Murre's Chow-Kunneth Conjectures

Murre's Chow-Kunneth Conjectures

OpenAI's claim

For every smooth complex projective surface with $p_g=0$, OpenAI claims that the Albanese map on integral degree-zero cycles is an isomorphism. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

For a smooth connected projective complex surface $S$ put $p_g=h^0(S,\omega_S)$ and $q=h^1(S,\mathcal O_S)$. OpenAI asserts the integral statements \[ p_g=q=0\ \Longrightarrow\ \deg:\operatorname{CH}_0(S)\xrightarrow{\sim}\mathbb Z, \qquad p_g=0\ \Longrightarrow\ \operatorname{alb}_S:\operatorname{CH}_0(S)^0\xrightarrow{\sim}\operatorname{Alb}(S)(\mathbb C), \] where $\operatorname{CH}_0(S)^0$ is the subgroup of degree-zero zero-cycles modulo rational equivalence. Under $p_g=0$, it also claims that the rational Chow motive is finite-dimensional in Kimura's sense and has zero transcendental degree-two summand. When $p_g=q=0$ it claims \[ h(S)_{\mathbb Q}\simeq\mathbf1\oplus\mathbb L^{\oplus b_2(S)}\oplus\mathbb L^2, \] where $\mathbb L$ is the Lefschetz motive. Under the joint hypothesis $p_g=q=0$, for each power $Y=S^m$ and each Hilbert scheme $Y=S^{[n]}$, the corresponding rational motive is pure Tate and $\operatorname{CH}^i(Y)_{\mathbb Q}\xrightarrow{\sim}H^{2i}(Y,\mathbb Q)$, with $H^{2i+1}(Y,\mathbb Q)=0$, for every $i\ge0$.

Outline of the argument

Manuscript 1 (family 040, 2026-09-24): A determinant-constrained Quot obstruction theory and torus localization produce a Chow relation involving the diagonal. A nested-Hilbert recursion leaves two surface factors unintegrated, retaining an actual correspondence identity; its external-product terms annihilate degree-zero cycles, and the diagonal coefficient gives the required vanishing after the standard reductions.

Formal verification

Family 040 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, Bloch's conjecture for surfaces with p_g=0, 2026-09-24; citation key OAI:Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: cor:tate-motives; Statement source: thm:bloch; Statement source: cor:bloch-pg-zero.
• OpenAI catalogue entry.

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.

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?

Research attempt

Scope check and current frontier (24 September 2026).

This entry uses the strong form of B in [S1], Conjecture 5.1: the forbidden indices include $i<j$, not only $i>2j$. Its requirement concerning every decomposition is retained. The full package is known for curves; for surfaces, constructing a decomposition with B and D must be distinguished from an assertion about all choices. Nilpotence and algebraicity of the cohomological Künneth projectors give existence and uniqueness of the motive summands up to isomorphism, not automatically equality of subspaces of Chow groups ([S1], Proposition 5.3 and Lemma 5.4; [E1], Theorems 8 and 14).

The recent paper of Laterveer--Vial, published in 2026 from a 2024 manuscript, records these distinctions and proves cycle-class injectivity on specified subrings for LLSS eightfolds; it does not prove the full package for arbitrary varieties ([E2], §1.1 and Theorem 2). A relevant established reduction is Vial's theorem that D for products of curves over a universal domain implies A--D for motives of abelian type ([E3], Theorem 3). It would be circular to invoke D for all such products without an independent proof.

Attack map.

A lifting approach separates A into algebraicity of the cohomological projectors and correction of their idempotence and orthogonality in the Chow ring of correspondences. The latter is accessible under nilpotence; the former is still a hypothesis for a general $X$. A second approach transports a splitting through correspondences with products of curves, but needs the corresponding vanishing and homological-kernel assertions on those products. A third approach attacks the universal quantifier directly: conjugate one splitting by a homologically trivial perturbation and test whether its filtration changes.

We pursue the third route in both directions. It gives a proof strategy if all relevant perturbations preserve the filtration, and an explicit recipe for a counterexample if one geometrically realizable perturbation moves it. The aim is to identify the exact extra condition beyond nilpotence, rather than mistake isomorphism of motive summands for equality of Chow subgroups.

Attempt I: an explicit conjugacy formula.

Fix $X$ of dimension $d$ and write \[ R_X=\mathrm{CH}^d(X\times X)_{{\mathbb{Q}}},\qquad I_X=\ker\left(R_X\longrightarrow H^{2d}(X\times X,{\mathbb{Q}})\right). \] Multiplication in $R_X$ means composition, its unit is $\Delta_X$, and $I_X$ is a two-sided ideal. Künneth and Poincaré duality identify the cohomology class of a degree-zero correspondence with its actions on all $H^i(X,{\mathbb{Q}})$. Thus two Chow--Künneth systems $p=(p_i)$ and $q=(q_i)$ satisfy $q_i-p_i\in I_X$.

For now make the following explicit assumption: \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-6e-69-6c} \text{Every $n\in I_X$ is nilpotent under composition.} \end{equation} No exponent uniform in $n$ is required here. This hypothesis holds when the Chow motive is Kimura finite-dimensional, by the nilpotence theorem recorded in [E1], Theorem 8. The universal finite-dimensionality assertion of Section 30006782 is not being assumed as a known theorem.

Set \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-63-6f-6e-6a-75-67-61-63-79} u=\sum_{i=0}^{2d}q_i p_i=1+n,\qquad n\in I_X. \end{equation} If $n^N=0$, then $u^{-1}=\sum_{a=0}^{N-1}(-n)^a$. Orthogonality, with the order of factors retained, gives \[ q_i u=q_i p_i=u p_i, \qquad q_i=u p_i u^{-1}. \] This is an explicit version of the conjugacy mechanism in [S1], Lemma 5.4. It uses only nilpotence of the particular $n$ in (ra30006755:conjugacy); it does not assume that every sufficiently long product of different members of $I_X$ vanishes.

Let $V_j=\mathrm{CH}^j(X)_{{\mathbb{Q}}}$ and let $K_j$ be its homologically trivial subgroup. Correspondences act on $V_j$; subscripts $*$ are suppressed below. For the filtration defined in the original statement, conjugacy gives the exact identity \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-74-72-61-6e-73-70-6f-72-74} F_q^r V_j=u(F_p^r V_j). \end{equation} Indeed, $q_i v=0$ if and only if $p_i u^{-1}v=0$, and the same equivalence holds for the entire intersection of kernels, including an empty one.

Suppose the reference system $p$ satisfies B and D on $V_j$. B transfers to $q$, because a conjugate of the zero operator is zero. D transfers too: $I_X V_j\subseteq K_j$ by compatibility of cycle classes with correspondences, so $u$ and $u^{-1}$ preserve $K_j$ and $F_q^1V_j=u(K_j)=K_j$. Thus under (ra30006755:nil), a single verified B,D splitting gives B,D for all splittings on that Chow group. The remaining problem is C, not transport of these two properties.

Attempt II: the precise obstruction to C.

Still assuming (ra30006755:nil) and B,D for $p$, the following conditions are equivalent for the fixed $j$: \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-63-72-69-74-65-72-69-6f-6e} \begin{aligned} &F_q^r V_j=F_p^r V_j\quad\text{for every Chow--K\"unneth system $q$ and $r\geq0$};\\ &I_X(F_p^r V_j)\subseteq F_p^r V_j\quad\text{for every $r\geq0$}. \end{aligned} \end{equation} For sufficiency, $u$ in (ra30006755:conjugacy) and its finite-series inverse preserve every $F_p^r$, and (ra30006755:transport) gives equality. Conversely, for each $n\in I_X$, conjugation by $1+n$ produces another valid system. Equality of its filtration with $F_p^r$ implies $(1+n)F_p^r=F_p^r$, hence $nF_p^r\subseteq F_p^r$. This checks the universal quantifier, rather than only a preferred class of splittings.

B supplies a finite direct-sum description, valid for every $r\geq0$: \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-73-70-6c-69-74-2d-66-69-6c-74-72-61-74-69-6f-6e} V_j=\bigoplus_{i=j}^{2j}p_iV_j,\qquad F_p^rV_j=\bigoplus_{j\leq i\leq 2j-r}p_iV_j, \end{equation} where an empty sum is zero. Moreover, D and $I_XV_j\subseteq K_j$ imply $p_{2j}I_XV_j=0$. Therefore the second line of (ra30006755:criterion) is equivalent to the following finite collection of types of vanishing conditions: \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-75-70-77-61-72-64} p_b\,I_X\,p_i(V_j)=0 \qquad\text{whenever}\qquad j\leq i<b<2j. \end{equation} Each condition still quantifies over all correspondences in $I_X$ and all cycles; this is not a finite computability assertion. To prove necessity, use the step $r=2j-i$, which contains $p_iV_j$ but no $p_bV_j$ with $b>i$. For sufficiency, expand any $n\in I_X$ as $\sum_{b,i}p_b n p_i$. The terms with $b>i$ vanish on $V_j$ by (ra30006755:upward), D, and B; every remaining term preserves all partial sums in (ra30006755:split-filtration).

In codimension $j=0$ or $1$ there is no pair in (ra30006755:upward), so no further condition is needed. In codimension two, for $d\geq2$, the entire additional condition is \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-63-6f-64-69-6d-2d-74-77-6f} p_3 I_X p_2\left(\mathrm{CH}^2(X)_{{\mathbb{Q}}}\right)=0. \end{equation} Thus a concrete remaining task is to rule out a correspondence moving a cycle from the $p_2$ part into the $p_3$ part. Nilpotence alone has no reason to forbid such a movement.

Attempt III: an explicit counterexample certificate, if realizable.

There is an especially small perturbation that would falsify C. Suppose $j\leq i<b<2j$ and choose $\gamma\in I_X$ with \[ h=p_b\gamma p_i, \qquad h(z)\ne0\text{ for some }z\in p_iV_j. \] Then $h^2=0$, because $p_i p_b=0$. No general nilpotence assumption is needed for this particular construction. Conjugating by $1+h$ gives \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-65-6c-65-6d-65-6e-74-61-72-79-2d-70-65-72-74-75-72-62-61-74-69-6f-6e} q_i=p_i+h,\qquad q_b=p_b-h,\qquad q_a=p_a\ (a\notin\{i,b\}). \end{equation} Direct multiplication verifies orthogonality and idempotence. Their sum and cohomological actions are unchanged. At $r=2j-i$, the cycle $z$ belongs to $F_p^r$, but $q_bz=-h(z)\ne0$, so $z\notin F_q^r$. This would be an actual counterexample to the printed universal C, not merely nonuniqueness of its individual projectors. No such geometric $h,z$ has been constructed here.

A small algebraic model shows that the obstacle is real at the ring level. Take the ring and its nilpotent ideal \[ R=\left\{\begin{pmatrix}a&0&0\\ b&c&0\\0&0&d\end{pmatrix}:a,b,c,d\in{\mathbb{Q}}\right\}, \qquad I={\mathbb{Q}} E_{21},\qquad I^2=0, \] acting on $V={\mathbb{Q}} e_2\oplus{\mathbb{Q}} e_3\oplus{\mathbb{Q}} e_4$, where the basis subscripts are weight labels, not matrix indices. Put $p_2=E_{11}$, $p_3=E_{22}$, $p_4=E_{33}$ and $p_0=p_1=0$. Give the degree-four cycle-class model the map $\mathrm{cl}(x e_2+y e_3+z e_4)=z$. Then \[ F_p^1=\ker(\mathrm{cl})={\mathbb{Q}} e_2\oplus{\mathbb{Q}} e_3, \qquad F_p^2={\mathbb{Q}} e_2, \qquad F_p^3=0. \] For $h=E_{21}$, (ra30006755:elementary-perturbation) gives \[ q_2=\begin{pmatrix}1&0&0\\1&0&0\\0&0&0\end{pmatrix},\quad q_3=\begin{pmatrix}0&0&0\\-1&1&0\\0&0&0\end{pmatrix},\quad q_4=p_4, \qquad F_q^2={\mathbb{Q}}(e_2+e_3)\ne F_p^2. \] Here $F_q^1=F_p^1$, so the analogue of D survives, as does B. Exact rational matrix multiplication verifies the displayed identities; the failure of C is also immediate by applying $q_3$ to $e_2$. This is not a smooth projective variety or a counterexample to Murre. It specifically refutes the proposed purely ring-theoretic step "nilpotence plus B,D makes the filtration independent."

Attempt IV: searching for a geometric perturbation.

The simplest construction to test is a sum of decomposable correspondences $\alpha\times\beta$, with $\alpha\in\mathrm{CH}^{d-s}(X)_{{\mathbb{Q}}}$ and $\beta\in\mathrm{CH}^{s}(X)_{{\mathbb{Q}}}$. On $V_j$ it acts as zero unless $s=j$, and for $s=j$ its action is \begin{equation}\label{wsref-0-72-61-33-30-30-30-36-37-35-35-3a-72-61-6e-6b-2d-6f-6e-65} (\alpha\times\beta)_*(z)=\deg(z\cdot\alpha)\,\beta. \end{equation} This follows directly by integrating the first factor in the pullback, intersection, and pushforward formula. If $z\in p_iV_j$ with $i<2j$, its cycle class is zero: the cycle class lies in $H^{2j}$ and $p_i$ projects to $H^i$. Hence $\deg(z\cdot\alpha)=0$ for every such $\alpha$. Thus every candidate built only from these external products kills the input in (ra30006755:upward). Choosing $\beta$ homologically trivial does not help. This construction fails by an exact intersection-number calculation, not by lack of a numerical example. A successful negative attack would need a non-decomposable correspondence acting nontrivially on homologically trivial cycles in the indicated direction.

For the positive attack, the known product argument explains why the missing vanishing is deep. Form the Chow--Künneth system on $X\times X$ \[ P_m=\sum_{a+c=m}{}^t p_{2d-a}\times p_c, \qquad 0\leq m\leq4d, \] with the factors rearranged in the usual way to make a correspondence on $X\times X$. Pull--push composition gives \[ ({}^t p_i\times p_b)_*(\gamma)=p_b\gamma p_i. \] Its total cohomological index is $2d-i+b$. B in codimension $d$ on $X\times X$, for this system, would kill it when $b>i$. D there would additionally kill $p_i I_X p_i$. Consequently $I_X$ would be strictly lower triangular, and $I_X^{2d+1}=0$ by counting successive decreases of indices. This is the mechanism of [S1], Proposition 5.8, not a new proof of its hypotheses. It would make every condition (ra30006755:upward) automatic, but assumes the requisite statements on the square rather than deriving them from statements on $X$.

Stress test and dependency check.

All equalities in the perturbation calculation take place modulo rational equivalence, except where explicitly passed to cohomology. The matrix model is used only to falsify a proposed abstract lemma; no geometric realization, Poincaré duality, or Chow-motive interpretation of that model is asserted. The calculations use finite sums and finite nilpotent inverses; there is no convergence or unproved completion of the correspondence ring. The perturbed systems need not be self-dual or multiplicative; neither extra condition is part of this entry.

The proof of the criterion handles every admissible $j$, every $r\geq0$, and every Chow--Künneth system. In particular, the indices $i<j$ in the strong B have not been omitted. Its assumptions still include one actual splitting with B,D. They do not establish A for arbitrary $X$, nor prove that the first splitting has B,D.

Assuming Kimura finite-dimensionality for all $X$ would use the unresolved Section 30006782; nilpotence here concerns composition, not the external powers in Section 30006776. Algebraizing every Künneth projector would itself establish the Künneth standard conjecture, not just a formal linear-algebra step. Finally, using a full Bloch--Beilinson filtration to justify invariance would invoke a conjectural structure equivalent to the universal Murre package in the framework of [S1], Theorem 5.2. None of these dependencies is discharged by the calculation above.

Outcome.

Outcome: Conditional reduction.

For a fixed $X$, suppose A holds, every element of $I_X$ is composition-nilpotent, and one chosen splitting satisfies strong B and D for every codimension. Then the full all-decompositions A--D package is equivalent to the vanishings (ra30006755:upward) for that splitting. The implication in each direction has been proved above; no priority claim is made for the underlying conjugacy and weight-filtration methods. In codimension two, the additional condition is exactly (ra30006755:codim-two). A nonzero geometric instance of that upward action would instead give the explicit counterexample certificate (ra30006755:elementary-perturbation).

Nilpotence alone is insufficient, as the matrix model shows, and the most obvious external-product realization of the bad perturbation is impossible by (ra30006755:rank-one). What remains is the genuinely geometric vanishing or construction of such actions, together with the separate existence and B,D hypotheses. No smooth projective counterexample or unconditional proof of the universal target has been obtained.

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.