Height Pairing on Higher Cycles and Mixed Hodge Structures II
J. I. Burgos Gil, S. Goswami, G. Pearlstein
TL;DR
This work extends the archimedean component of Beilinson’s height pairing to Bloch higher cycles by constructing a mixed Hodge structure attached to a properly intersecting pair of higher cycles and defining two distinct height pairings ht_1 and ht_2 for framed MHS. The construction hinges on forming a generalized biextension via a careful blow-up process that achieves a local-product setting, enabling a Deligne splitting-based height and a Deligne-differential map-based height. Key results include the independence of heights from choices of resolution, the duality ht_i(H^ opp) = -ht_i(H), and the link between these heights and single-valued polylogarithm-valuations, with applications to regulator currents and higher-cycle arithmetic. The framework integrates refined pre-cycles, regulator currents, slowly increasing/rapidly decreasing differential forms, and a robust functorial structure, broadening the scope of archimedean height pairings to higher-algebraic cycles. This provides a rigorous Hodge-theoretic mechanism to study archimedean contributions to Beilinson-type height pairings for higher Chow groups.
Abstract
We attach a mixed Hodge structure and associate two versions of heights to a pair of Bloch higher cycles. Both these heights generalize the biextension height attached to a pair of classical algebraic cycles homologous to zero. We also prove several salient properties of these heights.