# The holomorphic height pairing

# The holomorphic height pairing

In joint work with Mirel Caibar we show that the Beilinson-Bloch, Gillet-Soulé height pairing between algebraic (n-1)-cycles on a (2n-1)-dimensional complex projective manifold X is the imaginary part of a natural (multivalued) complex quantity that varies holomorphically on components of the Hilbert scheme of X. This pairing is intimately related to the Abel-Jacobi image of the respective cycles. Furthermore this 'holomorphic height pairing' can be extended to a C-infinity pairing on integral currents whose supports are real (2n-2)-dimensional oriented submanifolds of X. As an application in the holomorphic situation, Abel's theorem for Riemann surfaces, suitably interpreted, says that, for zero-cycles algebraically equivalent to zero, Abel-Jacobi equivalence is the same as incidence equivalence. Using the holomorphic height pairing , we propose a proof of Abel's theorem that generalizes to the case of (n-1)-cycles algebraically equivalent to zero on a complex projective (2n-1)-manifold.