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.
@article{AIF_2008__58_3_989_0,
author = {Kucharz, Wojciech},
title = {Homology classes of real algebraic sets},
journal = {Annales de l'Institut Fourier},
volume = {58},
year = {2008},
pages = {989-1022},
doi = {10.5802/aif.2376},
zbl = {1153.14035},
mrnumber = {2427517},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2008__58_3_989_0}
}
Kucharz, Wojciech. Homology classes of real algebraic sets. Annales de l'Institut Fourier, Tome 58 (2008) pp. 989-1022. doi : 10.5802/aif.2376. http://gdmltest.u-ga.fr/item/AIF_2008__58_3_989_0/
[1] Algebraic equivalence of real algebraic cycles, Ann. Inst. Fourier, Tome 49 (1999) no. 6, pp. 1797-1804 | Article | Numdam | MR 1738066 | Zbl 0932.14033
[2] Transversal Mappings and Flows, Benjamin Inc., New York (1967) | MR 240836 | Zbl 0171.44404
[3] The topology of real algebraic sets with isolated singularities, Ann. of Math., Tome 113 (1981), pp. 425-446 | Article | MR 621011 | Zbl 0494.57004
[4] The topology of real algebraic sets, Enseign. Math., Tome 29 (1983), pp. 221-261 | MR 719311 | Zbl 0541.14019
[5] Topology of Real Algebraic Sets, Springer, Math. Sci. Research Institute Publ., Tome 25 (1992) | MR 1225577 | Zbl 0808.14045
[6] Transcendental submanifolds of , Comment. Math. Helv., Tome 68 (1993) no. 2, pp. 308-318 | Article | MR 1214234 | Zbl 0806.57017
[7] Transplanting cohomology classes in complex projective space, Amer. J. Math., Tome 92 (1970), pp. 951-967 | Article | MR 287032 | Zbl 0206.50001
[8] Counter examples to representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Math., Tome 53 (1984), pp. 143-151 | Numdam | MR 766294 | Zbl 0547.14019
[9] On real algebraic vector bundles, Bull. Sci. Math., Tome 104 (1980) no. 2, pp. 89-112 | MR 560747 | Zbl 0421.58001
[10] Théorèmes d’approximation en géométrie algébrique réelle, Publ. Math. Univ. Paris VII, Tome 9 (1980), pp. 123-145 | Zbl 0576.14022
[11] Remarks and counterexamples in the theory of real vector bundles and cycles, Springer, Tome 959 (1982), pp. 198-211 | MR 683134 | Zbl 0498.14015
[12] Real Algebraic Geometry, Springer, Berlin Heidelberg New York, Ergebnisse der Math. und ihrer Grenzgeb. Folge (3), Tome 36 (1998) | MR 1659509 | Zbl 0912.14023
[13] Algebraic models of smooth manifolds, Invent. Math., Tome 97 (1989), pp. 585-611 | Article | MR 1005007 | Zbl 0687.14023
[14] Algebraic cycles and approximation theorems in real algebraic geometry, Trans. Amer. Math. Soc., Tome 337 (1993), pp. 463-472 | Article | MR 1091703 | Zbl 0809.57015
[15] Complete intersections in differential topology and analytic geometry, Bollettino U.M.I. (7), Tome 10-B (1996), pp. 1019-1041 | MR 1430164 | Zbl 0904.57013
[16] On homology classes represented by real algebraic varieties, Banach Center Publications, Tome 44 (1998), pp. 21-35 | MR 1677394 | Zbl 0915.14033
[17] La classe d’homologie fondamentále d’un espace analytique, Bull. Soc. Math. France, Tome 89 (1961), pp. 461-513 | Numdam | MR 149503 | Zbl 0102.38502
[18] Differentiable Periodic Maps, Springer, 2nd Edition, Lecture Notes in Math., Tome 738 (1979) | MR 548463 | Zbl 0417.57019
[19] Lectures on Algebraic Topology, Springer, Berlin Heidelberg New York, Grundlehren Math. Wiss., Tome 200 (1972) | MR 415602 | Zbl 0234.55001
[20] An analogue of Max Noether’s theorem, Duke Math. J., Tome 52 (1985) no. 3, pp. 689-706 | Article | Zbl 0589.14034
[21] Intersection Theory, Springer, Berlin Heidelberg New York, Ergebnisse der Math. und ihrer Grenzgeb. Folge (3), Tome 2 (1984) | MR 732620 | Zbl 0541.14005
[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 0229.14007
[23] Algebraic cycles and topology of real algebraic varieties, Centrum voor Wiscunde en informatica, Amsterdam, Dissertation, Vrije Universiteit Amsterdam. CWI Tract. 129, Stichting Mathematisch Centrum (2000) | MR 1824786 | Zbl 0986.14042
[24] Equivalence relations on algebraic cycles and subvarieties of small codimension, Amer. Math. Soc., Tome 29 (1975), pp. 129-164 | MR 369359 | Zbl 0314.14001
[25] Algebraic Geometry, Springer, New York Heidelberg Berlin, Graduate Texts in Math, Tome 52 (1977) | MR 463157 | Zbl 0367.14001
[26] Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math., Tome 79 (1964), pp. 109-326 | Article | MR 199184 | Zbl 0122.38603
[27] Differential Topology, Springer, New York Heidelberg Berlin, Graduate Texts in Math, Tome 33 (1976) | MR 448362 | Zbl 0356.57001
[28] Homotopy Theory, Academic Press, New York (1959) | MR 106454 | Zbl 0088.38803
[29] Algebraic equivalence and homology classes of real algebraic cycles, Math. Nachr., Tome 180 (1996), pp. 135-140 | Article | MR 1397672 | Zbl 0877.14003
[30] Algebraic morphisms into rational real algebraic surfaces, J. Algebraic Geometry, Tome 8 (1999), pp. 569-579 | MR 1689358 | Zbl 0973.14030
[31] Algebraic equivalence of real divisors, Math. Z., Tome 238 (2001), pp. 817-827 | Article | MR 1872575 | Zbl 1078.14537
[32] Algebraic cycles and algebraic models of smooth manifolds, J. Algebraic Geometry, Tome 11 (2002), pp. 101-127 | Article | MR 1865915 | Zbl 1060.14084
[33] Algebraic equivalence of cycles and algebraic models of smooth manifolds, Compositio Math., Tome 140 (2004), pp. 501-510 | Article | MR 2027201 | Zbl 1052.14071
[34] On the topology of complex projective manifolds, Invent. Math., Tome 19 (1973), pp. 251-260 | Article | MR 318511 | Zbl 0255.32004
[35] Characteristic Classes, Princeton Univ. Press, Princeton, New Jersey, Ann. of Math. Studies, Tome 76 (1974) | MR 440554 | Zbl 0298.57008
[36] Real algebraic manifolds, Ann. of Math., Tome 56 (1952) no. 2, pp. 405-421 | Article | MR 50928 | Zbl 0048.38501
[37] Functional Analysis, McGraw-Hill, Inc, New York, Second Edition (1991) | MR 1157815 | Zbl 0867.46001
[38] A bound on the order of on a real algebraic variety, Springer, Géometrie algébrique réelle et formes quadratiques. Lecture Notes in Math., Tome 959 (1982) | MR 683148 | Zbl 0558.14003
[39] Submanifolds of Abelain varieties, Math. Ann., Tome 233 (1978), pp. 229-256 | Article | MR 466647 | Zbl 0381.14007
[40] Algebraic Topology, McGraw-Hill, Inc, New York (1966) | MR 210112 | Zbl 0145.43303
[41] -dimensional manifolds without totally algebraic homology, Proc. Amer. Math. Soc., Tome 123 (1995), pp. 2909-2914 | MR 1264830 | Zbl 0858.57033
[42] Quelques propriétés globales de variétés différentiables, Comment. Math. Helvetici, Tome 28 (1954), pp. 17-86 | Article | MR 61823 | Zbl 0057.15502
[43] Su una congettura di Nash, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat., Tome 27 (1973) no. 3, pp. 167-185 | Numdam | MR 396571 | Zbl 0263.57011
[44] Algebraic approximation of manifolds and spaces, Lecture Notes in Math., Springer, Séminaire Bourbaki 32e année, 1979/1980, no. 548, Tome 842 (1981), pp. 73-94 | Numdam | MR 636518 | Zbl 0456.57012