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Math-11-A3 · revision r1 · on Math-11

OpenAI claim summary for Math-11

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 032, 2026-09-30): The main theorem supplies the coniveau statement for powers of K3 surfaces satisfying the additional quadratic-lattice embedding, rather than every smooth projective variety.

Manuscript 2 (family 032, 2026-09-30): The CM generalized-Hodge corollary establishes supports only for CM abelian varieties; our coniveau equality quantifies over every smooth projective complex variety.

Manuscript 3 (family 040, 2026-09-24): The pure-Tate cycle-class consequence supplies geometric support on the stated powers and Hilbert schemes of p_g=q=0 surfaces; the universal generalized-Hodge coniveau assertion remains outside this class.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Generalized Hodge Conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Manuscript 3.
  7. Outline of the argument
  8. Formal verification
  9. Status caveat
  10. Sources

Generalized Hodge Conjecture

OpenAI's claim

Manuscript 1 (family 032, 2026-09-30): On the specified quadratic-lattice locus of projective K3 surfaces, all rational Hodge classes on every power are claimed to be algebraic; an exact algebraic Kuga--Satake tensor also gives a conditional criterion beyond that locus. Manuscript 2 (family 032, 2026-09-30): The manuscripts claim algebraicity of rational Hodge classes on CM abelian varieties, with consequences for generalized Hodge classes and finite-field Tate classes. Manuscript 3 (family 040, 2026-09-24): For every smooth complex projective surface with $p_g=0$, OpenAI claims that the Albanese map on integral degree-zero cycles is an isomorphism.

Mathematical statement of the claimed result

Manuscript 1.
For a projective complex K3 surface $S$, write $T_S=\operatorname{NS}(S)_{\mathbb Q}^{\perp}$ with cup-product quadratic form $q$. Assume a rational isometric embedding \[ (T_S,q)\hookrightarrow\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4, \] where $\mathbb U$ is the hyperbolic plane with Gram matrix $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$. Choose any full even-Clifford realization $H^1(A,\mathbb Q)=C^+(T_S,q)$, any $w\in T_S$ with $q(w)\ne0$, and any rational weight-one polarization. The Kuga--Satake map is formed from $v\mapsto(a\mapsto vaw)$ and this polarization. OpenAI asserts existence of $\Gamma\in\operatorname{CH}^2(S\times A^2)_{\mathbb Q}$ inducing this exact prescribed map on $T_S$, also after transporting it to an isogenous realization. It further claims rational Hodge and generalized Hodge conjectures in all degrees of $S^m$ for every $m\ge1$. The quadratic embedding is a hypothesis, so this theorem alone is restricted to that locus.
Manuscript 2.
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 $0\le r\le g/2$.
Manuscript 3.
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 032, 2026-09-30): Clifford multiplication carries exterior powers of the transcendental space into degree-two cohomology of an abelian target. Transposed cycles and a polynomial inverse on the finite-dimensional image recover these summands, so divisor algebraicity supplies the required classes without an inverse Lefschetz correspondence.

Manuscript 2 (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.

Manuscript 3 (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 032 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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