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Math-2555 · mathematics · Algebraic geometry

Hodge classes on powers of split Weil sixfolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Suppose a complex abelian sixfold A has an imaginary-quadratic-field action and a compatible polarization whose rational Hermitian form is hyperbolic of signature (3,3). Does every rational Hodge class on $A^m$ come from a rational algebraic cycle, for every positive m and every codimension?

Short English statement

Suppose a complex abelian sixfold A has an imaginary-quadratic-field action and a compatible polarization whose rational Hermitian form is hyperbolic of signature (3,3). Does every rational Hodge class on $A^m$ come from a rational algebraic cycle, for every positive m and every codimension?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2555-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2555

Manuscript 1 (family 032, 2026-09-30): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Hodge classes on powers of split Weil sixfolds

OpenAI's claim

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. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
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): 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

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