  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 K\"uhn, in their extension of the arithmetic
intersection theory of Gillet and Soul\'e, aimed to deal with hermitian vector
bundles equipped with metrics admitting suitable logarithmic singularities. Our
results generalize the corresponding properties for the heights of
Bost-Gillet-Soul\'e, as well as the properties established by Faltings for
heights of points attached to hermitian line bundles whose metrics have
logarithmic singularities. We also discuss various geometric constructions
where such pre-log-log hermitian ample line bundles naturally arise.
