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We study the Arakelov intersection ring 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 ring CH(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 Kohler [KK] and the author [T3], [T4].

Arakelov theory of even orthogonal Grassmannians