Mixed Hodge structure in cyclic homology and algebraic K-theory. I (Q1106507)

The author studies mixed Hodge structures on multigraded chain complexes (up to quasi-isomorphism). He shows that, if X is a simply-connected quasi-projective complex variety, the reduced Waldhausen K-theory, \(\tilde A_*(X)\), possesses a natural mixed Hodge structure. When X is not simply-connected he constructs a mixed Hodge structure on a quotient of \(\tilde A_*(X)\) by powers of the augmentation ideal in \({\mathbb{Z}}[\pi_ 1(X)]\). Similar results are obtained for the Hochschild and cyclic homology of X. The results are derived from a paper by \textit{R. M. Hain} [Bull. Am. Math. Soc., New Ser. 14, 111-114 (1986; Zbl 0597.55009)] where a mixed Hodge structure is constructed on the homotopy Lie algebra of X.