| Résume | Since the foundational work of Néron and Tate in the 60s, various conjectural definitions of the height pairing for homologically trivial algebraic cycles on smooth, projective varieties over a number field, or more general global fields, have been proposed by Belinson, Bloch, Gillet-Soulé and Schneider, among others. The related concept of height functions on sub-varieties was extensively developed by Faltings and, in the early 2000s, extended by Moriwaki to varieties defined over finitely generated extensions of $\mathbb{Q}$, also called arithmetic function fields. More recently, Bruno Kahn has used algebraic intersection theory to give a geometric construction of a ``refined" height pairing over the function field of a smooth, projective variety $B$, taking its values in $\textrm{Pic}(B)$.
In this talk, I will present an Arakelov-geometric analogue of this construction over an arithmetic function field $K$, taking its values in the arithmetic Picard group of a normal arithmetic variety whose function field is $K$. The construction is in two parts: first, Kahn's definitions are extended and adapted to arithmetic varieties using the intersection theory on regular schemes of Gillet-Soulé, and then the Archimedean term is added by considering generalized Green currents satisfying some ``admissibility" conditions, inspired by the original work of Arakelov. If time allows, I will discuss the connection with Moriwaki's work and with the work of Brosnan and Pearlstein on the degeneration of the Archimedean height pairing. |
