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Math-2527 · mathematics · Algebraic geometry

Algebraicity of split Weil-type eightfold classes

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every imaginary quadratic field K and complex abelian eightfold A of split Weil type, with K acting with multiplicities (4,4) and a compatible split Hermitian polarization, is the Weil-class subspace $\wedge ^{8}_K$ $H^{1}(A,Q)\subset H^{8}(A,Q)$ contained in the image of $CH^{4}(A)\otimes Q$? All compatible polarization types and members with extra endomorphisms are included.

Short English statement

For every imaginary quadratic field K and complex abelian eightfold A of split Weil type, with K acting with multiplicities (4,4) and a compatible split Hermitian polarization, is the Weil-class subspace $\wedge ^{8}_K$ $H^{1}(A,Q)\subset H^{8}(A,Q)$ contained in the image of $CH^{4}(A)\otimes Q$? All compatible polarization types and members with extra endomorphisms are included.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2527-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2527

Manuscript 1 (family 032, 2026-09-18): 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.

Algebraicity of split Weil-type eightfold classes

OpenAI's claim

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. 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 $K=\mathbb Q(\sqrt{-d})$ 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.

Outline of the argument

Manuscript 1 (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.

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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