Computations of Bott-Chern classes on ℙ(E)

Abstract

We compute the Bott-Chern classes of the metrized relative Euler sequence describing the relative tangent bundle of the variety ℙ(E) of the hyperplanes in 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 $\overline{\mathscr{E}}$ and to the computation of the height of ℙ($\overline{\mathscr{E}}$) with respect to the tautological quotient bundle $\mathscr{O}_\overline{\mathscr{E}}$(1).