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We study the Arakelov intersection theory of the arithmetic scheme OG which parametrizes maximal isotropic subspaces in an even dimensional vector space, equipped with the standard hyperbolic quadratic form. We give a presentation of the Arakelov Chow ring of OG (when OG(C) is given its natural invariant hermitian metric) and formulate an `arithmetic Schubert calculus' which extends the classical one for the cohomology ring of OG. Our analysis leads to a computation of the Faltings height of OG with respect to its fundamental embedding in projective space, and a comparison of the resulting formula with previous ones, due to Kaiser and Koehler and the author. |