Grothendieck's Standard Conjecture D

Grothendieck's Standard Conjecture D

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's Standard Conjecture D

Definitions and mathematical statement

A Weil cohomology theory is a finite-dimensional graded cohomology theory on smooth projective varieties with cycle classes, products, Poincaré duality, and the standard compatibility axioms. A rational cycle $z$ of codimension $r$ on an $n$-fold is numerically trivial if $\deg(z\cdot w)=0$ for every complementary cycle $w$. It is homologically trivial if $\operatorname{cl}_H(z)=0$. The assertion is \[ \left[\forall w\in\operatorname{CH}^{n-r}(X)_{{\mathbb{Q}}},\ \deg(z\cdot w)=0\right] \quad\Longrightarrow\quad \operatorname{cl}_H(z)=0 \] for every field, available Weil cohomology theory, smooth projective variety, and codimension in the recorded scope.

Formulation and concept references: [S1].

Short English statement

If an algebraic cycle has zero intersection number with every complementary cycle, must its cohomology class vanish?

Research attempt

Current frontier.

The classical assertion in [S1], Conjecture 1.1, is the one retained here. Brown--Walker prove a noncommutative analogue for matrix factorizations of isolated hypersurface singularities in characteristic zero, not the displayed assertion for every variety and Weil theory. The relationship among algebraicity of Frobenius-invariant classes, semisimplicity, and numerical equivalence is developed in Milne's Sections 1 and 5 [E1]. The August 2026 manuscript of Matt Broe reports the Tate conjecture for abelian fourfolds over finite fields and consequences for standard conjectures on abelian fourfolds [S2]; we record its stated scope without treating a recent manuscript as an independently audited general theorem. In particular the 2025 paper [S3] assumes $D$ in some of its principal implications; those implications cannot be read backwards to prove $D$.

Attack route.

Three routes suggest themselves. One can construct explicit complementary cycles whose intersection matrix detects all algebraic cohomology; one can try to make the Poincaré dual of an algebraic class algebraic; or, over a finite field, one can isolate the Frobenius-invariant part and use Tate surjectivity. The missing inputs are respectively a spanning certificate, algebraicity of the needed duality operation, and both the appropriate Tate assertion and control of the eigenvalue-one Jordan blocks. The calculations below combine the first and third routes and show exactly why ambient duality alone is insufficient.

Attempt: a finite certificate and an algebraic projector.

Fix a Weil theory with coefficient field $E$, a smooth projective $n$-fold $X$, and codimension $r$. Twists are suppressed in the following notation. Choose rational cycles \[ z_1,\ldots,z_m\in\operatorname{CH}^r(X)_{{\mathbb{Q}}},\qquad w_1,\ldots,w_m\in\operatorname{CH}^{n-r}(X)_{{\mathbb{Q}}} \] and form the rational matrix $M_{ji}=\deg(z_iw_j)$. Suppose $\det M\ne0$. For every available Weil theory the classes $u_i=\operatorname{cl}(z_i)$ are $E$-linearly independent: a relation $\sum_i c_i u_i=0$ pairs with the $w_j$ to give $Mc=0$. This argument also proves independence of the complementary classes. No comparison isomorphism between cohomology theories is used.

There is more structure than just independence. Define the degree-zero correspondence \[ \pi=\sum_{i,j=1}^m(M^{-1})_{ij}\,[w_j\times z_i] \in\operatorname{CH}^{n}(X\times X)_{{\mathbb{Q}}}. \tag{44.1} \] It acts on degree $2r$ cohomology by \[ P(v)=\sum_{i,j}(M^{-1})_{ij}u_i\langle v,\operatorname{cl}(w_j)\rangle. \tag{44.2} \] In particular $P(u_k)=u_k$, and $P^2=P$. In fact idempotence holds already for correspondences modulo rational equivalence: composing $w_j\times z_i$ after $w_l\times z_k$ multiplies $w_l\times z_i$ by $\deg(z_kw_j)=M_{jk}$, and the coefficients contract to $M^{-1}MM^{-1}=M^{-1}$. This is a direct calculation using the intersection product and proper pushforward.

For any rational cycle $z$ define \[ z^\perp=z-\sum_{i,j}(M^{-1})_{ij}\deg(zw_j)z_i. \tag{44.3} \] It has zero intersection with all the chosen $w_j$. If $z$ is numerically trivial then $z^\perp=z$. If, in addition, \[ \operatorname{span}_E\{\operatorname{cl}(z):z\in \operatorname{CH}^r(X)_{{\mathbb{Q}}}\} =\operatorname{span}_E\{u_1,\ldots,u_m\}, \tag{44.4} \] then $P$ is the identity on every algebraic class. A numerically trivial $z$ has $P\operatorname{cl}(z)=0$ by $44.2$, hence $\operatorname{cl}(z)=0$. Thus $44.1$--$44.4$ give a finite conditional certificate for $D$ in the specified codimension and cohomology theory.

A useful strengthened certificate avoids checking $44.4$: if $m=\dim_E H^{2r}(X)$, the independent classes are automatically a basis of the whole cohomology group. This condition is much stronger than $D$, but is verifiable in some examples. For $X=({\mathbb{P}}^1)^n$, let $h_i$ be the hyperplane class from factor $i$. The monomials $h_I=\prod_{i\in I}h_i$, $|I|=r$, are a cohomology basis by the projective-space and Künneth axioms. Pair them with $h_{J^c}$, $|J|=r$. Since $h_i^2=0$ and $\deg(h_1\cdots h_n)=1$, the pairing matrix is exactly the identity. Consequently this familiar family satisfies $D$ for every available Weil theory. The accompanying script checks the subset pairing, but the argument proves it for every $n$.

Attempt: the eigenvalue-one obstruction over a finite field.

Now specialize, explicitly, to $X/{\mathbb{F}}_q$ and $\ell$-adic cohomology, $\ell\nmid q$. Put \[ V=H^{2r}(X_{\overline{{\mathbb{F}}}_q},{\mathbb{Q}}_\ell(r)),\qquad W=H^{2n-2r}(X_{\overline{{\mathbb{F}}}_q},{\mathbb{Q}}_\ell(n-r)). \] Their pairing $V\times W\to{\mathbb{Q}}_\ell$ is perfect, and Frobenius acts by mutually dual operators. Rational cycles defined over ${\mathbb{F}}_q$ give Frobenius-fixed classes. The Tate formulations and their relation to $D$ in this setting are discussed in [E1]; the elementary implication below is proved here with its hypotheses displayed.

Assume the following two statements for this $X$ and $r$: \[ V=\ker(F-1)\oplus\operatorname{im}(F-1), \tag{44.5} \] \[ W^{F=1}=\operatorname{span}_{{\mathbb{Q}}_\ell} \{\operatorname{cl}(w):w\in\operatorname{CH}^{n-r}(X)_{{\mathbb{Q}}}\}. \tag{44.6} \] The first is semisimplicity at the eigenvalue one, not necessarily semisimplicity at every eigenvalue. To verify this terminology, the generalized eigenvalue-one space has no nontrivial Jordan block exactly when its kernel has zero intersection with its image; rank--nullity then gives $44.5$. The second is complementary-codimension Tate surjectivity over the actual ground field.

For a perfect dual pairing, \[ W^{F=1}=(\operatorname{im}(F-1)\subseteq V)^\perp. \tag{44.7} \] Indeed $\langle(F-1)v,w\rangle =\langle v,(F^{-1}-1)w\rangle$; vanishing for all $v$ is equivalent to $Fw=w$. Let $0\ne v\in V^{F=1}$. By $44.5$ it is not in $\operatorname{im}(F-1)$. Perfect duality therefore supplies $w\in W^{F=1}$ with $\langle v,w\rangle\ne0$. By $44.6$ this $w$ is a finite ${\mathbb{Q}}_\ell$-linear combination of rational cycle classes. At least one of those rational cycles pairs nontrivially with $v$. Thus a numerically trivial rational cycle cannot have class $v\ne0$. This proves $D$ for this finite-field, $\ell$-adic instance under $44.5$--$44.6$, with no assumption that ${\mathbb{Q}}_\ell$ coefficients are rational.

Stress test: why the missing hypotheses matter.

Take $V={\mathbb{Q}}^2$ with $F=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$ and $W=V^*$ with Frobenius $(F^{-1})^t$. Then \[ V^{F=1}={\mathbb{Q}} e_1,\qquad W^{F=1}={\mathbb{Q}} e_2^*,\qquad \langle e_1,e_2^*\rangle=0. \tag{44.8} \] Both fixed spaces are nonzero, yet their restricted pairing is identically zero; $e_1$ is also in $\operatorname{im}(F-1)$. Calling all fixed vectors "algebraic" in this linear-algebra model would satisfy formal spanning assertions on both sides and would still not prove $D$. This is not a geometric counterexample. It proves that Poincaré duality plus the two spanning assertions, without further geometric input or semisimplicity, is not a valid linear-algebra proof.

An even simpler test uses the nondegenerate symmetric matrix $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ and restricts it to ${\mathbb{Q}} e_1$: the restriction is zero. Thus replacing a complementary algebraic class by an arbitrary cohomological dual loses the key condition. Conversely, an invertible intersection matrix certifies only the span it actually detects. It does not certify that all algebraic classes lie in that span. The coefficient field can have infinite degree over ${\mathbb{Q}}$, so no unproved assertion that rational cycle images have a finite ${\mathbb{Q}}$-basis is used in $44.4$.

The finite-field argument uses another major conjectural input, complementary Tate surjectivity, closely related to UnsolvedMath problem 30006680 of the collection. It does not establish the target over arbitrary fields or for arbitrary Weil theories. Numerical invariance under changing cohomology theories is exactly what one may not assume in order to prove the universal statement.

Outcome.

Outcome: Conditional reduction. Explicit nondegenerate intersection data construct an idempotent algebraic correspondence and certify $D$ provided the detected classes exhaust the algebraic part. Over finite fields, complementary Tate surjectivity and semisimplicity at eigenvalue one supply another sufficient criterion, proved above. The Jordan example identifies a precise failure in a shortcut that omits semisimplicity. These are auxiliary deductions and standard-type conditional implications, not claims of historical novelty. No argument here establishes their missing geometric hypotheses in the full scope of the original assertion.

Sources

• S1 Michael K. Brown and Mark E. Walker, Standard Conjecture D for matrix factorizations, classical Conjecture1.1 and surrounding introduction. https://arxiv.org/pdf/1905.04626. Catalogue formulation source.
• S2 The Tate conjecture for abelian fourfolds over finite fields. https://arxiv.org/abs/2608.28651. Catalogue research/status reference; not a proof endorsement.
• S3 Tate's question, Standard conjecture D, semisimplicity and Dynamical degree comparison conjecture. https://arxiv.org/abs/2503.09432. Catalogue research/status reference; not a proof endorsement.
• E1 James S. Milne, The Tate conjecture over finite fields (AIM talk), arXiv:0709.3040v2, Sections 1 (the Tate formulations) and 5 (equality of equivalence relations). https://arxiv.org/html/0709.3040v2. Used for the distinction between algebraicity, numerical equivalence and Frobenius semisimplicity; the implication needed here is also proved in the text. Consulted 27 September 2026.