On a class of motivic complexes and the Hermitian geometry of algebraic curves

The extension of the definition of Chow groups to the case of arithmetic schemes requires considering the analytic and hermitian geometry of their "fiber at (arithmetic) infinity," namely the smooth manifold determined by their complex points.

Green functions (currents) are prominently featured in the classical constructions pioneered by Gillet and Soulé, and there is a considerable interest in finding a purely sheaf-theoretic construction.

We review the construction of a class of complexes related to the construction of the archimedean part of these arithmetic Chow groups via cohomological techniques.

In particular, given the Riemann surface determined by an arithmetic surface, we recover a "symbol" map introduced by Deligne in his seminal paper Le déterminant de la Cohomologie, which assigns to two hermitian line bundles a real number built from their Chern forms.