Holmstrom Abel-Jacobi map [private]

Jan Neková\v r, $p$-adic Abel-Jacobi maps and $p$-adic heights (367–379)

MR1940667 (2003j:14060) Biswas, Jishnu(6-MATSCI); Dayal, Gautham; Paranjape, Kapil H.(6-MATSCI); Ravindra, G. V.(6-MATSCI) Higher Abel-Jacobi maps. (these could explain the Bloch-Beilinson filtration; see the review)

Conjecture: The complex Abel-Jacobi map is injective for smooth quasiprojective varieties defined over a number field. (see Lewis and Chen)

Toen: Champs affines. File Toen web publ chaff.pdf. Among other things, contains constructions of schematic homotopy types for the classical cohomologies (Betti, de Rham, Hodge, crystalline, l-adic), extending the classical notions of fundamental group. Also stuff on rational homotopy theory and p-adic homotopy theory, nonabelian Abel-Jacobi map and nonabelian period, and much more on schematic homotopy types and stacks.

Charles: On the zero locus of normal functions and the étale Abel-Jacobi map. http://arxiv.org/abs/0902.1948

arXiv:0907.3539 Projectors on the intermediate algebraic Jacobians from arXiv Front: math.AG by Charles Vial Let $X$ be a smooth projective variety over an algebraically closed subfield of $\C$. Under mild assumption, we construct projectors modulo rational equivalence onto the last step of the coniveau filtration on the cohomology of $X$. We obtain a “motivic” description of the Abel-Jacobi maps to the algebraic part of the intermediate Jacobians. As an application, this enables us to relate the injectivity of the Abel-Jacobi map in all degrees modulo torsion to finite dimensionality for the motive of $X$.

arXiv:1009.1431 Non-trivial elements in the Abel-Jacobi kernels of higher dimensional varieties from arXiv Front: math.AG by Sergey Gorchinskiy, Vladimir Guletskii The purpose of this paper is to construct non-trivial elements in the Abel-Jacobi kernels in any codimension by specializing correspondences with non-trivial Hodge-theoretical invariants at points with different transcendence degrees over a subfield in $\CC$.

nLab page on Abel-Jacobi map [private]