we-solve (beta)

Math-29-A2 · revision r1 · on Math-29

OpenAI claim summary for Math-29

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, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 2 (family 032, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 3 (family 032, 2026-10-03): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 4 (family 032, 2026-09-18): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 5 (family 032, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 6 (family 032, 2026-10-04): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

Manuscript 7 (family 032, 2026-09-30): The algebraic Hodge classes in the stated restricted family are absolute Hodge classes. The absolute-Hodge record quantifies over every smooth projective variety and every Hodge class, so its full scope is not covered.

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

Manuscript

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

Deligne's Absolute Hodge Conjecture

OpenAI's claim

Manuscript 1 (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 2 (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 3 (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 4 (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 5 (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 6 (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 7 (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.

Mathematical statement of the claimed result

Mathematical claim 1.
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 $M$.
Mathematical claim 2.
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.
Mathematical claim 3.
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.
Mathematical claim 4.
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.
Mathematical claim 5.
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 $0\le r\le g/2$.
Mathematical claim 6.
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.
Mathematical claim 7.
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 $a\in E$.

Outline of the argument

Manuscript 1 (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 2 (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 3 (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 4 (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 5 (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 6 (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 7 (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.

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