p-adic deformation of motivic Chow groups

Abstract

For a smooth projective scheme Y over W(k) we consider an element in the motivic Chow group of the reduction Ym over the truncated Witt ring Wm(k) and give a “Hodge” criterion - using the crystalline cycle class in relative crystalline cohomology - for the element to lift to the continuous Chow group of the associated p-adic formal scheme Y•. The result extends previous work of Bloch-EsnaultKerz on the p-adic variational Hodge conjecture to a relative setting. In the course of the proof we derive two new results on the relative de Rham-Witt complex and its Nygaard filtration, and work with a relative version of syntomic complexes to define relative motivic complexes for a smooth lifting of Ym over the ind-scheme Spec W•(Wm(k)).