Cohomology and products of real weight filtrations

Limoges, Thierry; Priziac, Fabien

Annales de l'Institut Fourier, Tome 65 (2015) no. 5, pp. 2235-2271.

On associe à toute variété algébrique réelle un complexe de cochaînes filtré, qui calcule la cohomologie à supports compacts et à coefficients dans ${\mathbb{Z}}_2$ 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.

We associate to each algebraic variety defined over $ℝ$ a filtered cochain complex, which computes the cohomology with compact supports and ${\mathbb{Z}}_2$-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.

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