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Dense Algebraic Loci of Hodge Classes

A mix of human-written and AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometrymath.KT — K-Theory and Homologymath.NT — Number Theorymath.RT — Representation Theory

Contributed by Deep Bhattacharjee ↗, Ushashi Bhattacharya ↗

Version 1 / Oct 17, 2026 / CC BY 4.0

Abstract

We prove unconditionally that the Hodge conjecture is equivalent to two statements about abelian varieties, that the locus on which either statement holds is dense in every family and is either the whole family or a meagre set, and that the real multiplications by 5\sqrt5, 6\sqrt6, 7\sqrt7 and by the cubic fields $\QQ(\zeta_{7})^{+}$ and $\QQ(\zeta_{9})^{+}$ are algebraic on explicit families of K3 surfaces, the cubic ones on the surfaces of Mumford type whose quaternion algebra is ramified at two real places. For each of the two statements we build the object it needs next, a complex on a product of two Weil fourfolds and a class on an abelian fourteenfold, prove what it does, and prove at which step it stops. Both statements remain open.

Provenance statement

The constructions on the two statements (F2) and (F3'), the search for a closure of the Hodge conjecture, the text, the figures, the Lean 4 file and the Python, Julia, C, C++ and shell programs were carried out by Deep Bhattacharjee with the help of AI assistance (Claude, Anthropic) in the Claude Code environment. Every hand proof was read against the cited literature; every finite quantity in the paper is recomputed by programs in at least two languages, and the combinatorial and arithmetic ones are proved in Lean 4 (forty-seven theorems, forty-five free of every axiom). No statement rests on a floating-point computation. The AI did not prove (F2), (F3') or the Hodge conjecture; the paper states at which step each construction stops. Three slips found by the machine checks were corrected before release (a Hasse invariant at 2, a Picard number of a twist family, the dimension of a dihedral Hecke algebra). Both authors approved the final version. Tools used: Anthropic - Claude (Claude Code); Lean community - Lean 4 (no Mathlib); Python; Julia; C; C++; shell. arXiv identifier: none (not posted on arXiv). Formalization: https://github.com/creelie/fermat-hodge-/blob/main/lean/HodgeClosure.lean (archived at Zenodo, release v1.0.0: https://doi.org/10.5281/zenodo.23439578)
AI provenance filesAI_PROVENANCE_HodgeConjecture.md · 3 KB

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 17, 2026