Height Pairing on Higher Cycles and Mixed Hodge Structures II

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.