One shows that W_mH ^ k (X;\\ mathbb) can actually be defined over\\ mathbb, inducing a mixed Hodge structure on H ^ k (X;\\ mathbb).
One shows that W_mH ^ k (X;\\ mathbb) can actually be defined over\\ mathbb. Thus the filtrations W --, F ^ give rise to a mixed Hodge structure on H ^ k (X;\\ mathbb).