Let X be a complex smooth projective variety of dimension d. Under some assumptions on the cohomology of X, we construct mutually orthogonal idempotents in CH_d(X × X) ⊗ Q whose action on algebraically trivial cycles coincides with the Abel-Jacobi map. Such a construction generalizes Murre's construction of the Albanese and Picard idempotents and makes it possible to give new examples of varieties admitting a self-dual Chow-Künneth decomposition as well as new examples of varieties having a Kimura finite-dimensional Chow motive. For instance, we prove that fourfolds with Chow group of zero-cycles supported on a curve (e.g., rationally connected fourfolds) have a self-dual Chow-Künneth decomposition. We also prove that hypersurfaces of very low degree are Kimura finite-dimensional.

This work was supported by a Nevile Research Fellowship at Magdalene College, Cambridge and an EPSRC Postdoctoral Fellowship under grant EP/H028870/1. I would like to thank both institutions for their support.