get for any . Now, as and , we have . We distinguish two cases.
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: there is a coordinate system such that
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: there is a coordinate system such that and .
The first case is known, the foliation is equivalent in a neighborhood of to a product of a dimension one foliation in a transversal section by a regular foliation of codimension two (Cerveau and Mattei 1982r10 CERVEAU D AND LINS NETO A. 1994. Codimension one Foliations in Cpn , n ≥ 3, with Kupka components. In: Lins A, Moussu R and Sad P (Eds), Complex Analytic Methods in Dynamical Systems. Camacho C, Astérisque 222, p. 93-133.).
In the second case, Loray’s preparation theorem (Loray 2006r16 LORAY F. 2006. A preparation theorem for codimension-one foliations. Ann of Math (2) 163(2): 709-722.), shows that there exists a coordinate system , a germ with , and germs such that the foliation is defined by the 1–form
Since and is a smooth point, we can assume that , where is the germ of at . Therefore,
Hence, either and , or and for all . In any case, we have and then , where is a germ of holomorphic function in the variable ; and does not divide . We have two possibilities:
case.–. In this case, we consider the biholomorphism
where is a branch of the root of , we get and
where , . Therefore, is equivalent to and moreover is given by
Since is zero identically on ,we get , so that , for some and such that does not divided . Using the above expression for in (3.2), we have
where. Consider defined by , then
case.–. We have and
therefore
Note that , otherwise would be contained in , but it is contradiction because . Furthermore , because otherwise .
Let be defined by
then from (3.4), we get that
where . Since , we deduce that has a non-constant holomorphic first integral such that . Therefore, is a non-constant holomorphic first integral for in a neighborhood of .∎
3.2 - Applications to foliations on
In order to give some applications of Theorem 1, we need the Baum-Bott index associated to singularities of foliations of codimension one.
Let be a complex manifold and let be a codimension one holomorphic foliation represented by . We have the exact sequence
Set and take . Then in a neighborhood of the foliation is induced by a holomorphic –form and there exists a differentiable –form such that
Let be an irreducible component of . Take a generic point ,that is, is a point where is smooth and disjoint from the other singular components. Pick a ball centered at sufficiently small, so that is a sub-ball of of codimension . Then the De Rhamclass can be integrated over an oriented -sphere positively linked with :
This complex number is the Baum-Bott residue of along Z.We have a particular case of the general Baum-Bott residues Theorem (Baum and Bott 1972r1 BAUM P AND BOTT R. 1972. Singularities of holomorphic foliations. J Differential Geometry 7: 279-342.), reproved by Brunella and Perrone (2011r3 BRUNELLA M AND PERRONE C. 2011. Exceptional singularities of codimension one holomorphic foliations. Publ Mathemàtique 55: 295-312.).
Theorem 3.1. Let be a codimension one holomorphic foliation on a complex manifold . Then
where is the normal sheaf of on and the sum is done over all irreducible components of .
In particular, if is a codimension one foliation on of degree , then the normal sheaf and the Baum-Bott Theorem looks as follows
Remark 3.2. If there exist a coordinates system around such that and . Moreover, if we assume that
is a holomorphic 1-form representing . Then we can consider the (1,0)-form on given by
Since, we get
where denotes the Grothendieck residue, is the Jacobian of the holomorphic map . It follows from Griffiths andHarris (1978r14 GRIFFITHS PH AND HARRIS J. 1978. Principles of Algebraic Geometry. In Pure & Appl Math, Wiley Interscience. ) that if is non-singular, then
In the situation explained above, the tangent sheaf is locally free and generated by the holomorphic vector fields
and the vector field carries the information of the Baum–Bott residues.
The next result, in an application of Theorem 1
Theorem 3.3. Let be a foliation and . Suppose that is smooth and for all , then .
Proof. We work in a small neighborhood of . According to Theorem 1 there exist a coordinate system at such that and one has three cases. In the first case, is the product of a dimension one foliation in a section transversal to by a regular foliation of codimension two and . In this case, it follows from (3.6) that . In the second case
where and it follows from Lemma 3.9 of Cerveau and Lins Neto (2013r9 CERVEAU D AND LINS NETO A. 2013. A structural theorem for codimension-one foliations on Pn , n ≥ 3, with application to degree three foliations. Ann Sc Norm Super Pisa Cl Sci (5) 12(1): 1-41.) that
Since is holomorphic at , we get In the third case has a holomorhic first integral in neighborhood of and is known that ∎
The Baum-Bott formula implies the following result.
Corollary 3.4. Let , , be a foliation with . Then there exists a smooth point such that .
Proof. If for all smooth point one has , the above theorem shows that for all irreducible components . By Baum–Bott’s theorem, we get
which is a contradiction. Therefore there exists a smooth point such that . ∎
In particular, if , , is a foliation with for any , then its Kupka set is not empty.
4 - The number of non-Kupka points
Through this section, we consider codimension one foliations on , but some results remain valid to codimension one foliations on others manifolds of dimension three.
4.1 - Simple singularities
Let be a germ of 1–form at . We define the rotational of as the unique vector field such that
moreover is integrable if and only if .
Let be a germ of an integrable 1–form at . We say that is a simple singularity of if and either or has an isolated singularity at . In the second case, these kind of singularities, are classified as follows
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Logarithmic. The second jet and the linear part of at has non zero eigenvalues.
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Degenerated. The rotational has a zero eigenvalue, the other two are non zero and necessarily satisfies the relation .
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Nilpotent. The rotational vector field is nilpotent as a derivation.
The structure near simple singularity is known (Calvo-Andrade et al. 2004r7 CALVO-ANDRADE O AND SOARES M. 1994. Chern numbers of a Kupka component. Ann Inst Fourier 44: 1219-1236.). If is a simple singularity and , then is a singular point of .
Theorem 4.1. Let , , be a germ of integrable 1-form such that has a simple singularity at then the tangent sheaf is locally free at and it is generated by, where has non zero linear part.
Proof.Let be a germ at of an integrable 1–form and a simple non-Kupka singularity. Then is an isolated singularity of . Consider the Koszul complex of the vector field at
Since , then that vanishes because has an isolated singularity at . Therefore, there exists such that . The mapis an isomorphism, hence
and then, the vector fields generate the sheaf in a neighborhood of . ∎
Let be a foliation and be a connected component of . Assume that is a local complete intersection and has only simple singularities. We will calculate the number of non-Kupka points in .
Proof of Theorem 2. Let be the ideal sheaf of . Since is a local complete intersection, consider the exact sequence
Taking and twisting by we get
Since , the singular set, we have seen before that
Now, from the equalities of sheaves
we have
the non-Kupka points of in satisfies , denoting
the associated divisor to , one has
as claimed. ∎
Remark 4.2. The method of the proof works also in projective manifolds, and does not depends on the integrability condition.
4.2 - Examples
We apply Theorem 2 for some codimension one holomorphic foliations on and determine the number of non-Kupka points.
Example 4.3 (Degree two logarithmic foliations). Recall that the canonical bundle of a degree two foliation of is trivial. There are two irreducible components of logarithmic foliations in the space of foliations of of degree two: and . We analyze generic foliations on each component.
Component : let be a generic element of and consider its singular scheme . By Theorem 3 of Cukierman et al. (2006r11 CUKIERMAN F, SOARES M AND VAINSENCHER I. 2006. Singularities of Logarithmic foliations. Compositio Math 142: 131-142.), we have . On the other hand, has three irreducible components, two quadratics and a line, the arithmetic genus is . Note that Theorem 2, implies that the number , of non-Kupka points in is
The non-Kupka points of the foliation are .
Component : let be a generic element of then the tangent sheaf is and the singular scheme (Giraldo and Pan-Collantes 2010r13 GIRALDO L AND PAN-COLLANTES AJ.. 2010. On the singular scheme of codimension one holomorphic foliations in P3 . Int J Math 21(7): 843-858.), moreover consists of 6 lines given the edges of a tetrahedron, obtained by intersecting any two of the four invariant hyperplanes . The arithmetic genus is , by Theorem 2, , corresponding to the vertices of the tetrahedron where there are simple singularities of logarithmic type.
Example 4.4 (The exceptional component ). The leaves of a generic foliation are the orbits of an action of and its tangent sheaf is (see Calvo-Andrade et al. 2004r7 CALVO-ANDRADE O AND SOARES M. 1994. Chern numbers of a Kupka component. Ann Inst Fourier 44: 1219-1236.; Giraldo and Pan-Collantes 2010r13 GIRALDO L AND PAN-COLLANTES AJ.. 2010. On the singular scheme of codimension one holomorphic foliations in P3 . Int J Math 21(7): 843-858.). Its singular locus has and three irreducible components: a line a conic tangent to at a point , and a twisted cubic with as an inflection line at . Then .
The arithmetic genus is and the canonical bundle of the foliation again is trivial, by Theorem 2, the number of non-Kupka points . Therefore the non-Kupka divisor .If represents the foliation at , then
ACKNOWLEDGMENTS
The first author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP Nº 2014/23594-6), CONACYT 262121 and thanks the Federal University of Minas Gerais (UFMG), Instituto de Matemática Pura e Aplicada (IMPA), and Instituto de Matemática, Estatística e Computação Científica da UNICAMP for the hospitality during the elaboration of this work. The second author was partially supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES-DGU 247/11), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq 300351/2012-3 and PPM-00169-13). The third author was partially supported by Bolsista/CAPES and thanks the Instituto de Matemática y Ciencias Afines (IMCA) for the hospitality. Finally, we would like to thank the referee by the suggestions, comments and improvements to the exposition.
APPENDIX
We prove Lemma 2.5.
Proof.First, we see that . By Riemann–Roch–Hirzebruch, we have
then
By Serre duality (Griffiths and Harris 1978r14 GRIFFITHS PH AND HARRIS J. 1978. Principles of Algebraic Geometry. In Pure & Appl Math, Wiley Interscience. ; Okonek et al. 1980r18 OKONEK CH, SCHNEIDER M AND SPINDLER H. 1980. Vector Bundles on Complex Projective spaces. Progress in Math Vol 3, Boston: Birkhauser, vii + 389 p.), we get . Moreover for all , hence .Let be a non zero section, consider the exact sequence
The sheaf is torsion free, therefore for some . The sequence (A.1), is a free resolution of the sheaf with vector bundles with zero Chern classes. From the definition of Chern classes for coherent sheaves (Baum and Bott 1972r1 BAUM P AND BOTT R. 1972. Singularities of holomorphic foliations. J Differential Geometry 7: 279-342.), we get , in particular , we conclude that and. Then is an extension of holomorphic line bundles, hence it splits (Okonek et al. 1980r18 OKONEK CH, SCHNEIDER M AND SPINDLER H. 1980. Vector Bundles on Complex Projective spaces. Progress in Math Vol 3, Boston: Birkhauser, vii + 389 p., p. 15).∎
REFERENCES
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r1BAUM P AND BOTT R. 1972. Singularities of holomorphic foliations. J Differential Geometry 7: 279-342.
-
r2BRUNELLA M. 2009. Sur les Feuilletages de l’espace projectif ayant une composante de Kupka. Enseig Math (2) 55(3-4): 227-234.
-
r3BRUNELLA M AND PERRONE C. 2011. Exceptional singularities of codimension one holomorphic foliations. Publ Mathemàtique 55: 295-312.
-
r4CALVO-ANDRADE O. 1999. Foliations with a Kupka Component on Algebraic Manifolds. Bull of the Brazilian Math Soc 30(2): 183-197.
-
r5CALVO-ANDRADE O. 2016. Foliations with a radial Kupka set on projective spaces. Bull of the Brazilian Math Soc, 13 p. doi 10.1007/s00574-016-0158-6
» 10.1007/s00574-016-0158-6 -
r7CALVO-ANDRADE O AND SOARES M. 1994. Chern numbers of a Kupka component. Ann Inst Fourier 44: 1219-1236.
-
r6CALVO-ANDRADE O, CERVEAU D, GIRALDO L AND LINS NETO A. 2004. Irreducible components of the space of foliations associated with the affine Lie algebra. Ergodic Theory and Dyn Sist 24: 987-1014.
-
r10CERVEAU D AND LINS NETO A. 1994. Codimension one Foliations in Cpn , n ≥ 3, with Kupka components. In: Lins A, Moussu R and Sad P (Eds), Complex Analytic Methods in Dynamical Systems. Camacho C, Astérisque 222, p. 93-133.
-
r9CERVEAU D AND LINS NETO A. 2013. A structural theorem for codimension-one foliations on Pn , n ≥ 3, with application to degree three foliations. Ann Sc Norm Super Pisa Cl Sci (5) 12(1): 1-41.
-
r8CERVEAU D AND MATTEI JF. 1982. Formes intégrables holomorphes singulières. Astérisque 97, Paris: Soc Math de France.
-
r11CUKIERMAN F, SOARES M AND VAINSENCHER I. 2006. Singularities of Logarithmic foliations. Compositio Math 142: 131-142.
-
r12DE MEDEIROS A. 1977. Structural stability of integrable differential forms. Palis J and do Carmo M (Eds), Geometry and Topology, Springer LNM 597, p. 395-428.
-
r13GIRALDO L AND PAN-COLLANTES AJ.. 2010. On the singular scheme of codimension one holomorphic foliations in P3 . Int J Math 21(7): 843-858.
-
r14GRIFFITHS PH AND HARRIS J. 1978. Principles of Algebraic Geometry. In Pure & Appl Math, Wiley Interscience.
-
r15KUPKA I. 1964. Singularities of structurally stable Pfaffian forms. Proc Nat Acad of Sc USA 52: 1431-1432.
-
r16LORAY F. 2006. A preparation theorem for codimension-one foliations. Ann of Math (2) 163(2): 709-722.
-
r17MALGRANGE B. 1976. Frobenius avec singularités. Codimension 1. IHES Publ Math 46: 163-173.
-
r18OKONEK CH, SCHNEIDER M AND SPINDLER H. 1980. Vector Bundles on Complex Projective spaces. Progress in Math Vol 3, Boston: Birkhauser, vii + 389 p.
Publication Dates
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Publication in this collection
Dec 2016
History
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Received
13 Jan 2016 -
Accepted
15 Aug 2016