%0 Journal Article
%T Cohomology and products of real weight filtrations
%+ Laboratoire Jean Alexandre Dieudonné (LJAD)
%+ Institut de Mathématiques de Marseille (I2M)
%A Limoges, Thierry
%A Priziac, Fabien
%Z 39 pages
%< avec comité de lecture
%@ 0373-0956
%J Annales de l'Institut Fourier
%I Association des Annales de l'Institut Fourier
%V 65
%N 5
%P 2235-2271
%8 2015-12-03
%D 2015
%Z 1403.0706
%R 10.5802/aif.2987
%K real algebraic varieties
%K weight filtrations
%K cohomology with compact supports
%K invariants
%K cross product
%K cup and cap products
%K Poincaré duality
%Z 2010 Mathematics Subject Classification : 14P25, 14P10, 55U25
%Z Mathematics [math]/Algebraic Geometry [math.AG]Journal articles
%X We associate to each algebraic variety defined over $\mathbb{R}$ 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 Parusi\'nski, and leads to a spectral sequence which computes the weight filtration on cohomology with compact supports. This spectral sequence is a natural invariant which contains the additive 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 in cohomology and homology are filtered with respect to the real weight filtrations.
%G English
%2 https://hal.science/hal-00955156v2/document
%2 https://hal.science/hal-00955156v2/file/coprodrwfil-tlimoges_fpriziac.pdf
%L hal-00955156
%U https://hal.science/hal-00955156
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