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Math-2526 · mathematics · Algebraic geometry

Algebraicity of the prescribed Kuga–Satake map

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let S be any smooth projective complex K3 surface, with rational transcendental cohomology T(S). Choose standard Kuga–Satake Clifford, polarization and realization data, giving an abelian variety A and its prescribed rational Hodge map $\kappa :T(S)\to H^{2}(A×A,Q)$. Is $\kappa $ induced on T(S) by a rational codimension-two cycle on S×A×A? The assertion includes an isogenous realization with the map transported by its chosen rational Hodge isomorphism.

Short English statement

Let S be any smooth projective complex K3 surface, with rational transcendental cohomology T(S). Choose standard Kuga–Satake Clifford, polarization and realization data, giving an abelian variety A and its prescribed rational Hodge map $\kappa :T(S)\to H^{2}(A×A,Q)$. Is $\kappa $ induced on T(S) by a rational codimension-two cycle on S×A×A? The assertion includes an isogenous realization with the map transported by its chosen rational Hodge isomorphism.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2526-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2526

Manuscript 1 (family 032, 2026-10-03): 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 the prescribed Kuga–Satake map

OpenAI's claim

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. 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 $S$ 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.

Outline of the argument

Manuscript 1 (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.

Formal verification

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

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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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