Standard Conjectures and Height Pairings (2009.07089v3)

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.