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Math-4-A6 · revision r1 · on Math-4

OpenAI claim summary for Math-4

Literature report · LLM-generated · unknown ·

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Claim: literature report

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

Manuscript 1 (family 032, 2026-09-10): A conditional geometric criterion for algebraicity of the Kuga–Satake tensor; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 2 (family 032, 2026-09-30): Hodge classes on powers of specified Hodge-generic abelian-cover Jacobians and diagonal complete intersections; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 3 (family 032, 2026-09-30): Hodge and generalized Hodge for powers of K3 surfaces whose transcendental lattice has the specified quadratic embedding; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 4 (family 032, 2026-10-03): Algebraicity of the prescribed Kuga–Satake tensor for every projective complex K3 surface; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 5 (family 032, 2026-09-18): Algebraicity of Weil classes for split Weil-type abelian eightfolds; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 6 (family 032, 2026-09-30): Rational Hodge for complex CM abelian varieties, with the stated CM generalized-Hodge and finite-field abelian consequences; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 7 (family 032, 2026-10-04): Rational cycle-class surjectivity on every product of projective complex K3 surfaces; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 8 (family 032, 2026-09-30): Hodge classes on powers of split Weil sixfolds and quadratic-action abelian varieties of dimension at most five; this problem quantify over every smooth projective complex variety, so the restricted object class or conditional tensor criterion leaves the general Hodge assertion open.

Manuscript 9 (family 040, 2026-09-24): The pure-Tate motive corollary proves Hodge for powers and Hilbert schemes of complex surfaces with p_g=q=0. It leaves the universal smooth projective Hodge statement untouched.

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

Manuscript

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

The Hodge Conjecture

OpenAI's claim

Manuscript 1 (family 032, 2026-09-10): A nonzero algebraic map from generic primitive K3 cohomology to an abelian target is claimed to recover the exact Kuga--Satake correspondence; existence of that starting map remains an assumption; Manuscript 2 (family 032, 2026-09-30): For tensor-Hodge-generic abelian-cover Jacobians and the specified very general diagonal complete intersections, OpenAI claims the rational Hodge assertion on every self-power; Manuscript 3 (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 4 (family 032, 2026-10-03): For every projective complex K3 surface, the precisely normalized Kuga--Satake embedding into the full even-Clifford abelian target is claimed to come from an algebraic cycle; Manuscript 5 (family 032, 2026-09-18): Every rational Weil class of a split eight-dimensional abelian variety is claimed to have an algebraic representative, uniformly over imaginary-quadratic fields and compatible polarizations; Manuscript 6 (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 7 (family 032, 2026-10-04): Rational Hodge classes on any finite product of projective complex K3 surfaces are claimed to be algebraic, even when the factors have unrelated periods or extra endomorphisms; Manuscript 8 (family 032, 2026-09-30): Algebraicity is claimed for Hodge classes on powers of split Weil sixfolds and of abelian varieties of dimension at most five carrying an imaginary-quadratic action, including special periods; Manuscript 9 (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.
Let $(S,h)$ be a primitively amply polarized projective complex K3 surface, put $V=h^\perp\subset H^2(S,\mathbb Q)$ with cup-product form $q$, and assume $\operatorname{Hg}(V)=\operatorname{SO}(V,q)$. This hypothesis implies $T(S)=\operatorname{NS}(S)_{\mathbb Q}^{\perp}=V$. For any stipulated rational quadratic polarization and full standard even-Clifford Kuga--Satake construction, including parity, weight-one polarization and isogenous realization, write $A_S$ for its abelian variety and $\kappa_S:V\to H^2(A_S^2,\mathbb Q)$ for the prescribed map. OpenAI asserts equivalence of these three existence statements: (i) some abelian variety $B$ admits a rational codimension-two cycle on $S\times B$ inducing a nonzero map from $V$ to $H^2(B,\mathbb Q)$; (ii) a smooth connected projective surface $Y$ admits a dominant generically finite map $f:Y\to S$ with \[ f_*\operatorname{im}\left(\bigwedge^2 H^0(Y,\Omega_Y^1)\longrightarrow H^0(Y,\Omega_Y^2)\right)\ne0; \] (iii) some $\Gamma\in\operatorname{CH}^2(S\times A_S^2)_{\mathbb Q}$ satisfies $\Gamma_*|_V=\kappa_S$. Thus algebraicity is conditional on either equivalent geometric input; this theorem does not assert that every K3 surface has that input.
Manuscript 2.
Fix $n\ge2$, $d\ge0$, $c\in\{0,1,2\}$ and $m=d+c+1$. For a rank-$c$ complex matrix $A=(a_{\alpha i})$ with every maximal coordinate minor nonzero, define the smooth diagonal complete intersection \[ X_A=\{[x_1:\cdots:x_m]\in\mathbb P^{m-1}:\ \sum_i a_{\alpha i}x_i^n=0\quad(1\le\alpha\le c)\}. \] OpenAI asserts surjectivity of $\operatorname{CH}^p(X_A^u)_{\mathbb Q}\to H^{2p}(X_A^u,\mathbb Q)\cap H^{p,p}(X_A^u)$ for every $u\ge1$ and every $p$, outside a union, indexed by a countable set, of proper algebraic closed subsets of this smooth parameter space. The exceptional set is unnecessary when $c\le1$ or $d=0$. A second assertion concerns a fixed connected finite abelian topological cover of a compact oriented surface with specified branch monodromy, allowing unbranched and trivial covers. Vary the full marked complex structure and branch points. At any parameter $s$ for which each rational tensor of type $(a,a)$ in $H^1(C_s,\mathbb Q)^{\otimes 2a}$ keeps its Hodge type at every point of the marked family, the same cycle-class surjectivity holds on \[ (\operatorname{Jac}(C_s)\times M)^N\quad(N\ge1),\qquad C_s^a\times M^b\quad(a,b\ge0), \] for every complex CM abelian variety $M$. CM means that its degree-one Hodge group is a torus. This is a Hodge-generic assertion, with the generic locus chosen independently of $M$.
Manuscript 3.
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 4.
Let $S$ be any smooth projective complex K3 surface and let $T(S)=\operatorname{NS}(S)_{\mathbb Q}^{\perp}$ in $H^2(S,\mathbb Q)$. Fix the full even-Clifford Kuga--Satake construction, including the rational quadratic polarization, parity identification and rational weight-one polarization, with associated abelian variety $A_S$. These choices specify $\kappa_S:T(S)\to H^2(A_S^2,\mathbb Q)$. OpenAI claims \[ \exists\Gamma_S\in\operatorname{CH}^2(S\times A_S^2)_{\mathbb Q}\quad (p_{A_S^2})_*\left(\operatorname{cl}(\Gamma_S)\smile p_S^*x\right)=\kappa_S(x) \quad\text{for all }x\in T(S). \] Here $\operatorname{CH}^2$ is the group of rational codimension-two cycles modulo rational equivalence and $p_S,p_{A_S^2}$ are the projections. The assertion includes every isogenous realization with the prescribed map transported by its rational Hodge isomorphism.
Manuscript 5.
Let $K=\mathbb Q(\sqrt{-d})$ be imaginary quadratic, $A$ a complex abelian variety of dimension eight with a $K$-action, and $\lambda$ a compatible polarization of Weil type, with eigenspace multiplicities $(4,4)$. Put $V=H_1(A,\mathbb Q)$, let $\psi$ be the rational Riemann form and $J$ multiplication by $\sqrt{-d}$. Assume the associated Hermitian form \[ h(x,y)=\psi(Jx,y)+\sqrt{-d}\,\psi(x,y) \] vanishes on a four-dimensional $K$-subspace of $V$ (the split condition). Define the rational Weil space $W_K(A)=\bigwedge_K^8H^1(A,\mathbb Q)\subset H^8(A,\mathbb Q)$ using the two conjugate $K$-eigenspace exterior powers. OpenAI claims \[ W_K(A)\subset\operatorname{im}\left(\operatorname{CH}^4(A)_{\mathbb Q}\xrightarrow{\operatorname{cl}}H^8(A,\mathbb Q)\right). \] The field and compatible polarization type are arbitrary; extra endomorphisms are allowed. The statement concerns these Weil classes, rather than all Hodge classes on all varieties.
Manuscript 6.
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 7.
Choose any integer $m\ge1$ and any projective complex K3 surfaces $S_1,\ldots,S_m$, allowing repeated factors. For $X=\prod_{j=1}^m S_j$, OpenAI claims \[ \operatorname{CH}^p(X)_{\mathbb Q}\xrightarrow{\operatorname{cl}} H^{2p}(X,\mathbb Q)\cap H^{p,p}(X) \quad\text{is surjective for every }0\le p\le2m. \] There are no Picard-number, period or endomorphism-field restrictions. The quantified varieties are products of K3 surfaces, so this statement has smaller scope than the Hodge conjecture for arbitrary smooth projective complex varieties.
Manuscript 8.
Let $E=\mathbb Q(u)$ be imaginary quadratic with $u^2=-d$, $d>0$, and let $A$ be a complex abelian variety with a unital $E$-action. OpenAI claims rational cycle-class surjectivity \[ \operatorname{CH}^p(A^m)_{\mathbb Q}\longrightarrow H^{2p}(A^m,\mathbb Q)\cap H^{p,p}(A^m) \quad(m\ge1,\ p\ge0) \] in either of these two cases: $\dim A\le5$; or $\dim A=6$ and the rational Hermitian form associated to a compatible polarization is hyperbolic with signature $(3,3)$. In the second case, with rational Riemann form $e$, this means $h_E(x,y)=e(x,uy)+u e(x,y)$ on $H_1(A,\mathbb Q)$ has a totally isotropic three-dimensional $E$-subspace. Compatibility means $e(ax,y)=e(x,\bar a y)$ for all $a\in E$.
Manuscript 9.
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, 2026-09-10): Acting independently on each weight-one factor reconstructs the Clifford multiplicities, and rational descent treats nonsplit algebras. Hilbert parameter spaces carry the specified flat cycle class to special periods, while signed square-root comparisons keep the polarization normalization fixed. Manuscript 2 (family 032, 2026-09-30): Deck-character projections and equivariant period calculations identify the necessary monodromy invariants of curve covers. Correspondences dominated by CM sources supply determinant tensors, while Gale duality transfers the character-level construction to diagonal complete intersections with the required supports and degree shifts. Manuscript 3 (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 4 (family 032, 2026-10-03): The construction engineers a Lagrangian whose second cohomology has a nineteen-dimensional deficit, yielding a corresponding bound on self-extensions after mirror realization. The resulting characteristic class is transported through K3 moduli by semiregularity and specialization, retaining the prescribed Clifford tensor. Manuscript 5 (family 032, 2026-09-18): An embedded spin Lagrangian detects the exceptional middle-cohomology component and makes two homological obstruction maps injective. Finite-cutoff weighted Floer operations are assembled into algebraic module identities; cancellation in alternating specialization then transports the desired Chern character to an algebraic cycle. Manuscript 6 (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 7 (family 032, 2026-10-04): A Lagrangian immersion is used to produce prescribed Euler pairings while bounding the second self-extension space of the associated complex. Clifford-algebra calculations then identify its transcendental cohomology component; algebraic Kuga--Satake tensors and their contractions supply the classes needed on products. Manuscript 8 (family 032, 2026-09-30): A coherent seed and its Fourier--Mukai transform produce the split-sixfold Weil plane through an obstruction calculation and deformation. A determinant description of the actual Hodge group, combined with the CM result, extends these algebraic generators to powers and to the stated low-dimensional cases. Manuscript 9 (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