Projectors on the intermediate algebraic Jacobians

Details Projectors on the intermediate algebraic Jacobians Projectors on the intermediate algebraic Jacobians

2009-07-21

arxiv.org/abs/0907.3539v2

Mathematics

Algebraic Geometry

29 pages

Scientific paper

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 $CH_d(X \times X) \otimes \Q$ 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.

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