Projectors on the intermediate algebraic Jacobians

Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $C{H}_d(X\times X)\otimes \text{Q}$ 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-K"unneth decomposition satisfying the motivic Lefschetz conjecture 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-K"unneth decomposition which satisfies the motivic Lefschetz conjecture and consequently Grothendieck’s standard conjectures. We also prove that hypersurfaces of very low degree are Kimura finite dimensional.