Standard Conjectures and Height Pairings

2020/09/15 by Shou-Wu Zhang, Zhang, Shou-Wu
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2009.07089

openalex publication_date 2020/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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--Soulé. In the local setting, our decompositions provide non-archimedean analogues of "harmonic forms" on Kähler 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.

Citations

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

Related

• A relation between standard conjectures and their arithmetic analogues
• The Business of Height Pairings
• Topological versions of Abel-Jacobi, the height pairing, and the Poincaré bundle
• Degeneration of the archimedean height pairing of algebraically trivial cycles
• A generalization of Beilinson's geometric height pairing