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_53 (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_54 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". Here the induced filtration on