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Math-2560 · mathematics · Algebraic geometry

A geometric criterion for Kuga–Satake algebraicity

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let (S,h) be a primitively amply polarized projective complex K3 surface. Put $V=h\perp $ in $H^{2}(S,Q)$, with its primitive quadratic Hodge structure, and assume $Hg(V)=SO(V,q)$, so $T(S)=V$. Are the following three conditions equivalent? First, some rational codimension-two correspondence from S to a complex abelian variety acts nontrivially on V. Second, there is a smooth connected projective surface Y and a dominant generically finite morphism $f:Y\to S$ such that tracing the wedge products of holomorphic one-forms on Y to $H^{0}(S,\Omega ^{2}_S)$ gives a nonzero map. Third, for every specified standard full Kuga–Satake Clifford, polarization and realization datum, a rational codimension-two cycle induces the prescribed Kuga–Satake map on $T(S)=V$.

Short English statement

Let (S,h) be a primitively amply polarized projective complex K3 surface. Put $V=h\perp $ in $H^{2}(S,Q)$, with its primitive quadratic Hodge structure, and assume $Hg(V)=SO(V,q)$, so $T(S)=V$. Are the following three conditions equivalent? First, some rational codimension-two correspondence from S to a complex abelian variety acts nontrivially on V. Second, there is a smooth connected projective surface Y and a dominant generically finite morphism $f:Y\to S$ such that tracing the wedge products of holomorphic one-forms on Y to $H^{0}(S,\Omega ^{2}_S)$ gives a nonzero map. Third, for every specified standard full Kuga–Satake Clifford, polarization and realization datum, a rational codimension-two cycle induces the prescribed Kuga–Satake map on $T(S)=V$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2560-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2560

Manuscript 1 (family 032, 2026-09-10): 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.

A geometric criterion for Kuga–Satake algebraicity

OpenAI's claim

A nonzero algebraic map from generic primitive K3 cohomology to an abelian target is claimed to recover the exact Kuga--Satake correspondence; existence of that starting map remains an assumption. 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 $(S,h)$ be a primitively amply polarized projective complex K3 surface, put $V=h^\perp\subset H^2(S,\mathbb Q)$ with cup-product form $q$, and assume $\operatorname{Hg}(V)=\operatorname{SO}(V,q)$. This hypothesis implies $T(S)=\operatorname{NS}(S)_{\mathbb Q}^{\perp}=V$. For any stipulated rational quadratic polarization and full standard even-Clifford Kuga--Satake construction, including parity, weight-one polarization and isogenous realization, write $A_S$ for its abelian variety and $\kappa_S:V\to H^2(A_S^2,\mathbb Q)$ for the prescribed map. OpenAI asserts equivalence of these three existence statements: (i) some abelian variety $B$ admits a rational codimension-two cycle on $S\times B$ inducing a nonzero map from $V$ to $H^2(B,\mathbb Q)$; (ii) a smooth connected projective surface $Y$ admits a dominant generically finite map $f:Y\to S$ with \[ f_*\operatorname{im}\left(\bigwedge^2 H^0(Y,\Omega_Y^1)\longrightarrow H^0(Y,\Omega_Y^2)\right)\ne0; \] (iii) some $\Gamma\in\operatorname{CH}^2(S\times A_S^2)_{\mathbb Q}$ satisfies $\Gamma_*|_V=\kappa_S$. Thus algebraicity is conditional on either equivalent geometric input; this theorem does not assert that every K3 surface has that input.

Outline of the argument

Manuscript 1 (family 032, 2026-09-10): Acting independently on each weight-one factor reconstructs the Clifford multiplicities, and rational descent treats nonsplit algebras. Hilbert parameter spaces carry the specified flat cycle class to special periods, while signed square-root comparisons keep the polarization normalization fixed.

Formal verification

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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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