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Math-2552 · mathematics · Algebraic geometry

Hodge support on a specified K3 quadratic locus

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let S be a smooth projective complex K3 surface whose transcendental quadratic space, with its cup form, embeds isometrically over Q in $U_Q\oplus U_Q\oplus \langle -1\rangle ^{4}$. For every $m\geq 1$, are all rational (p,p) classes on $S^m$ algebraic, and is every rational Hodge substructure with Hodge coniveau at least r supported on an algebraic subset of codimension at least r? U denotes the hyperbolic plane.

Short English statement

Let S be a smooth projective complex K3 surface whose transcendental quadratic space, with its cup form, embeds isometrically over Q in $U_Q\oplus U_Q\oplus \langle -1\rangle ^{4}$. For every $m\geq 1$, are all rational (p,p) classes on $S^m$ algebraic, and is every rational Hodge substructure with Hodge coniveau at least r supported on an algebraic subset of codimension at least r? U denotes the hyperbolic plane.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2552-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2552

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 support on a specified K3 quadratic locus

OpenAI's claim

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

Mathematical statement of the claimed result

Mathematical claim 1.
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.

Outline of the argument

Manuscript 1 (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.

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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