Skip to main content
SpringerLink
Log in

Algebraic geometric methods in real algebraic geometry

  • Survey Papers
  • Conference paper
  • First Online:
Real Algebraic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1524))

  • 877 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Akbulut, H. King, A relative Nash theorem, Trans. Amer. Math. Soc. 267 (2), 465–481 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  2. J.Kr. Arason, R. Elman, B. Jacob, The graded Witt ring and Galois cohomology, CMS.Conf. Proc. Vol.4, Providence: AMS, 1984, 17–50.

    MATH  Google Scholar 

  3. G. Ayoub, Le groupe de Witt d'une surface réelle. Comment. Math. Helv. 62, 74–105 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Bass, Unitary algebraic K-theory, Hermitian K-theory and Geometric Applications, Battelle Institute Conference 1972. Algebraic K-theory III, SLN 343, 1973, pp. 57–265.

    Google Scholar 

  5. J. Barge, M. Ojanguren. Fibrés algébriques sur une surface réelle. Comment. Math. Helv. 62, 616–629 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Benedetti, M. Dedo, Counterexamples to representing homology classes by real algebraic subvarities up to homeomorphism. Compositio Math. 53, 143–151 (1984).

    MathSciNet  MATH  Google Scholar 

  7. R. Benedetti, A. Tognoli, Sur les fibrés vectoriels algébriques réels. C.R. Acad. Sc. Paris, t. 287, Série A-831-833 (1978).

    Google Scholar 

  8. R. Benedetti, A. Tognoli, Remarks and counterexamples in the theory of real algebraic vector bundles and cycles, Géométrie algébrique réelle et formes quadratiques, Rennes 1981, SLN 959, pp. 198–211.

    Google Scholar 

  9. J. Bochnak, Topology of real analytic sets-Some open problems. Rennes 1981, SLN 959, pp. 212–217.

    Google Scholar 

  10. J. Bochnak, M. Buchner and W. Kucharz, Vector bundles on real algebraic varieties, K-theory, 3, 271–298 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Bochnak, M. Coste, M-F.Roy, Géométrie algébrique réelle, Ergeb. der Math. 3. Folge, Bd. 12, Springer 1987.

    Google Scholar 

  12. J. Bochnak, W. Kucharz, Representation of homotopy classes by algebraic mappings, J. Reine. Angew. Math. 377, 159–169 (1987).

    MathSciNet  MATH  Google Scholar 

  13. J. Bochnak, W. Kucharz, On real algebraic morphisms into even dimensional spheres, Ann. of Math. 128, 415–433 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Bochnak, W. Kurcharz, K-theory of real algebraic surfaces and threefolds, Math. Proc. Camb. Phil. Soc. 106, 471–480 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Bochnak, W. Kucharz, Algebraic models of smooth manifolds, Invent. Math. 588–611 (1989).

    Google Scholar 

  16. A. Borel, A. Haefliger, La classe d'homologie fondamentale d'un espace analytique. Bull. Soc. Math. France, 89, 461–513 (1961).

    MathSciNet  MATH  Google Scholar 

  17. L. Bröcker, Reelle Divisoren, Arch. Math. 35, 140–143 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Bröcker, Minimale Erzeugung von Positivbereich. Geom. Dedication 16. 335–350 (1984).

    MATH  Google Scholar 

  19. L. Bröcker, Spaces of orderings and semialgebraic sets, Quadratic and hermitian forms, CMS Conf. Proc. Vol. 4, Providence: AMS, 1984, pp. 231–248.

    Google Scholar 

  20. G.W. Brumfiel, Witt rings and K-theory. Rocky Mountain J. of Math. 14, 733–765 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  21. J.-L. Colliot-Thélène, Abstract of a talk at Oberwolfach, 1990.

    Google Scholar 

  22. J.-L. Colliot-Thélène, Ischebeck, L'équivalence rationnelle sur les cycles de dimension zéro des variétés algébriques réelles. C.R. Acad. Sci. Paris, Serie I, 292, 723–725 (1981).

    MATH  Google Scholar 

  23. J.-L. Colliot-Thélène, R. Parimala, Real components of algebraic varieties and étale cohomology, Invent. Math. 101, 81–99 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  24. J.-L. Colliot-Thélène, J.-J. Sansuc, Fibrés quadratiques et composantes connexes réelles, Math. Ann. 244, 105–134 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  25. D.A. Cox, The étale homotopy type of varieties over ℝ, Proc. Amer. Math. Soc. 76, 17–22 (1979).

    MathSciNet  MATH  Google Scholar 

  26. H. Delfs, M. Knebusch, Semialgebraic topology over a real closed field II: Basic theory of semialgebraic spaces. Math. Z. 178, 175–213 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Fossum, Vector bundles over spheres are algebraic. Invent. Math. 8, 222–225 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  28. A.V. Geramita, L.G. Roberts, Algebraic vector bundles on projective space. Invent. Math. 10, 298–304 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  29. W.D. Geyer, Ein algebraischer Beweis des Satzes von Weichold über reelle algebraische Funktionenkörper. In: Algebraische Zahlentheorie, Oberwolfach, 1964.

    Google Scholar 

  30. W.D. Geyer, Dualität bei abelschen Varietäten über reelle abgeschlossen Körpern, J. reine. angew. Math. 293–294, 62–66 (1977).

    MathSciNet  MATH  Google Scholar 

  31. J. Houdebine, L. Mahé, Séparation des composantes connexs réelles dans le cas des varietés projectives, Géométrie algébrique réelle et formes quadratiques, Rennes 1981, SLN 959, p. 358–370.

    Google Scholar 

  32. F. Ischebeck, H-W. Schülting, Rational and homological equivalence for real cycles, Invent. Math. 94, 307–316 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Knebusch, On algebraic curves over real closed fields I. Math. Z. 150, 49–70 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Knebusch, On algebraic curves over real closed fields II. Math. Z. 151, 189–205 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Knebusch, Some open problems, Queen's Papers in Pure and Appl. Math. 46, 361–370 (1977).

    MathSciNet  MATH  Google Scholar 

  36. L. Mahé, Signatures et componentes connexes. Math. Ann. 260, 191–210 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  37. L. Mahé, Théorème de Pfister pour les variétés et anneaux de Witt réduits, Invent. Math. 85, 53–72 (1986).

    Article  MathSciNet  Google Scholar 

  38. M. Ojanguren, R. Parimala and R. Sridharan, Symplectic bundles over affine surfaces, Comment. Math. Helv. 61, 491–500 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  39. W. Pardon, A relation between Witt groups and zero cycles in a regular ring, Algebraic K-theory, Geometry and Analysis, SLN 1046, 1984, pp. 261–328.

    Google Scholar 

  40. R. Parimala, Witt groups of affine threefolds, Duke Math. J. 57, 947–954 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  41. C. Scheiderer, A remark on the paper, Real components of algebraic varieties and étale cohomology by J.-L. Colliot-Thélène, R. Parimala, Preprint, 1990.

    Google Scholar 

  42. C. Scheiderer, Real and étale cohomology, Preprint, 1991.

    Google Scholar 

  43. H.W. Schülting, Algebraische und topologische reelle Zykeln unter birationalen Transformationen, Math. Ann. 272, 441–448 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Silhol, Cohomologie de Galois et cohomologie des variétés algébriques réelles; Applications aux surfaces rationnelles, Bull. Soc. Math. France 115, 107–125 (1987).

    MathSciNet  MATH  Google Scholar 

  45. R. Sujatha, Witt group of real projective surfaces, Math. Ann. 288, 89–101 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Swan, Vector bundles and projective modules. Trans. Amer. Math. Soc. 105, 264–277 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Swan, Topological examples of projective modules, Trans. Amer. Math. Soc. 230, 201–234 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  48. R. Swan, K-theory of quadric hypersurfaces, Ann. of Math. 122, 113–154 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Tognoli, Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa 27, 167–185 (1973).

    MathSciNet  MATH  Google Scholar 

  50. E. Witt, Zerlegung reeller algebraischer Funktionen in Quadrate, Schiefkörper über reelle Funktionenkörper. J. Reine Angew. Math. 171, 4–11 (1934).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Tata Institute of Fundamental Research, 400005, Bombay, India

    R. Parimala

Authors
  1. R. Parimala
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Michel Coste Louis Mahé Marie-Françoise Roy

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Parimala, R. (1992). Algebraic geometric methods in real algebraic geometry. In: Coste, M., Mahé, L., Roy, MF. (eds) Real Algebraic Geometry. Lecture Notes in Mathematics, vol 1524. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084607

Download citation

  • DOI: https://doi.org/10.1007/BFb0084607

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55992-4

  • Online ISBN: 978-3-540-47337-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation