Height Pairing on Higher Cycles and Mixed Hodge Structures II

Abstract

To a pair of Bloch higher cycles that intersect properly and have complementary codimensions, we attach a mixed Hodge structure with some extra data (a framed mixed Hodge structure). Using this mixed Hodge structure, we define two different archimedean local height pairings. Both constructions generalize the biextension archimedean height attached to a pair of classical algebraic cycles homologous to zero. When applied to the polylogarithm variation of mixed Hodge structures, we recover both the single-valued polylogarithm of Bloch, Wigner et al. and the one defined by Brown. We also prove several salient properties of these heights, including various vanishing results.