Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T02:47:04.839Z Has data issue: false hasContentIssue false

Complex vector bundles on real algebraic varieties of small dimension

Published online by Cambridge University Press:  17 April 2009

Wojciech Kucharz
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131, United States of America.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be an affine real algebraic variety. In this paper, assuming that dim X ≤ 7 and that X satisfies some other reasonable conditions, we give a characterisation of those continuous complex vector bundles on X which are topologically isomorphic to algebraic complex vector bundles on X.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bochnak, J., Buchner, M. and Kucharz, W., ‘Vector bundles over real algebraic varieties’, (preprint) (University of New Mexico).Google Scholar
[2]Bochnak, J., Coste, M. and Roy, M.-F., Géométrie Algébrique Réelle (Erge. Math. Grenzgeb (3) 12 Springer, 1987).Google Scholar
[3]Bochnak, J. and Kucharz, W., ‘Sur les classes d'homologie représentables par des hypersurfaces algébriques réelles’, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 609611.Google Scholar
[4]Bochnak, J. and Kucharz, W., ‘On homology classes represented by real algebraic hypersurfaces’, (preprint) (University of New Mexico).Google Scholar
[5]Bochnak, J. and Kucharz, W., ‘Morphismes algébriques réelles à valeurs dana S 2k et K-theorie algébrique’, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 545547.Google Scholar
[6]Borel, A. and Haefliger, A., ‘La classe d'homologie fondamental d'un espace analytique’, Bull. Soc. Math. France 89 (1961), 461513.CrossRefGoogle Scholar
[7]Buchner, M. and Kucharz, W., ‘Algebraic vector bundles over real algebraic varieties’, Bull. Amer. Math. Soc. 17 (1987), 279282.CrossRefGoogle Scholar
[8]Fossum, R., ‘Vector bundles over spheres are algebraic’, Invent. Math. 8 (1969), 222225.CrossRefGoogle Scholar
[9]Fulton, W., Intersection Theory 2 (Erge. Math. Grenzgeb (3) Springer, 1984).CrossRefGoogle Scholar
[10]Geramita, A.V. and Roberts, L. G., ‘Algebraic vector bundles on projective spaces’, Invent. Math. 10 (1970), 298304.CrossRefGoogle Scholar
[11]Jouanolou, J.-P., ‘Comparison des K-théories algebrique et topologique de quelque variétés algébrique’, C. R. Acad. Sci. Paris, Sér A, Ser. A 272 (1971), 13731375.Google Scholar
[12]Kucharz, W., ‘Vector bundles over real algebraic surfaces and threefolds’, Compositio Math. 60 (1986), 209225.Google Scholar
[13]Kucharz, W., ‘Topology of real algebraic threefolds’, Duke Math. J. 53 (1986), 10731079.CrossRefGoogle Scholar
[14]Milnor, J. and Stasheff, J., Characteristic Classes (Princeton Univ. Press, Princeton, 1974).CrossRefGoogle Scholar
[15]Peterson, F.P., ‘Some remarks on Chern classes’, Ann. of Math. (2) 60 (1959), 414420.CrossRefGoogle Scholar
[16]Roberts, L.G., ‘Comparison of algebraic and topological K-theory’, Algebraic K-Theory II. Lecture Notes in Math. 342, pp. 7478 (Springer-Verlag, Berlin and New York, 1973).Google Scholar
[17]Swan, R.G., ‘Vector bundles and projective modules’, Trans. Amer. Math. Soc. 105 (1962), 264277.CrossRefGoogle Scholar
[18]Swan, R.G., ‘Topological examples of projective modules’, Trans. Amer. Math. Soc. 230 (1977), 201234.CrossRefGoogle Scholar