Standard Conjectures and Height Pairings
Abstract
In this article, we extend Grothendieck's standard conjectures to cycles on degenerated fibers and use them to define some decompositions for the arithmetic Chow group of Gillet--Soul\'e. In the local setting, our decompositions provide non-archimedean analogues of "harmonic forms" on K\"ahler manifolds. In the global setting, our decompositions provide canonical arithmetic liftings called "L- liftings" of algebraic cycles on varieties over number fields, and thus provide a new height pairing called the L-height pairing as one extension of Beilinson--Bloch's pairing of homologically trivial cycles.
Cite
@article{arxiv.2009.07089,
title = {Standard Conjectures and Height Pairings},
author = {Shou-Wu Zhang},
journal= {arXiv preprint arXiv:2009.07089},
year = {2022}
}
Comments
This is the final version of the paper before publication. Significant changes include moving the first section to an appendix and adding materials for perverse sheaves calculations