Heights and metrics with logarithmic singularities

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 Kühn in [J. I. Burgos, J. Kramer and U. Kühn, Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), 1–172.], in their extension of the arithmetic intersection theory of Gillet and Soulé [H. Gillet and C. Soulé, Arithmetic intersection theory, Publ. Math. IHES 72 (1990), 94–174.], 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 [J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), 903–1027.], as well as the properties established by Faltings [G. Faltings, Finiteness theorems for abelian varieties over number fields, Arithmetic Geometry, G. Cornell and J. H. Silverman, eds., Springer-Verlag (1986), 9–27.] 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.