OpenAI claim summary for Math-2553

Hodge classes on generic abelian-cover Jacobian powers

OpenAI's claim

For tensor-Hodge-generic abelian-cover Jacobians and the specified very general diagonal complete intersections, OpenAI claims the rational Hodge assertion on every self-power. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

Fix $n\ge2$, $d\ge0$, $c\in\{0,1,2\}$ and $m=d+c+1$. For a rank-$c$ complex matrix $A=(a_{\alpha i})$ with every maximal coordinate minor nonzero, define the smooth diagonal complete intersection \[ X_A=\{[x_1:\cdots:x_m]\in\mathbb P^{m-1}:\ \sum_i a_{\alpha i}x_i^n=0\quad(1\le\alpha\le c)\}. \] OpenAI asserts surjectivity of $\operatorname{CH}^p(X_A^u)_{\mathbb Q}\to H^{2p}(X_A^u,\mathbb Q)\cap H^{p,p}(X_A^u)$ for every $u\ge1$ and every $p$, outside a union, indexed by a countable set, of proper algebraic closed subsets of this smooth parameter space. The exceptional set is unnecessary when $c\le1$ or $d=0$. A second assertion concerns a fixed connected finite abelian topological cover of a compact oriented surface with specified branch monodromy, allowing unbranched and trivial covers. Vary the full marked complex structure and branch points. At any parameter $s$ for which each rational tensor of type $(a,a)$ in $H^1(C_s,\mathbb Q)^{\otimes 2a}$ keeps its Hodge type at every point of the marked family, the same cycle-class surjectivity holds on \[ (\operatorname{Jac}(C_s)\times M)^N\quad(N\ge1),\qquad C_s^a\times M^b\quad(a,b\ge0), \] for every complex CM abelian variety $M$. CM means that its degree-one Hodge group is a torus. This is a Hodge-generic assertion, with the generic locus chosen independently of $M$.

Outline of the argument

Manuscript 1 (family 032, 2026-09-30): Deck-character projections and equivariant period calculations identify the necessary monodromy invariants of curve covers. Correspondences dominated by CM sources supply determinant tensors, while Gale duality transfers the character-level construction to diagonal complete intersections with the required supports and degree shifts.

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

• OpenAI, Abelian covers, Gale correspondences, and the Hodge conjecture for powers, 2026-09-30; citation key OAI:Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: p4:abelian-covers.
• OpenAI catalogue entry.