The rational homotopy of Thom spaces and the smoothing of isolated singularities

Stefan Papadima

Annales de l'institut Fourier (1985)

  • Volume: 35, Issue: 3, page 119-135
  • ISSN: 0373-0956

Abstract

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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).

How to cite

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Papadima, Stefan. "The rational homotopy of Thom spaces and the smoothing of isolated singularities." Annales de l'institut Fourier 35.3 (1985): 119-135. <http://eudml.org/doc/74680>.

@article{Papadima1985,
abstract = {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).},
author = {Papadima, Stefan},
journal = {Annales de l'institut Fourier},
keywords = {smoothing of complex algebraic isolated singularities; topological smoothing; conical singularities; normal Chern degrees; rational homotopy of Thom complexes},
language = {eng},
number = {3},
pages = {119-135},
publisher = {Association des Annales de l'Institut Fourier},
title = {The rational homotopy of Thom spaces and the smoothing of isolated singularities},
url = {http://eudml.org/doc/74680},
volume = {35},
year = {1985},
}

TY - JOUR
AU - Papadima, Stefan
TI - The rational homotopy of Thom spaces and the smoothing of isolated singularities
JO - Annales de l'institut Fourier
PY - 1985
PB - Association des Annales de l'Institut Fourier
VL - 35
IS - 3
SP - 119
EP - 135
AB - 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).
LA - eng
KW - smoothing of complex algebraic isolated singularities; topological smoothing; conical singularities; normal Chern degrees; rational homotopy of Thom complexes
UR - http://eudml.org/doc/74680
ER -

References

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  1. [1] O. BURLET, Cobordismes de plongements et produits homotopiques, Comm. Math. Helv., 46 (1971), 277-288. Zbl0221.57017MR45 #4433
  2. [2] Y. FELIX, D. TANRE, Sur la formalité des applications, Publ. IRMA, Lille, 3-2 (1981). 
  3. [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. Zbl0483.32007MR82i:32023
  4. [4] R. HARTSHORNE, Topological conditions for smoothing algebraic singularities, Topology, 13 (1974), 241-253. Zbl0288.14006MR50 #2170
  5. [5] R. HARTSHORNE, E. REES, E. THOMAS, Nonsmoothing of algebraic cycles on Grassmann varieties, BAMS, 80(5) (1974), 847-851. Zbl0289.14011MR50 #9870
  6. [6] S. HALPERIN, J.D. STASHEFF, Obstructions to homotopy equivalences, Adv. in Math., 32 (1979), 233-279. Zbl0408.55009MR80j:55016
  7. [7] M.L. LARSEN, On the topology of complex projective manifolds, Inv. Math., 19 (1973), 251-260. Zbl0255.32004MR47 #7058
  8. [8] J. MILNOR, Singular points of complex hypersurfaces, Princeton University Press, 1968. Zbl0184.48405MR39 #969
  9. [9] S. PAPADIMA, The rational homotopy of Thom spaces and the smoothing of homology classes, to appear Comm. Math. Helv. Zbl0592.57025
  10. [10] E. REES, E. THOMAS, Cobordism obstructions to deforming isolated singularities, Math. Ann., 232 (1978), 33-53. Zbl0381.57011MR58 #18475
  11. [11] H. SHIGA, Notes on links of complex isolated singular points, Kodai Math. J., 3 (1980), 44-47. Zbl0442.57015MR81f:57039
  12. [12] A.J. SOMMESE, Non-smoothable varieties, Comm. Math. Helv., 54 (1979), 140-146. Zbl0394.14016MR80k:32027
  13. [13] D. SULLIVAN, Infinitesimal computations in topology, Publ. Math. IHES, 47 (1977), 269-331. Zbl0374.57002MR58 #31119
  14. [14] R. THOM, Quelques propriétés globales des variétés différentiable, Comm. Math. Helv., 28 (1954), 17-86. Zbl0057.15502MR15,890a

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