Projectors on the intermediate algebraic Jacobians

TL;DR

This paper constructs special projectors on algebraic cycles of smooth projective varieties, generalizing previous work, and provides new examples of varieties with desirable motivic properties, including those satisfying the motivic Lefschetz conjecture.

Contribution

It introduces a method to construct mutually orthogonal idempotents in the Chow group that align with the Abel-Jacobi map, extending Murre's work and producing new examples of varieties with self-dual Chow-K"unneth decompositions.

Findings

Fourfolds with zero-cycles supported on a curve have a self-dual Chow-K"unneth decomposition.

Hypersurfaces of very low degree are Kimura finite dimensional.

The construction satisfies the motivic Lefschetz conjecture and supports Grothendieck's standard conjectures.

Abstract

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.