  Our goal is to prove that the Leray spectral sequence associated to a map of
algebraic varieties is motivic in the following sense: If the singular
cohomology groups of the category of quasiprojective varieties defined over a
subfield of C can be canonically endowed with some additional structure, such
as its mixed Hodge structure or its motive (in Nori's sense or as a product of
realizations). Then the Leray spectral sequence for a projective map in this
category is automatically compatible with this structure. As byproduct, we get
a relatively cheap construction of a mixed Hodge structure on the cohomology of
variation of Hodge structure of geometric origin.
