Projectors on the intermediate algebraic Jacobians

الأصل · EN

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 CHd(X × X) ⊗ 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 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-Künneth 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.