Abstract
After a short review on foliations, we prove that a codimension 1 holomorphic foliation on \({\mathbb P^3_{\mathbb C}}\) with simple singularities is given by a closed rational 1-form. The proof uses Hironaka-Matsumura prolongation theorem of formal objects.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Brunella M.: Feuilletages holomorphes sur les surfaces complexes compactes. Ann. Sci. École Norm. Sup (4)(30(5)), 569–594 (1997)
Calvo-Andrade O.: Irreducible components of the space of holomorphic foliations. Math. Ann. 299(4), 751–767 (1994)
Calvo-Andrade O., Cerveau D., Giraldo L., Lins Neto A.: Irreducible components of the space of foliations associated to the affine Lie algebra. Ergodic Theory Dynam. Syst. 24(4), 987–1014 (2004)
Camacho C., Sad P.: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. (2)(115(3)), 579–595 (1982)
Cano F.: Reduction of the singularities of codimension one singular foliations in dimension three. Ann. Math. (2)(160(3)), 907–1011 (2004)
Cano F., Cerveau D.: Desingularization of nondicritical holomorphic foliations and existence of separatrices. Acta Math. 169(1-2), 1–103 (1992)
Cano F., Mattei J.F.: Hypersurfaces intégrales des feuilletages holomorphes. Ann. Inst. Fourier Grenoble 42(1–2), 49–72 (1992)
Cartan H.: Sur le premier problème de Cousin. C. R. Acad. Sci. Paris 207, 558–560 (1939)
Cerveau D., Lins Neto A.: Irreducible components of the space of holomorphic foliations of degree two in CP(n), n ≥ 3. Ann. Math. (2)(143(3)), 577–612 (1996)
Cerveau, D., Lins Neto, A.: A structural theorem for codimension one foliations on Pn, n ≥ 3, with application to degree three foliations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (accepted)
Cerveau, D., Lins-Neto, A., Loray, F., Pereira, J.V., Touzet, F.: Complex codimension one singular foliations and Godbillon-Vey sequences. Mosc. Math. J. 7(1):21–54 (2007) (166)
Hironaka H.: On some formal imbeddings. Illinois J. Math. 12, 587–602 (1968)
Hironaka H., Matsumura H.: Formal functions and formal embeddings. J. Math. Soc. Japan 20, 52–82 (1968)
Jouanolou, JP.: Équations de Pfaff algébriques, Volume 708 of Lecture Notes in Mathematics. Springer, Berlin (1979)
Lins Neto, A.: Componentes irredutí veis dos espaços de folheações. Publicações Matemáticas do IMPA. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2007. 26o Colóquio Brasileiro de Matemática.
Loray, F., Pereira, JV., Touzet, F.: Singular foliations with trivial canonical class, arxiv. 1107.1538, 2011.
Malgrange B.: Frobenius avec singularités. I. Codimension un. Inst. Hautes Études Sci. Publ. Math. (46), 163–173 (1976)
Saito, K.: On a generalization of de-Rham lemma. Ann. Inst. Fourier Grenoble. 26(2), 165–170 (1976) (vii)
Seidenberg A.: Reduction of singularities of the differential equation A dy = B dx. Amer. J. Math. 90, 248–269 (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
En l’honneur de H. Hironaka pour son 80ème anniversaire.
Rights and permissions
About this article
Cite this article
Cerveau, D. Codimension one holomorphic foliations on \({\mathbb P^n_{\mathbb C}}\): problems in complex geometry. RACSAM 107, 69–77 (2013). https://doi.org/10.1007/s13398-012-0087-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-012-0087-1