Weights in cohomology and the Eilenberg-Moore spectral sequence

We show that in the category of complex algebraic varieties, the Eilenberg–Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all spaces involved have pure cohomology. As application, we compute the rational cohomology of an algebraic G-variety X (G being a connected algebraic group) in terms of its equivariant cohomology provided that H∗ G(X) is pure. This is the case, for example, if X is smooth and has only finitely many orbits. We work in the category of mixed sheaves; therefore our results apply equally to (equivariant) intersection homology.