Math Conjectures

Bloch–Beilinson Filtration Conjecture

OPENLandmarkConjectureProposed 1979–1987 · Standard version

Canonical statement

For every smooth projective complex variety XX and every j≥0j\ge0, the rational Chow group CH⁡j(X)Q\operatorname{CH}^j(X)_{\mathbb Q} admits a finite descending filtration F0⊇F1⊇⋯F^0\supseteq F^1\supseteq\cdots such that F0=CH⁡j(X)QF^0=\operatorname{CH}^j(X)_{\mathbb Q}, F1=CH⁡homj(X)QF^1=\operatorname{CH}^j_{\mathrm{hom}}(X)_{\mathbb Q}, pullbacks, proper pushforwards, and algebraic correspondences preserve FF, and products satisfy FrCH⁡i⋅FsCH⁡j⊆Fr+sCH⁡i+jF^r\operatorname{CH}^i\cdot F^s\operatorname{CH}^j\subseteq F^{r+s}\operatorname{CH}^{i+j}. Moreover, Gr⁡FνCH⁡j(X)Q\operatorname{Gr}_F^\nu\operatorname{CH}^j(X)_{\mathbb Q} depends only on the homological-motive summand h2j−ν(X)h^{2j-\nu}(X), and Fν=0F^\nu=0 for all sufficiently large ν\nu.
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For every smooth projective complex variety \(X\) and every \(j\ge0\), the rational Chow group \(\operatorname{CH}^j(X)_{\mathbb Q}\) admits a finite descending filtration \(F^0\supseteq F^1\supseteq\cdots\) such that \(F^0=\operatorname{CH}^j(X)_{\mathbb Q}\), \(F^1=\operatorname{CH}^j_{\mathrm{hom}}(X)_{\mathbb Q}\), pullbacks, proper pushforwards, and algebraic correspondences preserve \(F\), and products satisfy \(F^r\operatorname{CH}^i\cdot F^s\operatorname{CH}^j\subseteq F^{r+s}\operatorname{CH}^{i+j}\). Moreover, \(\operatorname{Gr}_F^\nu\operatorname{CH}^j(X)_{\mathbb Q}\) depends only on the homological-motive summand \(h^{2j-\nu}(X)\), and \(F^\nu=0\) for all sufficiently large \(\nu\).
AG-CYCLE-025:[Bloch1980AlgebraicCycles][Murre1993ChowFiltration][Jannsen1994MotivicFiltration]
Algebraic geometryAlgebra and representation theory

Notes

Rational equivalence on algebraic cycles is much finer than homological equivalence. Bloch and Beilinson predicted that the Chow group should carry a finite descending filtration whose first nontrivial step consists of homologically trivial cycles and whose successive quotients are controlled by individual cohomological pieces of the motive [Bloch1980AlgebraicCycles]. Such a filtration would organize Abel–Jacobi phenomena and higher obstructions to a cycle being rationally trivial.

There are several closely related formulations. Murre expresses the expected structure through Chow–Künneth projectors [Murre1993ChowFiltration], while Jannsen gives functorial axioms involving correspondences, products, and dependence of Gr⁡FνCH⁡j\operatorname{Gr}_F^\nu\operatorname{CH}^j on h2j−νh^{2j-\nu} [Jannsen1994MotivicFiltration]. Special varieties and restricted motivic categories admit pieces of the expected structure, but these constructions do not supply the full filtration in general.

The unresolved task is to construct one finite filtration for every smooth projective variety that satisfies all of the stated functorial, multiplicative, and motivic-dependence conditions. The entry uses Jannsen's axiomatic core, not every stronger consequence commonly grouped under the broad name “Bloch–Beilinson conjectures.”

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.