Papadima, Stefan
The rational homotopy of Thom spaces and the smoothing of isolated singularities
Annales de l'institut Fourier, Tome 35 (1985) no. 3 , p. 119-135
Zbl 0563.57010 | MR 87b:55009
doi : 10.5802/aif.1021
URL stable : http://www.numdam.org/item?id=AIF_1985__35_3_119_0

On utilise les méthodes de l’homotopie rationnelle pour étudier le problème du lissage topologique des singularités algébriques complexes isolées. On montre que, dans toutes les situations, un revêtement convenable peut être lissé. On considère ensuite le problème du lissage topologique (la structure complexe normale y compris) pour les singularités coniques. On établit des liaisons entre l’existence de certaines relations entre les degrés de Chern normaux d’une variété projective lisse et la question de sa réalisation comme section linéaire (pas nécessairement hyperplane).
Rational homotopy methods are used for studying the problem of the topological smoothing of complex algebraic isolated singularities. It is shown that one may always find a suitable covering which is smoothable. The problem of the topological smoothing (including the complex normal structure) for conical singularities is considered in the sequel. A connection is established between the existence of certain relations between the normal Chern degrees of a smooth projective variety and the question of its realization as a linear section (not necessarily hyperplane).

Bibliographie

[1] O. Burlet, Cobordismes de plongements et produits homotopiques, Comm. Math. Helv., 46 (1971), 277-288. MR 45 #4433 | Zbl 0221.57017

[2] Y. Felix, D. Tanre, Sur la formalité des applications, Publ. IRMA, Lille, 3-2 (1981).

[3] H. Hamm, On the vanishing of local homotopy groups for isolated singularities of complex spaces, Journal für die reine und ang. Math., 323 (1981), 172-176. MR 82i:32023 | Zbl 0483.32007

[4] R. Hartshorne, Topological conditions for smoothing algebraic singularities, Topology, 13 (1974), 241-253. MR 50 #2170 | Zbl 0288.14006

[5] R. Hartshorne, E. Rees, E. Thomas, Nonsmoothing of algebraic cycles on Grassmann varieties, BAMS, 80(5) (1974), 847-851. MR 50 #9870 | Zbl 0289.14011

[6] S. Halperin, J.D. Stasheff, Obstructions to homotopy equivalences, Adv. in Math., 32 (1979), 233-279. MR 80j:55016 | Zbl 0408.55009

[7] M.L. Larsen, On the topology of complex projective manifolds, Inv. Math., 19 (1973), 251-260. MR 47 #7058 | Zbl 0255.32004

[8] J. Milnor, Singular points of complex hypersurfaces, Princeton University Press, 1968. MR 39 #969 | Zbl 0184.48405

[9] S. Papadima, The rational homotopy of Thom spaces and the smoothing of homology classes, to appear Comm. Math. Helv. Zbl 0592.57025

[10] E. Rees, E. Thomas, Cobordism obstructions to deforming isolated singularities, Math. Ann., 232 (1978), 33-53. MR 58 #18475 | Zbl 0381.57011

[11] H. Shiga, Notes on links of complex isolated singular points, Kodai Math. J., 3 (1980), 44-47. MR 81f:57039 | Zbl 0442.57015

[12] A.J. Sommese, Non-smoothable varieties, Comm. Math. Helv., 54 (1979), 140-146. MR 80k:32027 | Zbl 0394.14016

[13] D. Sullivan, Infinitesimal computations in topology, Publ. Math. IHES, 47 (1977), 269-331. Numdam | MR 58 #31119 | Zbl 0374.57002

[14] R. Thom, Quelques propriétés globales des variétés différentiable, Comm. Math. Helv., 28 (1954), 17-86. MR 15,890a | Zbl 0057.15502