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

Abstract

This paper generalizes Yu. Manin's approach toward a geometrical interpretation of Arakelov theory at infinity to linear cycles in projective spaces. We show how to interpret certain non-Archimedean Arakelov intersection numbers of linear cycles on ∙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.