Computations of Bott-Chern classes on P(E)

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 and to the computation of the height of P(E) with respect to the tautological quotient bundle O_E(1).