Statement copy in the gpt 6 astra pro collection (Math-167)

Murre's Chow-Kunneth Conjectures

Definitions and mathematical statement

For a smooth projective complex $d$-fold $X$, $\mathrm{CH}^j(X)_{{\mathbb{Q}}}$ is the rational vector space of codimension-$j$ cycles modulo rational equivalence. Degree-zero correspondences belong to $\mathrm{CH}^d(X\times X)_{{\mathbb{Q}}}$ and act by pullback, intersection, and pushforward. A Chow--Künneth decomposition consists of orthogonal idempotents $\pi_i$, summing to the diagonal, whose cohomological actions project onto $H^i(X,{\mathbb{Q}})$. Murre's package asks for such a decomposition, vanishing of $(\pi_i)_*$ on $\mathrm{CH}^j$ outside $j\le i\le2j$, and a decomposition-independent filtration \[ F^r\mathrm{CH}^j=\bigcap_{i=\max(0,2j-r+1)}^{2j}\ker(\pi_i)_*, \qquad F^1\mathrm{CH}^j=\mathrm{CH}^j_{\mathrm{hom}}. \] An empty intersection is the whole group; the homologically trivial subgroup is the kernel of the Betti cycle-class map. The universal quantification over every decomposition in B--D is retained, not replaced by existence of one specially chosen decomposition.

Formulation and concept references: [S1].

Short English statement

Can algebraic cycles be split into cohomological pieces so that the resulting filtration is independent of the splitting and starts precisely with the homologically trivial cycles?

Sources

• S1 Uwe Jannsen, Motivic Sheaves and Filtrations on Chow Groups, Proc.Symp.Pure Math.55$1994$,Part1,245-302. https://epub.uni-regensburg.de/26639/1/ubr13239_ocr.pdf. Catalogue formulation source.
• E1 Vladimir Guletskii and Claudio Pedrini, Finite dimensional motives and the conjectures of Beilinson and Murre, K-Theory 30$3$ $2003$, 243--263. Theorems 8 (Kimura's nilpotence theorem) and 14 (isomorphism of the summands) in the consulted manuscript: https://arxiv.org/html/math/0303170. Consulted 24 September 2026; isomorphism of motives is not identified with equality of their images in a fixed Chow group.
• E2 Robert Laterveer and Charles Vial, The Beauville--Voisin--Franchetta conjecture and LLSS eightfolds, Indagationes Mathematicae 37 $2026$, 250--270; arXiv:2404.10465v1 (16 April 2024), §1.1, Theorem 2, Proposition 1.7 and Corollary 1.10. https://arxiv.org/html/2404.10465v1. Publication information: https://www.math.uni-bielefeld.de/~vial/. Manuscript and author's publication list consulted 24 September 2026. The result on selected subrings is not used as an injectivity theorem for the entire Chow ring.
• E3 Charles Vial, Remarks on motives of abelian type, arXiv:1112.1080v3 (27 July 2015), Theorem 3 and §2; published in Tohoku Mathematical Journal 69$2$ $2017$. https://arxiv.org/html/1112.1080v3. Consulted 24 September 2026. The assumption of Murre (D) for products of curves over a universal domain is retained.