Il existe un vaste programme de recherche portant sur la comparaison entre catégories topologiques et algébriques, dont l’origine remonte à 1952 avec les travaux célèbres de J. Nash sur les variétés algébriques réelles lisses. Ce papier est une contribution à ce programme. Il contient l’étude des classes d’homologie et de cohomologie représentées par des ensembles algébriques réels. En particulier, de telles classes sont étudiées dans les modèles algébriques de variétés lisses.
There is a large research program focused on comparison between algebraic and topological categories, whose origins go back to 1952 and the celebrated work of J. Nash on real algebraic manifolds. The present paper is a contribution to this program. It investigates the homology and cohomology classes represented by real algebraic sets. In particular, such classes are studied on algebraic models of smooth manifolds.
Keywords: Real algebraic variety, algebraic cycles, cohomology
Mot clés : Variété algébrique réelle, cycles algébriques, cohomologie
@article{AIF_2008__58_3_989_0, author = {Kucharz, Wojciech}, title = {Homology classes of real algebraic sets}, journal = {Annales de l'Institut Fourier}, pages = {989--1022}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {3}, year = {2008}, doi = {10.5802/aif.2376}, zbl = {1153.14035}, mrnumber = {2427517}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2376/} }
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Kucharz, Wojciech. Homology classes of real algebraic sets. Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 989-1022. doi : 10.5802/aif.2376. http://archive.numdam.org/articles/10.5802/aif.2376/
[1] Algebraic equivalence of real algebraic cycles, Ann. Inst. Fourier, Volume 49 (1999) no. 6, pp. 1797-1804 | DOI | Numdam | MR | Zbl
[2] Transversal Mappings and Flows, Benjamin Inc., New York, 1967 | MR | Zbl
[3] The topology of real algebraic sets with isolated singularities, Ann. of Math., Volume 113 (1981), pp. 425-446 | DOI | MR | Zbl
[4] The topology of real algebraic sets, Enseign. Math., Volume 29 (1983), pp. 221-261 | MR | Zbl
[5] Topology of Real Algebraic Sets, Math. Sci. Research Institute Publ., 25, Springer, 1992 | MR | Zbl
[6] Transcendental submanifolds of , Comment. Math. Helv., Volume 68 (1993) no. 2, pp. 308-318 | DOI | MR | Zbl
[7] Transplanting cohomology classes in complex projective space, Amer. J. Math., Volume 92 (1970), pp. 951-967 | DOI | MR | Zbl
[8] Counter examples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Math., Volume 53 (1984), pp. 143-151 | Numdam | MR | Zbl
[9] On real algebraic vector bundles, Bull. Sci. Math., Volume 104 (1980) no. 2, pp. 89-112 | MR | Zbl
[10] Théorèmes d’approximation en géométrie algébrique réelle, Publ. Math. Univ. Paris VII, Volume 9 (1980), pp. 123-145 | Zbl
[11] Remarks and counterexamples in the theory of real vector bundles and cycles, Springer, Volume 959 (1982), pp. 198-211 | MR | Zbl
[12] Real Algebraic Geometry, Ergebnisse der Math. und ihrer Grenzgeb. Folge (3), 36, Springer, Berlin Heidelberg New York, 1998 | MR | Zbl
[13] Algebraic models of smooth manifolds, Invent. Math., Volume 97 (1989), pp. 585-611 | DOI | MR | Zbl
[14] Algebraic cycles and approximation theorems in real algebraic geometry, Trans. Amer. Math. Soc., Volume 337 (1993), pp. 463-472 | DOI | MR | Zbl
[15] Complete intersections in differential topology and analytic geometry, Bollettino U.M.I. (7), Volume 10-B (1996), pp. 1019-1041 | MR | Zbl
[16] On homology classes represented by real algebraic varieties, Banach Center Publications, Volume 44 (1998), pp. 21-35 | EuDML | MR | Zbl
[17] La classe d’homologie fondamentále d’un espace analytique, Bull. Soc. Math. France, Volume 89 (1961), pp. 461-513 | EuDML | Numdam | MR | Zbl
[18] Differentiable Periodic Maps, 2nd Edition, Lecture Notes in Math., 738, Springer, 1979 | MR | Zbl
[19] Lectures on Algebraic Topology, Grundlehren Math. Wiss., 200, Springer, Berlin Heidelberg New York, 1972 | MR | Zbl
[20] An analogue of Max Noether’s theorem, Duke Math. J., Volume 52 (1985) no. 3, pp. 689-706 | DOI | Zbl
[21] Intersection Theory, Ergebnisse der Math. und ihrer Grenzgeb. Folge (3), 2, Springer, Berlin Heidelberg New York, 1984 | MR | Zbl
[22] Technique de descente et théorèmes d’existence en géométrie algebrique, I - VI, Séminaire Bourbaki (1959-1962), pp. 190, 195, 212, 221, 232, 236 Ergebnisse der Math. und ihrer Grenzgeb. Folge (3) | Numdam | Zbl
[23] Algebraic cycles and topology of real algebraic varieties, Dissertation, Vrije Universiteit Amsterdam. CWI Tract. 129, Stichting Mathematisch Centrum, Centrum voor Wiscunde en informatica, Amsterdam, 2000 | MR | Zbl
[24] Equivalence relations on algebraic cycles and subvarieties of small codimension, Amer. Math. Soc., Volume 29 (1975), pp. 129-164 | MR | Zbl
[25] Algebraic Geometry, Graduate Texts in Math, 52, Springer, New York Heidelberg Berlin, 1977 | MR | Zbl
[26] Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., Volume 79 (1964), pp. 109-326 | DOI | MR | Zbl
[27] Differential Topology, Graduate Texts in Math, 33, Springer, New York Heidelberg Berlin, 1976 | MR | Zbl
[28] Homotopy Theory, Academic Press, New York, 1959 | MR | Zbl
[29] Algebraic equivalence and homology classes of real algebraic cycles, Math. Nachr., Volume 180 (1996), pp. 135-140 | DOI | MR | Zbl
[30] Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geometry, Volume 8 (1999), pp. 569-579 | MR | Zbl
[31] Algebraic equivalence of real divisors, Math. Z., Volume 238 (2001), pp. 817-827 | DOI | MR | Zbl
[32] Algebraic cycles and algebraic models of smooth manifolds, J. Algebraic Geometry, Volume 11 (2002), pp. 101-127 | DOI | MR | Zbl
[33] Algebraic equivalence of cycles and algebraic models of smooth manifolds, Compositio Math., Volume 140 (2004), pp. 501-510 | DOI | MR | Zbl
[34] On the topology of complex projective manifolds, Invent. Math., Volume 19 (1973), pp. 251-260 | DOI | MR | Zbl
[35] Characteristic Classes, Ann. of Math. Studies, 76, Princeton Univ. Press, Princeton, New Jersey, 1974 | MR | Zbl
[36] Real algebraic manifolds, Ann. of Math., Volume 56 (1952) no. 2, pp. 405-421 | DOI | MR | Zbl
[37] Functional Analysis, Second Edition, McGraw-Hill, Inc, New York, 1991 | MR | Zbl
[38] A bound on the order of on a real algebraic variety, Géometrie algébrique réelle et formes quadratiques. Lecture Notes in Math., 959, Springer, 1982 | MR | Zbl
[39] Submanifolds of Abelain varieties, Math. Ann., Volume 233 (1978), pp. 229-256 | DOI | MR | Zbl
[40] Algebraic Topology, McGraw-Hill, Inc, New York, 1966 | MR | Zbl
[41] -dimensional manifolds without totally algebraic homology, Proc. Amer. Math. Soc., Volume 123 (1995), pp. 2909-2914 | MR | Zbl
[42] Quelques propriétés globales de variétés différentiables, Comment. Math. Helvetici, Volume 28 (1954), pp. 17-86 | DOI | MR | Zbl
[43] Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat., Volume 27 (1973) no. 3, pp. 167-185 | Numdam | MR | Zbl
[44] Algebraic approximation of manifolds and spaces, Lecture Notes in Math., Volume 842, Springer, Séminaire Bourbaki 32e année, 1979/1980, no. 548 (1981), pp. 73-94 | Numdam | MR | Zbl
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