Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T20:07:04.074Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPLEX CYCLES AS OBSTRUCTIONS ON REAL ALGEBRAIC VARIETIES

Published online by Cambridge University Press:  19 December 2014

WOJCIECH KUCHARZ
WOJCIECH KUCHARZ*
Affiliation:
Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Profesora Łojasiewicza 6, 30-348 Kraków, Poland E-mail: Wojciech.Kucharz@im.uj.edu.pl
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 Y be a compact nonsingular real algebraic variety of positive dimension. Then one can find a compact connected nonsingular real algebraic variety X, which admits a continuous map into Y that is not homotopic to any regular map. It is hard to determine the minimum dimension of such a variety X. In this paper, new upper bounds for dim X are obtained. The main role in the constructions is played by complex algebraic cycles on Y.

MSC classification

Primary: 14P05: Real algebraic sets
Secondary: 14P25: Topology of real algebraic varieties
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 2 , May 2015 , pp. 343 - 347
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Benedetti, R. and Tognoli, A., Remarks and counterexamples in the theory of real vector bundles and cycles, in Géométrie algebraique réelle et formes quadratiques, Lecture Notes in Mathematics, vol. 959 (Springer, 1982), 198–211.Google Scholar
2.Bochnak, J., Buchner, M. and Kucharz, W., Vector bundles over real algebraic varieties, K-Theory 3 (1989), 271–298.CrossRefGoogle Scholar
3.Bochnak, J., Buchner, M. and Kucharz, W., Vector bundles over real algebraic varieties, Erratum: K-Theory 4 (1990), 103.Google Scholar
4.Bochnak, J., Coste, M. and Roy, M.-F., Real algebraic geometry, Ergeb. Math. Grenzgeb., vol. 36 (Berlin, Springer, 1998).Google Scholar
5.Bochnak, J. and Kucharz, W., On real algebraic morphisms into even-dimensional spheres Ann. Math. 128 (1988), 415–433.Google Scholar
6.Bochnak, J. and Kucharz, W., On homology classes represented by real algebraic varieties, in Singularities Symposium – Łojasiewicz 70, Banach Center Publ. 44, Inst. Math. Polish Acad. Sci. 21–35 (1998).Google Scholar
7.Bochnak, J. and Kucharz, W., Real algebraic morphisms represent few homotopy classes Math. Ann. 337 (2007), 909–921.CrossRefGoogle Scholar
8.Borel, A. and Haefliger, A., La classe d'homologie fondamentale d'un espace analytique Bull. Soc. Math. France 89 (1961), 461–513.Google Scholar
9.Fulton, W., Intersection theory, Ergeb. Math. Grenzgeb., vol. 2 (Berlin, Springer), 1984.Google Scholar
10.Ghiloni, R., Second order homological obstructions on real algebraic manifolds Topol. Appl. 154 (2007), 3090–3094.Google Scholar
11.Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero Ann. Math. 79 (1964), 109–326.Google Scholar
12.Kollár, J., Lectures on resolutuon of singularities, Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, NJ, 2007).Google Scholar
13.Kucharz, W., Algebraic cycles and realification of complex projecive varieties Geom. Dedicata 54 (1995), 317–322.Google Scholar
14.Thom, R., Quelques propriétiés globales des variétés différentiables Comment. Math. Helv. 28 (1954), 17–86.Google Scholar
15.Wall, C. T. C., Determinations of the cobordism ring Ann. Math. 72 (1960), 292–311.Google Scholar