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The Hodge conjecture for CM abelian varieties

Let XX be a smooth projective complex variety. The rational Hodge conjecture asserts that every class in H2p(X,Q)∩Hp,p(X)H^{2p}(X,\mathbb Q)\cap H^{p,p}(X) is a rational linear combination of classes of codimension-pp algebraic subvarieties. For abelian varieties it is known in codimension one (Lefschetz) and in scattered cases; Deligne proved every Hodge class on an abelian variety is absolute Hodge, and Deligne and Andre reduced Hodge classes on CM abelian varieties to Weil-type classes on auxiliary CM varieties, without producing the cycles. Let AA be a complex abelian variety with complex multiplication, so that End(A)⊗Q\mathrm{End}(A)\otimes\mathbb Q contains a commutative semisimple algebra of dimension 2dim⁡A2\dim A. Is every rational Hodge class on AA, in every codimension, algebraic?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry: Hodge theory and algebraic cycles on abelian varieties
Posed by
W. V. D. Hodge (1950 ICM address) for the conjecture; the CM case was singled out through Deligne (1982), Andre (1992) and Milne
Year posed
1950
Years open
76y
Solved
2026-09-10
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
72 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims the rational Hodge conjecture in every codimension for every complex CM abelian variety, and hence for finite products and powers of such varieties. Consequences stated in the same paper: the generalized Hodge conjecture for CM abelian varieties, and via Milne's theorems the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties. It does not prove the Hodge conjecture for arbitrary abelian varieties (a remark gives only algebraicity of specialized Hodge classes on good-reduction special fibers) and says nothing about integral Hodge classes.

What the AI did

The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The README names the proof of the Hodge conjecture for CM abelian varieties as one of two exceptions to that fixed procedure. It does not say what was done differently or whether people intervened, so the three-hour figure does not apply to this result and the human role is undisclosed.

Verification

No independent mathematician has checked this yet. Checked here: the main theorem as stated in the TeX source (surjectivity of the Betti cycle-class map CHp(A)Q→H2p(A,Q)∩Hp,p(A)CH^p(A)_\mathbb Q \to H^{2p}(A,\mathbb Q)\cap H^{p,p}(A) for every complex CM abelian variety and every pp) read against the posed problem; it is the full rational Hodge conjecture for that class. The proof was not refereed. It builds four-factor Hodge classes from nonzero mixed periods of theta one-forms on compact two-ball quotients and needs an arithmetic Frobenius and Hecke argument to identify their CM sources; this is long and new. No Lean formalization is listed for this manuscript. The README marks this result as produced outside the fixed procedure, without saying how.

Sources

Changelog1 change

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