Unfoldings and deformations of rational and logarithmic foliations
[Déploiements et déformations de feuilletages rationnels et logarithmiques]
Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1583-1613.

Nous étudions des feuilletages de codimension un dans l’espace projectif n sur en regardant leurs perturbations du premier ordre : déploiements et déformations. Nous prêtons une attention particulière aux feuilletages rationnels et logarithmiques.

Pour une forme différentielle ω définissant un feuilletage de codimension un, nous présentons un module gradué 𝕌(ω), lié aux déploiements du premier ordre de ω. Si ω est une forme générique de type rationnel ou logarithmique, comme une première application de la construction de 𝕌(ω), nous classifions les déformations du premier ordre qui apparaissent à partir des déploiements du premier order. Ensuite, nous comptons le nombre de points isolés dans l’ensemble singulier de ω, en termes d’un polynôme de Hilbert associé à 𝕌(ω).

Nous revoyons la notion de régularité de ω en termes d’un complexe long de modules gradués que nous introduisons dans ce travail. Nous utilisons ce complexe pour prouver que, pour des feuilletages rationnels et logarithmiques génériques, ω est régulièr si et seulement si tout déploiement est trivial modulo isomorphisme.

We study codimension one foliations in projective space n over by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type.

For a differential form ω defining a codimension one foliation, we present a graded module 𝕌(ω), related to the first order unfoldings of ω. If ω is a generic form of rational or logarithmic type, as a first application of the construction of 𝕌(ω), we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in the singular set of ω, in terms of a Hilbert polynomial associated to 𝕌(ω).

We review the notion of regularity of ω in terms of a long complex of graded modules that we also introduce in this work. We use this complex to prove that, for generic rational and logarithmic foliations, ω is regular if and only if every unfolding is trivial up to isomorphism.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3044
Classification : 37F75, 14D20, 14B10
Keywords: foliations, codimension one, unfoldings, deformations
Mot clés : feuilletages, codimension un, déploiements, déformations
Molinuevo, Ariel 1

1 Departamento de Matemática, FCEyN Universidad de Buenos Aires Ciudad Universitaria, Pabellón I CP C1428EGA Buenos Aires (Argentina)
@article{AIF_2016__66_4_1583_0,
     author = {Molinuevo, Ariel},
     title = {Unfoldings and deformations of rational and logarithmic foliations},
     journal = {Annales de l'Institut Fourier},
     pages = {1583--1613},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {4},
     year = {2016},
     doi = {10.5802/aif.3044},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3044/}
}
TY  - JOUR
AU  - Molinuevo, Ariel
TI  - Unfoldings and deformations of rational and logarithmic foliations
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 1583
EP  - 1613
VL  - 66
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3044/
DO  - 10.5802/aif.3044
LA  - en
ID  - AIF_2016__66_4_1583_0
ER  - 
%0 Journal Article
%A Molinuevo, Ariel
%T Unfoldings and deformations of rational and logarithmic foliations
%J Annales de l'Institut Fourier
%D 2016
%P 1583-1613
%V 66
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3044/
%R 10.5802/aif.3044
%G en
%F AIF_2016__66_4_1583_0
Molinuevo, Ariel. Unfoldings and deformations of rational and logarithmic foliations. Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1583-1613. doi : 10.5802/aif.3044. https://aif.centre-mersenne.org/articles/10.5802/aif.3044/

[1] Calvo-Andrade, Omegar Irreducible components of the space of holomorphic foliations, Math. Ann., Volume 299 (1994) no. 4, pp. 751-767 | DOI

[2] Calvo-Andrade, Omegar El espacio de foliaciones holomorfas de codimensión uno, Monografías del Seminario Iberoamericano de Matemáticas [Monographs of the Seminario Iberoamericano de Matemáticas], 2, Instituto Interuniversitario de Estudios de Iberoamerica y Portugal, Tordesillas, 2003, 139 pages

[3] Camacho, César; Lins Neto, Alcides The topology of integrable differential forms near a singularity, Inst. Hautes Études Sci. Publ. Math. (1982) no. 55, pp. 5-35

[4] Cerveau, D.; Lins Neto, A. Irreducible components of the space of holomorphic foliations of degree two in CP(n), n3, Ann. of Math. (2), Volume 143 (1996) no. 3, pp. 577-612 | DOI

[5] Cukierman, Fernando; Gargiulo, J.; Massri, C. D. On the stability of logarithmic differential one-forms (To appear)

[6] Cukierman, Fernando; Pereira, Jorge Vitório Stability of holomorphic foliations with split tangent sheaf, Amer. J. Math., Volume 130 (2008) no. 2, pp. 413-439 | DOI

[7] Cukierman, Fernando; Pereira, Jorge Vitório; Vainsencher, I. Stability of foliations induced by rational maps, Ann. Fac. Sci. Toulouse Math. (6), Volume 18 (2009) no. 4, pp. 685-715 http://afst.cedram.org/item?id=AFST_2009_6_18_4_685_0

[8] Cukierman, Fernando; Soares, Marcio G.; Vainsencher, Israel Singularities of logarithmic foliations, Compos. Math., Volume 142 (2006) no. 1, pp. 131-142 | DOI

[9] Decker, W; Greuel, G.-M.; Pfister, G.; Schönemann, H. A computer algebra system for polynomial computations (http://www.singular.uni-kl.de.)

[10] Dieudonné, J. Éléments d’analyse. Tome I, Cahiers Scientifiques [Scientific Reports], XXVIII, Gauthier-Villars, Paris, 1981, xxi+390 pages (Fondements de l’analyse moderne. [Foundations of modern analysis], Translated from the English by D. Huet, With a foreword by Gaston Julia)

[11] Dubinsky, M.; Massri, C. D.; Molinuevo, A. diffAlg, a differential algebra library (Available at https://savannah.nongnu.org/projects/diffalg/)

[12] Eisenbud, David Commutative algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, xvi+785 pages (With a view toward algebraic geometry) | DOI

[13] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008, x+523 pages (Reprint of the 1994 edition)

[14] Gómez-Mont, Xavier; Lins Neto, Alcides Structural stability of singular holomorphic foliations having a meromorphic first integral, Topology, Volume 30 (1991) no. 3, pp. 315-334 | DOI

[15] Grayson, D. R.; Stillman, M. E. Macaulay2, a software system for research in algebraic geometry (Available at http://www.math.uiuc.edu/Macaulay2/)

[16] Grothendieck, Alexander Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2), Volume 9 (1957), pp. 119-221

[17] Grothendieck, Alexander Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. (1964) no. 20, 259 pages

[18] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, xvi+496 pages (Graduate Texts in Mathematics, No. 52)

[19] Hartshorne, Robin Deformation theory, Graduate Texts in Mathematics, 257, Springer, New York, 2010, viii+234 pages | DOI

[20] Herrera, M.; Lieberman, D. Duality and the de Rham cohomology of infinitesimal neighborhoods, Invent. Math., Volume 13 (1971), pp. 97-124

[21] Huybrechts, Daniel Complex geometry, Universitext, Springer-Verlag, Berlin, 2005, xii+309 pages (An introduction)

[22] Jouanolou, J. P. Équations de Pfaff algébriques, Lecture Notes in Mathematics, 708, Springer, Berlin, 1979, v+255 pages

[23] Lins Neto, Alcides Componentes irredutíveis dos espaços de folheações, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2007, iv+204 pages (26 o Colóquio Brasileiro de Matemática. [26th Brazilian Mathematics Colloquium])

[24] Malgrange, B. Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math. (1976) no. 46, pp. 163-173

[25] Malgrange, B. Frobenius avec singularités. II. Le cas général, Invent. Math., Volume 39 (1977) no. 1, pp. 67-89

[26] Mattei, J.-F. Modules de feuilletages holomorphes singuliers. I. Équisingularité, Invent. Math., Volume 103 (1991) no. 2, pp. 297-325 | DOI

[27] Michor, Peter W. Topics in differential geometry, Graduate Studies in Mathematics, 93, American Mathematical Society, Providence, RI, 2008, xii+494 pages | DOI

[28] Moerdijk, I.; Mrčun, J. Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003, x+173 pages | DOI

[29] Mumford, David The red book of varieties and schemes, Lecture Notes in Mathematics, 1358, Springer-Verlag, Berlin, 1999, x+306 pages Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico Arbarello | DOI

[30] Serre, Jean-Pierre Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, 11, Springer-Verlag, Berlin-New York, 1965, vii+188 pp. (not consecutively paged) pages

[31] Suwa, Tatsuo Unfoldings of complex analytic foliations with singularities, Japan. J. Math. (N.S.), Volume 9 (1983) no. 1, pp. 181-206

[32] Suwa, Tatsuo Unfoldings of foliations with multiform first integrals, Ann. Inst. Fourier (Grenoble), Volume 33 (1983) no. 3, pp. 99-112

[33] Suwa, Tatsuo Unfoldings of meromorphic functions, Math. Ann., Volume 262 (1983) no. 2, pp. 215-224 | DOI

[34] Suwa, Tatsuo Unfoldings of codimension one complex analytic foliation singularities, Singularity theory (Trieste, 1991), World Sci. Publ., River Edge, NJ, 1995, pp. 817-865

[35] Warner, Frank W. Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 94, Springer-Verlag, New York-Berlin, 1983, ix+272 pages (Corrected reprint of the 1971 edition)

Cité par Sources :