Refining the Abel--Jacobi maps

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Given a smooth projective variety X over a field k of characteristic zero, we consider the composition of the de Rham cohomology cycle class map over k from the Chow group CHq(X×ₖK), where K is the field of fractions of henselization Aʰ of the local ring of a smooth closed point of a variety over the field k with an appropriate projection: CHq(X×ₖK)ₚ₌₁qgrFq⁻ᵖNq⁻ᵖ H²q⁻ᵖdR/k(X)⊗ₖΩᵖAʰ/k, closed, where F and N are the Hodge and the coniveau filtrations on the de Rham cohomology, respectively. The classical Abel--Jacobi map corresponds to the composition of this homomorphism with the projection to the summand p=1. This homomorphism is not injective, however, its composition with the embedding into the space ₚ₌₁qgrFq⁻ᵖNq⁻ᵖH²q⁻ᵖdR/k(X)⊗ₖ ₘd(Ωᵖ⁻¹ₐₘ/ₖ), where Aₘ=Aʰ/ mᵐ and m is the maximal ideal, is dominant for any q for which the inverse Lefschetz operator H2 X-q(X)(X) Hq(X)(q) is induced by a correspondence.