[Chern classes for analytic sets]
Let V be a compact complex analytical subset of a holomorphic manifold M. We shall define classes in homology, which coincide, when V is non-singular, with the Poincaré duals and of the Chern classes of the normal bundle NV and of the tangent bundle TV. However, these definitions depend in general on the data on a desingularization of V, except in some particular cases, as complex curves or sets which are locally complete intersection (LCI). These classes make possible to generalize some theories already known for LCI, such as the various indices of foliations relatively to invariant subsets, or the Minor numbers and classes.
Soit V un sous-ensemble analytique complexe compact d'une variété holomorphe M. Nous allons définir des classes en homologie, qui coı̈ncident, lorsque V est sans singularité, avec les duales de Poincaré et des classes de Chern des fibrés normal NV et tangent TV. Cependant ces définitions dépenderont en général de la donnée d'une désingularisation de V, excepté dans quelques cas particuliers tels ceux des courbes complexes ou des ensembles qui sont localement intersection complète (LCI). Ces classes permettent de généraliser des théories déjà connues pour les LCI, telle celle des indices de feuilletages relatifs à un sous-ensemble analytique invariant, ou celle des nombres et classes de Milnor.
Accepted:
Published online:
Vincent Cavalier 1; Daniel Lehmann 1; Marcio Soares 2
@article{CRMATH_2004__338_11_879_0, author = {Vincent Cavalier and Daniel Lehmann and Marcio Soares}, title = {Classes de {Chern} des ensembles analytiques}, journal = {Comptes Rendus. Math\'ematique}, pages = {879--884}, publisher = {Elsevier}, volume = {338}, number = {11}, year = {2004}, doi = {10.1016/j.crma.2004.03.009}, language = {fr}, }
Vincent Cavalier; Daniel Lehmann; Marcio Soares. Classes de Chern des ensembles analytiques. Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 879-884. doi : 10.1016/j.crma.2004.03.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.03.009/
[1] Analytic cycles on complex manifolds, Topology, Volume 1 (1961), pp. 25-45
[2] Singularities of holomorphic foliations, J. Differential Geom., Volume 7 (1972), pp. 279-342
[3] Resolution of singularities, Several Complex Variables, Berkeley CA, 1995–1996, Math. Sci. Res. Inst. Publ., Cambridge University Press, 1999, pp. 43-78
[4] Milnor classes of local complete intersections, Trans. Amer. Math. Soc., Volume 354 (2001) no. 4, pp. 1351-1371
[5] Sur les classes de Chern d'un ensemble analytique complexe, Caractéristique d'Euler–Poincaré, Astérisque, vol. 82–83, Société Mathématique de France, 1981, pp. 93-147
[6] Localisation des résidus de Baum–Bott, courbes généralisées, et K-théorie, Comment. Math. Helv., Volume 76 (2001), pp. 665-683
[7] V. Cavalier, D. Lehmann, M. Soares, Classes de Chern des ensembles analytiques, et applications, Prépublication, Département des Sciences mathématiques, Université de Montpellier II, 2003
[8] Resolution of singularities of an algebraic variety over a field of characteristic zero I, II, Ann. of Math., Volume 79 (1964), pp. 109-326
[9] Residue of holomorphic vector fields relative to singular invariant subvarieties, J. Differential Geom., Volume 42 (1995) no. 1, pp. 165-192
[10] On the index of a holomorphic vector field tangent to a singular variety, Bol. Soc. Brasil Mat., Volume 26 (1995), pp. 183-199
[11] Generalization of variations and Baum–Bott residues for holomorphic foliations on singular varieties, Int. J. Math., Volume 10 (1999) no. 3, pp. 367-384
Cited by Sources:
Comments - Policy