Computations of Bott-Chern classes on \(\mathbb{P}(E)\) (Q1881732)

Bott-Chern classes of the metrized relative Euler sequence describing the relative tangent bundle of the variety \(\mathbb P(E)\) of hyperplanes in a holomorphic Hermitian vector bundle \((E,h)\) on a complex manifold are computed. Applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle \({\overline{E}}\) and to the computation of the height of \(\mathbb P({\overline{E}})\) with respect to the tautological quotient bundle \({\mathcal O}_{\overline{E}}(1)\) are also presented.