HEXAGONbeta

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 2 / 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, that the real multiplications by 6\sqrt6 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, and that the Hodge conjecture holds on every power and every Hilbert scheme of points of the K3 surfaces of the one-dimensional Fricke families of levels 55 and 77, which carry complex multiplication by $\QQ(\sqrt{-5})$ and $\QQ(\sqrt{-7})$. 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; among the steps, every convolution of the Weil line bundles on E08E_{0}^{8} whose shifts lie in seven consecutive integers fails the polarised criterion in eighteen of the twenty-eight arrangements of the shifts, an explicit curve of genus four makes the Prym map of cyclic triple covers \'etale onto the Weil domain of $\QQ(\sqrt{-3})$, and the conjecture for a class is equivalent to the vanishing of the mass gap of a multiple of it plus a complete intersection, which Koll\'ar's hypersurfaces show cannot be dropped for integral classes. 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 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 (sixty-two theorems, fifty-two free of every axiom, two on propext and Quot.sound, eight on propext). No proved statement rests on a floating-point computation; the enumerations (Proposition VI.13, Remark VI.25) and the linear-programming bounds of Remark VI.39 are stated as computations of the archive. The AI did not prove (F2), (F3') or the Hodge conjecture; the paper states at which step each construction stops. Both authors approved the final version. Code and Lean: release v2.7.0, https://github.com/creelie/fermat-hodge-/releases/tag/v2.7.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 17, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 22, 2026