Characteristic Classes of Adelic Vector Bundles and Applications
Abstract
We define numerical adelic characteristic classes for vector bundles on projective varieties over number fields. The metric data are carried by the tautological quotient line bundle on the associated projective bundle. The construction uses the established intersection theory of adelic line bundles. Higher characteristic classes are multilinear intersection functionals. They are continuous for simultaneously controlled model sequences that are Cauchy for the supremum norm. We construct the resulting tautological numerical intersection algebra. We also prove controlled numerical Bogomolov--Gieseker inequalities. We first treat the case of a curve over K, where the numerical inequality is combined with the Deligne pairing and the adelic Hodge index theorem. We then pass to higher dimension, using Moriwaki's model-level dimension induction before taking the controlled Zhang limit. The curve theorem includes equality and uniform-gap criteria.
Related
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BibTeX
@misc{gao2026characteristic,
title = {{Characteristic Classes of Adelic Vector Bundles and Applications}},
author = {Jiahui Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12138},
eprint = {2609.12138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12138},
}
RIS
TY - UNPB
TI - Characteristic Classes of Adelic Vector Bundles and Applications
AU - Jiahui Gao
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12138
UR - https://arxiv.org/abs/2609.12138
AB - We define numerical adelic characteristic classes for vector bundles on projective varieties over number fields. The metric data are carried by the tautological quotient line bundle on the associated projective bundle. The construction uses the established intersection theory of adelic line bundles. Higher characteristic classes are multilinear intersection functionals. They are continuous for simultaneously controlled model sequences that are Cauchy for the supremum norm. We construct the resulting tautological numerical intersection algebra. We also prove controlled numerical Bogomolov--Gieseker inequalities. We first treat the case of a curve over K, where the numerical inequality is combined with the Deligne pairing and the adelic Hodge index theorem. We then pass to higher dimension, using Moriwaki's model-level dimension induction before taking the controlled Zhang limit. The curve theorem includes equality and uniform-gap criteria.
ER -