On homology classes represented by real algebraic varieties

Jacek Bochnak; Wojciech Kucharz

Banach Center Publications (1998)

  • Volume: 44, Issue: 1, page 21-35
  • ISSN: 0137-6934

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Bochnak, Jacek, and Kucharz, Wojciech. "On homology classes represented by real algebraic varieties." Banach Center Publications 44.1 (1998): 21-35. <http://eudml.org/doc/208886>.

@article{Bochnak1998,
author = {Bochnak, Jacek, Kucharz, Wojciech},
journal = {Banach Center Publications},
keywords = {real algebraic variety; fundamental homology class; algebraic cycles},
language = {eng},
number = {1},
pages = {21-35},
title = {On homology classes represented by real algebraic varieties},
url = {http://eudml.org/doc/208886},
volume = {44},
year = {1998},
}

TY - JOUR
AU - Bochnak, Jacek
AU - Kucharz, Wojciech
TI - On homology classes represented by real algebraic varieties
JO - Banach Center Publications
PY - 1998
VL - 44
IS - 1
SP - 21
EP - 35
LA - eng
KW - real algebraic variety; fundamental homology class; algebraic cycles
UR - http://eudml.org/doc/208886
ER -

References

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  1. [1] R. Benedetti and M. Dedò, Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Math. 53 (1984), 143-151. Zbl0547.14019
  2. [2] R. Benedetti and A. Tognoli, Approximation theorems in real algebraic geometry, Boll. Un. Mat. Ital. Suppl. 1980, no. 2, 209-228. Zbl0465.14011
  3. [3] R. Benedetti and A. Tognoli, On real algebraic vector bundles, Bull. Sci. Math. (2) 104 (1980), 89-112. Zbl0421.58001
  4. [4] R. Benedetti and A. Tognoli, Remarks and counterexamples in the theory of real algebraic vector bundles and cycles, in: Géométrie algébrique réelle et formes quadratiques, Lecture Notes in Math. 959, Springer, Berlin, 1982, 198-211. 
  5. [5] E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207-302. Zbl0896.14006
  6. [6] J. Bochnak, M. Coste and M.-F. Roy, Géométrie algébrique réelle, Ergeb. Math. Grenzgeb. (3) 12, Springer, Berlin, 1987. Zbl0633.14016
  7. [7] J. Bochnak and J. Huisman, When is a complex elliptic curve the product of two real algebraic curves?, Math. Ann. 293 (1992), 469-474. Zbl0734.14018
  8. [8] J. Bochnak and W. Kucharz, Algebraic approximation of mappings into spheres, Michigan Math. J. 34 (1987), 119-125. Zbl0631.14019
  9. [9] J. Bochnak and W. Kucharz, Algebraic models of smooth manifolds, Invent. Math. 97 (1989), 585-611. Zbl0687.14023
  10. [10] J. Bochnak and W. Kucharz, K-theory of real algebraic surfaces and threefolds, Math. Proc. Cambridge Philos. Soc. 106 (1989), 471-480. Zbl0707.14006
  11. [11] J. Bochnak and W. Kucharz, Algebraic cycles and approximation theorems in real algebraic geometry, Trans. Amer. Math. Soc. 337 (1993), 463-472. Zbl0809.57015
  12. [12] J. Bochnak and W. Kucharz, Real algebraic hypersurfaces in complex projective varieties, Math. Ann. 301 (1995), 381-397. Zbl0813.14044
  13. [13] J. Bochnak, W. Kucharz and M. Shiota, The divisor class groups of some rings of global real analytic, Nash or regular functions, in: Géométrie algébrique réelle et formes quadratiques, Lecture Notes in Math. 959, Springer, Berlin, 1982, 218-248. Zbl0519.14019
  14. [14] J. Bochnak, W. Kucharz and M. Shiota, On algebraicity of global real analytic sets and functions, Invent. Math. 70 (1982), 115-156. Zbl0502.32021
  15. [15] A. Borel and A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513. Zbl0102.38502
  16. [16] N. Bourbaki, Algèbre commutative, Hermann, Paris, 1961-1965. 
  17. [17] L. Bröcker, Reelle Divisoren, Arch. Math. (Basel) 35 (1980), 140-143. 
  18. [18] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1984. Zbl0541.14005
  19. [19] J. van Hamel, Real algebraic cycles on complex projective varieties, Math. Z. 225 (1997), 177-198. Zbl0874.14005
  20. [20] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. (2) 79 (1964), 109-203, 205-326. Zbl0122.38603
  21. [21] S. T. Hu, Homotopy Theory, Academic Press, New York, 1959. 
  22. [22] J. Huisman, Real abelian varieties with complex multiplication, Ph.D. Thesis, Vrije Universiteit Amsterdam, 1992. 
  23. [23] J. Huisman, The underlying real algebraic structure of complex elliptic curves, Math. Ann. 294 (1992), 19-35. Zbl0734.14019
  24. [24] J. Huisman, A real algebraic vector bundle is strongly algebraic whenever its total space is affine, in: Real Algebraic Geometry and Topology, Contemp. Math. 182, Amer. Math. Soc., Providence, 1995, 117-119. Zbl0868.14009
  25. [25] F. Ischebeck and H.-W. Schülting, Rational and homological equivalence for real cycles, Invent. Math. 94 (1988), 307-316. Zbl0663.14002
  26. [26] N. Ivanov, Approximation of smooth manifolds by real algebraic sets, Russian Math. Surveys 37 (1982), 1-59. Zbl0571.14013
  27. [27] W. Kucharz, On homology of real algebraic sets, Invent. Math. 82 (1985), 19-25. Zbl0547.14018
  28. [28] W. Kucharz, Algebraic equivalence and homology classes of real algebraic varieties, Math. Nachr. 180 (1996), 135-140. Zbl0877.14003
  29. [29] F. Mangolte, Cycles algébriques sur les surface K3 réelles, Math. Z. 225 (1997), 559-576. Zbl0914.14019
  30. [30] F. Mangolte and J. van Hamel, % Algebraic cycles on real Enriques surfaces, %preprint, 1996. Algebraic cycles and topology of real Enriques surfaces, Compositio Math. 110 (1998), 215-237. Zbl0920.14029
  31. [31] J.-J. Risler, Sur l'homologie des surfaces algébriques réelles, in: Géométrie algébrique réelle et formes quadratiques, Lecture Notes in Math. 959, Springer, Berlin, 1982, 381-385. 
  32. [32] H.-W. Schülting, Algebraische und topologische reelle Zyklen unter birationalen Transformationen, Math. Ann. 272 (1985), 441-448. Zbl0585.14018
  33. [33] M. Shiota, Real algebraic realization of characteristic classes, Publ. Res. Inst. Math. Sci. 18 (1982), 995-1008. Zbl0518.14011
  34. [34] M. Shiota, Equivalence of differentiable functions, rational functions and polynomials, Ann. Inst. Fourier (Grenoble) 32 no. 4 (1982), 167-204. Zbl0466.58006
  35. [35] R. Silhol, A bound on the order of H ( a ) n - 1 ( X , / 2 ) on a real algebraic variety, in: Géométrie algébrique réelle et formes quadratiques, Lecture Notes in Math. 959, Springer, Berlin, 1982, 443-450. 
  36. [36] R. Silhol, Real Algebraic Surfaces, Lecture Notes in Math. 1392, Springer, Berlin, 1989. 
  37. [37] P. Teichner, 6-dimensional manifolds without totally algebraic homology, Proc. Amer. Math. Soc. 123 (1995), 2909-2914. Zbl0858.57033
  38. [38] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. Zbl0057.15502
  39. [39] A. Tognoli, Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 27 (1973), 167-185. 
  40. [40] E. Witt, Zerlegung reeler algebraischer Funktionen in Quadrate, Schiefkörper über reellen Funktionenkörper, J. Reine Angew. Math. 171 (1934), 4-11. Zbl60.0099.01

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