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--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.

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

References in corpus (1)

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

Cited by in corpus (1)

• Modularity of arithmetic special divisors for unitary Shimura varieties (with an appendix by Yujie Xu)