Refining the Abel--Jacobi maps

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 $C{H}^q(X{\times }_{k}K)$, where $K$ is the field of fractions of henselization $A^h$ of the local ring of a smooth closed point of a variety over the field $k$ with an appropriate projection: $CH^q(X\times_kK)\longrightarrow\bigoplus_{p=1}^qgr_F^{q-p}N^{q-p} H^{2q-p}{dR/k}(X)\otimes_k\Omega^p{A^h/k,{\rm closed}},$where$F^{\bullet}$and$N^{\bullet}$ 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 $\bigoplus_{p=1}^qgr_F^{q-p}N^{q-p}H^{2q-p}{dR/k}(X)\otimes_k \lim{\longleftarrow_M}d(\Omega^{p-1}_{A_M/k}),$where$A_M=A^h/{\frak m}^M$and${\frak m}$isthemaximalideal,isdominantforany$q$ for which the inverse Lefschetz operator $H^{2\dim X-q}(X)(\dim X)\stackrel{\sim}{\longrightarrow}H^q(X)(q)$ is induced by a correspondence.