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We compute the Bott-Chern classes of the metric Euler sequence describing the relative tangent bundle of the variety P(E) of hyperplans of a holomorphic hermitian vector bundle (E,h) on a complex manifold. We give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle E and to the computation of the height of P(E) with respect to the tautological quotient bundle OE(1). |