 This suggests that "H"1("X") admits an increasing filtration

whose successive quotients "Wn"/"W""n"−1 originate from the cohomology of smooth complete varieties, hence admit (pure) Hodge structures, albeit of different weights.Further examples can be found in "A Naive Guide to Mixed Hodge Theory".A mixed Hodge structure on an abelian group formula_1 consists of a finite decreasing filtration "Fp" on the complex vector space "H" (the complexification of formula_1), called the Hodge filtration and a finite increasing filtration "Wi" on the rational vector space formula_35 (obtained by extending the scalars to rational numbers), called the weight filtration, subject to the requirement that the "n"-th associated graded quotient of formula_36 with respect to the weight filtration, together with the filtration induced by "F" on its complexification, is a pure Hodge structure of weight "n", for all integer "n".