Height pairing on higher cycles and mixed Hodge structures

For a smooth, projective complex variety, we introduce several mixed Hodge structures associated to higher algebraic cycles. Most notably, we introduce a mixed Hodge structure for a pair of higher cycles which are in the refined normalized complex and intersect properly. In a special case, this mixed Hodge structure is an oriented biextension, and its height agrees with the higher archimedean height pairing introduced in a previous paper by the first two authors. We also compute a non-trivial example of this height given by Bloch–Wigner dilogarithm function. Finally, we study the variation of mixed Hodge structures of Hodge–Tate type, and show that the height extends continuously to degenerate situations.