Heights and metrics with logarithmic singularities

We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and Kühn in [2], in their extension of the arithmetic intersection theory of Gillet and Soulé [8];, aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities. Our results generalize the corresponding properties for the heights of cycles in Bost-Gillet-Soulé [1], as well as the properties established by Faltings [7] for heights of points attached to hermitian ample line bundles whose metrics have logarithmic singularities. We also discuss various geometric constructions where such pre-log-log hermitian ample line bundles naturally arise.