Equidistribution for semiabelian varieties over number fields

Abstract

K{ü}hne established equidistribution for canonical adelic line bundles on semiabelian varieties, a setting which need not lie in the quasi-canonical range of Yuan-Zhang's theorem. We study adelic line bundles on semiabelian compactifications whose toric part is governed by a toric metrized divisor. The main unconditional result is generic equidistribution when the toric metric is monocritical and arithmetically T-effective. The proof isolates the asymptotic estimates in K{ü}hne's argument and reinterprets them as estimates along explicit compression paths. This gives a comparison mechanism between canonical, quasi-canonical, and more general toric metrics. For the Bogomolov application, the quasi-canonical case is handled by a K{ü}hne local-trivialization transport package. After fixing a single theta-factor convention for the local trivializations, the proof checks the Picard-zero theta factors under K{ü}hne's operations and obtains the Bogomolov theorem for the metric class treated in this paper, namely the monocritical and arithmetically T-effective toric metrics, through the quasi-canonical replacement argument.