Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 5

Special Issue in honor of Professor Benedict Gross’s 70th birthday

Guest Editors: Zhiwei Yun, Shouwu Zhang, and Wei Zhang

Standard conjectures and height pairings

Pages: 2221 – 2278

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n5.a7

Author

Shou-Wu Zhang (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

Abstract

In this article, we extend Grothendieck’s standard conjectures [20, Conjectures 1, 2] to cycles on degenerated fibers and use them to define some decompositions for the arithmetic Chow group of Gillet–Soulé. In a local setting, our decompositions provide non-archimedean analogs of “harmonic forms” on Kähler manifolds. In a global setting, our decompositions provide canonical arithmetic liftings called “$\mathsf{L}$‑liftings” of algebraic cycles on varieties over number fields and thus provide a new height pairing called the $\mathsf{L}$‑height pairing as one extension of Beilinson–Bloch’s pairing of homologically trivial cycles.

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This work is partially supported by a grant from the National Science Foundation.

Received 8 July 2021

Received revised 28 April 2022

Accepted 9 May 2022

Published 12 January 2023