Standard conjectures and height pairings

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 man-ifolds. In a 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.