Approximately holomorphic geometry and estimated transversality on 2-calibrated manifolds

Ibort, Alberto; Martínez Torres, David

doi:10.1016/j.crma.2004.03.003

Differential Geometry/Differential Topology

Approximately holomorphic geometry and estimated transversality on 2-calibrated manifolds
[Géométrie approximativement holomorphe et transversalité quantitative pour les variétés 2-calibrées]

Présenté par : Étienne Ghys

Ibort, Alberto ¹ ; Martínez Torres, David ¹

¹ Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Spain

Comptes Rendus. Mathématique, Tome 338 (2004) no. 9, pp. 709-712.

Résumé
Abstract

On définit la notion de structure 2-calibrée, qui généralise celle de structure de contact, feuilletage tendu différentiable, etc. La géométrie approximativement holomorphe, introduite par S. Donaldson pour les variétés symplectiques est généralisée pour les variétés 2-calibrées. On démontre aussi un résultat de transversalité quantitative qui permet d'étudier la géométrie de ces variétés.

The notion of 2-calibrated structure, generalizing contact structures, smooth taut foliations, etc., is defined. Approximately holomorphic geometry as introduced by S. Donaldson for symplectic manifolds is extended to 2-calibrated manifolds. An estimated transversality result that enables to study the geometry of such manifolds is presented.

Reçu le : 2003-11-11
Accepté le : 2004-03-03
Publié le : 2004-04-08

DOI : 10.1016/j.crma.2004.03.003

Affiliations des auteurs :

Ibort, Alberto ¹ ; Martínez Torres, David ¹

¹ Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Spain

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Ibort, Alberto; Martínez Torres, David. Approximately holomorphic geometry and estimated transversality on 2-calibrated manifolds. Comptes Rendus. Mathématique, Tome 338 (2004) no. 9, pp. 709-712. doi : 10.1016/j.crma.2004.03.003. http://www.numdam.org/articles/10.1016/j.crma.2004.03.003/

Bibliographie
Cité par

[1] Auroux, D. Symplectic 4-manifolds as branched coverings of ${ℂℙ}^{2}$ , Invent. Math., Volume 139 (2000), pp. 551-602

[2] Auroux, D. Estimated transversality in symplectic geometry and projective maps, Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 1-30

[3] B. Deroin, Laminations par variétés complexes, Thèse, École normale supérieure de Lyon, 2003

[4] Donaldson, S.K. Symplectic submanifolds and almost-complex geometry, J. Differential Geom., Volume 44 (1996), pp. 666-705

[5] Donaldson, S.K. Lefschetz fibrations in symplectic geometry, J. Differential Geom., Volume 53 (1999) no. 2, pp. 205-236

[6] Ghys, E. Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997), ix, xi, Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. 49-95

[7] Giroux, E. Géométrie de contact : de la dimension trois vers les dimensions supérieures, Proceedings of the International Congress of Mathematicians, vol. II, Beijing, 2002, pp. 405-414

[8] Gromov, M. Topological invariants of dynamical systems and spaces of holomorphic maps, I, Math. Phys. Anal. Geom., Volume 2 (1999) no. 4, pp. 323-415

[9] A. Ibort, D. Martínez Torres, in preparation

[10] Ibort, A.; Martínez-Torres, D.; Presas, F. On the construction of contact submanifolds with prescribed topology, J. Differential Geom., Volume 56 (2000) no. 2, pp. 235-283

[11] D. Martínez Torres, Geometries with topological character, Ph.D. Thesis, Universidad Carlos III de Madrid, 2003

[12] J.P. Mohsen, Transversalité quantitative et sous-variétés isotropes, Ph.D. Thesis, École Norm. Sup. Lyon, 2001

[13] Muñoz, V.; Presas, F.; Sols, I. Almost holomorphic embeddings in Grassmannians with applications to singular symplectic submanifolds, J. Reine Angew. Math., Volume 547 (2002), pp. 149-189

[14] Ohsawa, T.; Sibony, N. Kähler identity on Levi flat manifolds and application to the embedding, Nagoya Math. J., Volume 158 (2000), pp. 87-93

[15] Presas, F. Lefschetz type pencils on contact manifolds, Asian J. Math., Volume 6 (2002) no. 2, pp. 277-301

[16] Sullivan, D. Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., Volume 36 (1976), pp. 225-255

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