Cohomology and products of real weight filtrations
Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2235-2271.

We associate to each algebraic variety defined over a filtered cochain complex, which computes the cohomology with compact supports and 2 -coefficients of the set of its real points. This filtered complex is additive for closed inclusions and acyclic for resolution of singularities, and is unique up to filtered quasi-isomorphism. It is represented by the dual filtration of the geometric filtration on semialgebraic chains with closed supports defined by McCrory and Parusiński, and induces a spectral sequence which computes the weight filtration on cohomology with compact supports. This spectral sequence is a natural invariant which contains the virtual Betti numbers.

We then show that the cross product of two varieties allows us to compare the product of their respective weight complexes and spectral sequences with those of their product, and prove that the cup and cap products are filtered with respect to the real weight filtrations.

On associe à toute variété algébrique réelle un complexe de cochaînes filtré, qui calcule la cohomologie à supports compacts et à coefficients dans 2 de l’ensemble de ses points réels. Unique à quasi-isomorphisme filtré près, il est additif pour les inclusions fermées et acyclique pour la résolution des singularités, est représenté par la filtration duale de la filtration géométrique sur les chaînes semi-algébriques à supports fermés définie par McCrory et Parusiński, et induit une suite spectrale calculant la filtration par le poids sur la cohomologie à supports compacts. Cette suite spectrale est un invariant naturel qui contient les nombres de Betti virtuels.

On montre ensuite que le produit de deux variétés nous permet de comparer le produit des complexes et suites spectrales de poids avec ceux du produit, et on prouve que les produits cup et cap sont filtrés par rapport aux filtrations par le poids réelles.

DOI: 10.5802/aif.2987
Classification: 14P25, 14P10, 55U25
Keywords: real algebraic varieties, weight filtrations, cohomology with compact supports, invariants, cross product, cup and cap products, Poincaré duality.
Mot clés : variétés algébriques réelles, filtrations par le poids, cohomologie à supports compacts, invariants, produit de variétés, produits cup et cap, dualité de Poincaré
Limoges, Thierry 1; Priziac, Fabien 2

1 Laboratoire Jean-Alexandre Dieudonné Université de Nice - Sophia Antipolis 06108 Nice Cedex 02 (France)
2 Institut de Mathématiques de Marseille (UMR 7373 du CNRS) Aix-Marseille Université 39, rue F. Joliot Curie 13453 Marseille Cedex 13 (France)
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Limoges, Thierry; Priziac, Fabien. Cohomology and products of real weight filtrations. Annales de l'Institut Fourier, Volume 65 (2015) no. 5, pp. 2235-2271. doi : 10.5802/aif.2987. http://www.numdam.org/articles/10.5802/aif.2987/

[1] Coste, M.; Mahé, L.; Roy, M.-F. Real algebraic geometry, Lecture Notes in Mathematics, 1524 (1992), pp. viii+418 | DOI | MR

[2] Deligne, Pierre Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971) no. 40, pp. 5-57 | EuDML | Numdam | MR | Zbl

[3] Deligne, Pierre Poids dans la cohomologie des variétés algébriques, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que. (1975), pp. 79-85 | MR | Zbl

[4] Fichou, Goulwen Motivic invariants of arc-symmetric sets and blow-Nash equivalence, Compos. Math., Volume 141 (2005) no. 3, pp. 655-688 | DOI | MR | Zbl

[5] Greenberg, Marvin J. Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967, pp. x+235 | MR | Zbl

[6] Guillén, F.; Navarro Aznar, V.; Pascual Gainza, P.; Puerta, F. Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, 1335, Springer-Verlag, Berlin, 1988, pp. xii+192 (Papers from the Seminar on Hodge-Deligne Theory held in Barcelona, 1982) | MR | Zbl

[7] Guillén, Francisco; Navarro Aznar, Vicente Un critère d’extension des foncteurs définis sur les schémas lisses, Publ. Math. Inst. Hautes Études Sci. (2002) no. 95, pp. 1-91 | DOI | EuDML | Numdam | MR | Zbl

[8] Kurdyka, Krzysztof Ensembles semi-algébriques symétriques par arcs, Math. Ann., Volume 282 (1988) no. 3, pp. 445-462 | DOI | EuDML | MR | Zbl

[9] Kurdyka, Krzysztof; Parusiński, Adam Arc-symmetric sets and arc-analytic mappings, Arc spaces and additive invariants in real algebraic and analytic geometry (Panor. Synthèses), Volume 24, Soc. Math. France, Paris, 2007, pp. 33-67 | Zbl

[10] McCrory, Clint; Parusiński, Adam Algebraically constructible functions, Ann. Sci. École Norm. Sup. (4), Volume 30 (1997) no. 4, pp. 527-552 | DOI | Numdam | Zbl

[11] McCrory, Clint; Parusiński, Adam Virtual Betti numbers of real algebraic varieties, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 9, pp. 763-768 | DOI | Zbl

[12] McCrory, Clint; Parusiński, Adam Algebraically constructible functions: real algebra and topology, Arc spaces and additive invariants in real algebraic and analytic geometry (Panor. Synthèses), Volume 24, Soc. Math. France, Paris, 2007, pp. 69-85 | Zbl

[13] McCrory, Clint; Parusiński, Adam The weight filtration for real algebraic varieties, Topology of stratified spaces (Math. Sci. Res. Inst. Publ.), Volume 58, Cambridge Univ. Press, Cambridge, 2011, pp. 121-160 | Zbl

[14] McCrory, Clint; Parusiński, Adam The weight filtration for real algebraic varieties II: classical homology, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, Volume 108 (2014) no. 1, pp. 63-94 | DOI

[15] Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 52, Springer-Verlag, Berlin, 2008, pp. xiv+470 | Zbl

[16] Spanier, Edwin H. Algebraic topology, Springer-Verlag, New York, 1966, pp. xvi+528 (Corrected reprint of the 1966 original) | Zbl

[17] Totaro, B. Topology of singular algebraic varieties, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing (2002), pp. 533-541 | Zbl

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