The projectors of the decomposition theorem are motivic

We prove that the projectors arising from the decomposition theorem applied to a projective map of quasi projective varieties are absolute Hodge, Andr\'e motivated, Tate and Ogus classes. As a by-product, we introduce, in characteristic zero, the notions of algebraic de Rham intersection cohomology groups of a quasi projective variety and of intersection cohomology motive of a projective variety.