Grothendieck Standard Conjecture of Hodge Type

Grothendieck Standard Conjecture of Hodge Type

OpenAI's claim

The manuscripts claim algebraicity of rational Hodge classes on CM abelian varieties, with consequences for generalized Hodge classes and finite-field Tate classes. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

A complex abelian variety $A$ is CM if $\operatorname{End}(A)\otimes\mathbb Q$ contains a commutative semisimple algebra of dimension $2\dim A$. For every such $A$ and every $p\ge0$, OpenAI claims \[ \operatorname{CH}^p(A)_{\mathbb Q}\longrightarrow H^{2p}(A,\mathbb Q)\cap H^{p,p}(A) \quad\text{is surjective}. \] This also applies to arbitrary finite products of CM abelian varieties, including their powers. If $W\subset H^k(A,\mathbb Q)$ is a rational Hodge substructure and $W(r)$ is effective (all Hodge bidegrees are nonnegative), it claims a closed algebraic $Z\subset A$ of codimension at least $r$ with $W\subset\ker[H^k(A,\mathbb Q)\to H^k(A\setminus Z,\mathbb Q)]$. The manuscript additionally asserts that, for every abelian variety $A/\mathbb F_q$, every $r\ge0$ and prime $\ell\ne\operatorname{char}\mathbb F_q$, \[ \operatorname{CH}^r(A)\otimes\mathbb Q_\ell\longrightarrow H^{2r}_{\mathrm{\acute et}}(A_{\overline{\mathbb F}_q},\mathbb Q_\ell(r))^{\operatorname{Gal}(\overline{\mathbb F}_q/\mathbb F_q)} \quad\text{is surjective}. \] Numerical and $\ell$-adic homological equivalence are claimed to agree on rational cycles. Over any algebraically closed field, for an abelian $g$-fold and rational ample class $h$, the form $(-1)^r\deg(xyh^{g-2r})$ is claimed positive definite on the real extension of $\ker[h^{g-2r+1}:A^r_{\mathrm{num}}\to A^{g-r+1}_{\mathrm{num}}]$ for $0\le r\le g/2$.

Outline of the argument

Manuscript 1 (family 032, 2026-09-30): The argument realizes four-factor balanced tensors using unitary theta constructions. Elementary switches between equal-sum lists, followed by algebraic polarization contractions and descent of coefficients, build arbitrary balanced CM tensors from these elementary cycle classes.

Formal verification

Family 032 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, The rational Hodge conjecture for CM abelian varieties, 2026-09-30; citation key OAI:The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:cm; Statement source: cc:ghc; Statement source: cc:tate; Statement source: cc:standard.
• OpenAI catalogue entry.

Grothendieck Standard Conjecture of Hodge Type

Definitions and mathematical statement

Let $N^i(X)_{{\mathbb{Q}}}$ be codimension-$i$ rational cycles modulo numerical equivalence, and let $L$ be an ample divisor class. Define \[ P_L^i=\ker\left(L^{n-2i+1}:N^i(X)_{{\mathbb{Q}}}\to N^{n-i+1}(X)_{{\mathbb{Q}}}\right), \qquad q_L^i(a,b)=(-1)^i\deg(a\cdot b\cdot L^{n-2i}). \] For $0\le i\le n/2$, require $q_L^i(a,a)>0$ for every nonzero $a\in P_L^i\otimes{\mathbb{R}}$. Groups outside codimensions $0,\ldots,n$ are zero. This is numerical primitive positivity over an algebraically closed field of any characteristic, not a silent identification with the primitive subspace of an arbitrary cohomology theory.

Formulation and concept references: [S1].

Short English statement

Is the signed intersection form positive definite on primitive numerical cycle classes in every characteristic?

Research attempt

Prepared by GPT-6 Astra Ultra on 27 September 2026, with primary-source review, explicit example checks, and examination of the stated conditional reductions. This is an unresolved research attempt.

Scope and checked context (27 September 2026).

Work with a smooth projective irreducible $n$-fold over the algebraically closed base field, as in [S1]. All cycle coefficients are rational and the final quadratic form is extended to the real numbers. The primitive condition is imposed in numerical groups. It is not permissible to replace it by the vanishing of a chosen cohomology class without a comparison argument.

Agugliaro [S1] constructs classes of examples and records the classical divisor and characteristic-zero context. Ancona [S2] proves the abelian fourfold case. The later paper [S3] treats powers of abelian threefolds and other specified abelian families, rather than arbitrary varieties. The attempt below gives exact intersection calculations and a finite certificate for any variety whose numerical groups can actually be determined. Its cohomological comparison lemma states its additional hypotheses explicitly. These elementary deductions are not new standard conjectures or proofs of the unrestricted positive-characteristic case.

The pairing descends, but positivity does not follow from that.

Numerical equivalence is compatible with intersection products on a smooth projective variety. If $a$ is numerically trivial, then \[ \deg(a b L^{n-2i})=0 \] for every codimension-$i$ cycle $b$, because $bL^{n-2i}$ is an algebraic complementary cycle. Thus $q_L^i$ is well-defined on $N^i$. Similarly multiplication by $L$ is well-defined in the numerical quotient. This justifies the displayed definitions, but says nothing about the sign of a nonzero square.

For $i=0$, one has $N^0=\mathbb Q[X]$ and the target of $L^{n+1}$ is zero. Hence \[ q_L^0(c[X],c[X])=c^2\deg L^n>0\qquad(c\ne0). \] For $X=\mathbb P^n$, write $h$ for the hyperplane class and $L=ah$ with $a>0$. For every $1\le i\le n/2$, multiplication by $L^{n-2i+1}$ sends $h^i$ to a nonzero multiple of $h^{n-i+1}$. The primitive group is therefore zero. This example checks the codimension endpoint, but cannot test any nontrivial primitive square.

On $X=\mathbb P^1\times\mathbb P^1$, put $L=ah_1+bh_2$ with $a,b>0$. The relations are $h_1^2=h_2^2=0$ and $\deg h_1h_2=1$. The primitive divisor space is generated by $ah_1-bh_2$, and \[ -\deg(ah_1-bh_2)^2=2ab>0. \] The minus sign in codimension one is essential. More generally the Hodge index theorem gives the known divisor case in every characteristic; it is an external geometric input, recalled in [S1], not deduced merely from the preceding two-dimensional calculation.

A nontrivial middle-dimensional calculation in every characteristic.

Take $X=(\mathbb P^1)^4$ and $L=h_1+h_2+h_3+h_4$. Its cycle ring is \[ \mathbb Q[h_1,h_2,h_3,h_4]/(h_1^2,h_2^2,h_3^2,h_4^2), \qquad \deg h_1h_2h_3h_4=1. \] The six squarefree degree-two monomials form a numerical basis: each pairs to one with its complementary monomial and to zero with the others. A class $\sum_{i<j}x_{ij}h_ih_j$ is primitive if and only if \[ x_{ij}+x_{ik}+x_{jk}=0 \quad\text{for each three-element set }\{i,j,k\}. \] The four equations are independent over $\mathbb Q$. For instance, a linear relation between their rows would require the coefficients of the two triples containing each fixed pair to sum to zero; these relations force all four row coefficients to vanish.

The kernel is thus two-dimensional, with basis \[ u=h_1h_2+h_3h_4-h_1h_3-h_2h_4,\qquad v=h_1h_2+h_3h_4-h_1h_4-h_2h_3. \] Their intersection matrix is \begin{equation}\label{wsref-0-6e-68-3a-67-72-61-6d} \begin{pmatrix}\deg u^2&\deg uv\\\deg uv&\deg v^2\end{pmatrix} =\begin{pmatrix}4&2\\2&4\end{pmatrix}. \end{equation} In codimension two the sign is positive. The eigenvalues are $2,6$, so the required positivity holds on this entire primitive group, including real linear combinations. The computation uses an explicit cycle ring and perfect complementary pairings, and is valid in every characteristic because the coefficient field for cycles is $\mathbb Q$. It is a genuine complete subcase, not an inference from a few positive self-intersections on an unknown space.

A false primitive class obtained by testing too few cycles.

The same variety gives an exact warning about incomplete computations. Put $w=h_1-h_2$ and $z=Lw\in N^2(X)$. Direct multiplication gives \[ \deg(wL^3)=0,\qquad \deg(w^2L^2)=-4. \] Thus $\deg((Lz)L)=0$. If one tested the codimension-three class $Lz$ only against the ample divisor $L$, one would incorrectly declare $z$ primitive. Its square is $\deg z^2=-4$, apparently contradicting the conjecture. The missing test is the divisor $w$: \[ \deg((Lz)w)=\deg(L^2w^2)=-4\ne0. \] Hence $Lz$ is not numerically trivial and $z$ is not primitive. This does not expose a counterexample to positivity. It exposes the failure to test numerical equivalence against all complementary cycles. Increasing the accuracy of the first intersection number would not repair that omission.

An exact finite certificate when the numerical groups are known.

Suppose rational numerical bases have been proved for $N^i$ and $N^{n-i+1}$, and all relevant intersections are known. Let $B$ be the matrix of multiplication by $L^{n-2i+1}$ in these bases and let \[ Q_{ab}=(-1)^i\deg(e_ae_bL^{n-2i}). \] Compute a rational matrix $K$ whose columns form a basis of $\ker B$. The required quadratic form is represented exactly by \begin{equation}\label{wsref-0-6e-68-3a-63-65-72-74-69-66-69-63-61-74-65} G=K^{\mathsf T}QK. \end{equation} It is positive definite if and only if its leading principal minors are positive, by Sylvester's criterion. For a zero-dimensional kernel the condition is vacuous. This certificate proves positivity over $\mathbb R$, not only on the finitely many rational basis vectors. Checking $G_{aa}>0$ alone would fail: the matrix $\left(\begin{smallmatrix}1&2\\2&1\end{smallmatrix}\right)$ has positive diagonal but takes the value $-2$ on $(1,-1)$.

There are two separate completeness obligations. An invertible intersection matrix with complementary cycles proves that a proposed list is independent. It proves the list spans only if its size also meets a justified upper bound on the numerical dimension. Likewise a class killed by the computed multiplication matrix is truly primitive only if the target coordinates detect every numerical class. The preceding $z=Lw$ example shows exactly what can go wrong otherwise. No point-counting or finite-cycle search performed here provides those dimension bounds for a general variety.

A conditional bridge to cohomological primitive positivity.

For a complex variety, let $A^j$ be the rational span of algebraic classes in $H^{2j}(X,\mathbb Q)$. Homologically trivial cycles are numerically trivial, so there is a surjection $A^j\to N^j$. If numerical and homological equivalence agree in the relevant degrees, then a numerical primitive class has a unique algebraic cohomology class, and its multiplication by $L^{n-2i+1}$ is zero in cohomology. The Hodge--Riemann relation on real primitive classes of type $(i,i)$ then gives exactly the sign $(-1)^i$ required here. This implication is immediate only after the comparison hypothesis has been checked.

Here is another sufficient bridge with that checking made explicit. Assume the cohomological Lefschetz decomposition preserves algebraic classes in every degree, and the inverse hard-Lefschetz maps send algebraic classes to algebraic classes. Then, for $i\le n/2$, \[ A^i=\bigoplus_{r=0}^i L^r A^{i-r}_{\mathrm{prim}}. \] The summands are orthogonal for $B_i(a,b)=\int_X abL^{n-2i}$: when $r>s$, the factor multiplying the primitive class of codimension $i-s$ contains at least $L^{n-2(i-s)+1}$ and vanishes. On the $r$-th summand the form is definite of sign $(-1)^{i-r}$ by Hodge--Riemann. It is therefore nondegenerate on $A^i$. A numerically trivial class pairs to zero with $L^{n-2i}A^i$, so it must vanish. Inverse hard Lefschetz transfers the same conclusion to complementary degrees. The numerical and cohomological spaces can now be identified, and primitive positivity follows. The stated algebraicity properties are sufficient inputs; they have not been established here for arbitrary varieties.

This explains the logical role of a cohomological argument without identifying numerical primitivity with cohomological primitivity by notation alone. In positive characteristic there is the further issue that a chosen Weil cohomology theory does not itself provide an ordered real Hodge--Riemann form. The rational intersection numbers have an ordinary real sign, but an $\ell$-adic absolute value is not that sign.

Verification, gap, and outcome.

Exact rational calculations checked the full multiplication matrix on $(\mathbb P^1)^4$, its kernel dimension, (nh:gram), and the two different tests of $Lz$. They also checked Sylvester's criterion in the displayed positive and indefinite examples. No numerical sampling of real vectors was used to certify definiteness.

The first remaining task in the general numerical approach is to determine the complete primitive numerical space and show that its intersection matrix has the required sign. A useful continuation would establish numerical dimension bounds matching explicit cycle lists, construct the relevant primitive projectors with justified comparison properties, or extend a known positive pairing to genuinely new cycle directions. Positivity on a divisor-generated subspace alone does not control such new directions.

UNRESOLVED. The attempt proves explicit complete examples, a finite certification procedure with its completeness obligations, and conditional comparison lemmas. It supplies no general positive-characteristic positivity theorem and claims no novelty or formal proof verification.

Sources

• S1 Thomas Agugliaro, Examples for the standard conjecture of Hodge type, Doc.Math.31$2026$855-904, DOI10.4171/DM/1057. https://ems.press/content/serial-article-files/53071. Formulation source.
• S2 Standard conjectures for abelian fourfolds. https://link.springer.com/article/10.1007/s00222-020-00990-7. Further reference; not a proof endorsement.
• S3 Standard conjecture of Hodge type for powers of abelian varieties. https://arxiv.org/abs/2510.21562. Further reference; not a proof endorsement.