Arakelov intersection indices of linear cycles and the geometry of buildings and symmetric spaces

Abstract

This paper generalizes Manin's approach towards a geometrical interpretation of Arakelov theory at infinity to linear cycles on projective spaces. We show how to interpret certain non-Archimedean Arakelov intersection numbers of linear cycles on $P^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to PGL(n). This geometric setting has an Archimedean analogue, namely the Riemannian symmetric space associated to SL(n,C), which we use to interpret analogous Archimedean intersection numbers of linear cycles in a similar way.