Heights and metrics with logarithmic singularities

TL;DR

This paper establishes lower bounds and finiteness results for Arakelov heights associated with pre-log-log hermitian line bundles, extending previous theories to more singular metrics.

Contribution

It generalizes known properties of heights to include line bundles with logarithmic singularities, broadening the scope of Arakelov geometry.

Findings

Proves lower bound properties for heights with singular metrics

Establishes finiteness properties for these heights

Connects geometric constructions with pre-log-log hermitian line bundles

Abstract

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.