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.Comment: 39 page

We show the existence of a weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology to the weight complex of McCrory and Parusiński. If the group is of even order, we can not extract additive invariants directly from the induced spectral sequence.Nevertheless, we construct finite additive invariants in terms of bounded long exact sequences, recovering Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these additive invariants by using globally invariant chains and the equivariant version of Guillén and Navarro Aznar's extension criterion.

RésuméEn utilisant la fonctorialité du complexe de poids de C. McCrory et A. Parusiński -qui induit un analogue de la filtration par le poids pour les variétés algébriques complexes sur l'homologie de Borel-Mooreà coefficients dans Z 2 des variétés algébriques réelles-, on définit un complexe de poids avec action sur les variétés algébriques réelles munies d'une action d'un groupe fini. Mettant l'accent sur le groupeà deuxéléments, onétablit ensuite une version filtrée de la suite courte de Smith pour une involution, tenant compte de la filtration Nash-constructible qui réalise le complexe de poids avec action. Son exactitude est impliquée par le découpage d'une variété Nash munie d'une involution algébrique le long d'un sous-ensemble symétrique par arcs.

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef-Loeser formulae for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence. IntroductionA crucial issue in the study of real analytic germs is the choice of a good equivalence relation by which we can distinguish them. One may think about C r -equivalence, r = 0, 1, . . . , ∞, ω. However, the topological equivalence seems, unlike the complex case, not fine enough : for example, all the germs of the form x 2m +y 2n are topologically equivalent. On the other hand, the C 1 -equivalence has already moduli : consider the Whitney family f t (x, y) = xy(y − x)(y − tx), t > 1, then f t and f t ′ are C 1 -equivalent if and only if t = t ′ . In [15], T.-C. Kuo proposed an equivalence relation for real analytic germs named the blow-analytic equivalence for which, in particular, analytically parametrized family of isolated singularities have a locally finite classification. Roughly speaking, two real analytic germs are said blow-analytically equivalent if they become analytically equivalent after composition with real modifications (e.g., finite successions of blowings-up along smooth centers). With respect to this equivalence relation, Whitney family has only one equivalence class. Slightly stronger versions of blow-analytic equivalence have been proposed so far, by S. Koike and A. Parusiński in [13] and T. Fukui and L. Paunescu in [8] for example. An important feature of blow-analytic equivalence is also that we have invariants for this equivalence relation, like the Fukui invariants ([7]) and the zeta functions ([13]) inspired by the motivic zeta functions of J. Denef and F. Loeser ([5]) using the Euler characteristic with compact supports as a motivic measure.The present paper is interested in the study of Nash germs, that is real analytic germs with semialgebraic graph. In [10], G. Fichou defined an analog adapted to Nash germs of the