Grothendieck's Hodge standard conjecture for abelian varieties
Let be an abelian variety of dimension over an algebraically closed field , an ample class, and the primitive part of the space of codimension- cycles modulo numerical equivalence, . Grothendieck's Hodge standard conjecture asserts that the form is positive definite on . In characteristic zero it follows from Hodge theory; in positive characteristic it is open beyond surfaces and some low-dimensional cases. Milne (2002) showed that the Hodge conjecture for CM abelian varieties would imply it for abelian varieties. Does the Hodge standard conjecture hold for all abelian varieties in every characteristic?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Algebraic geometry: standard conjectures in positive characteristic
- Posed by
- Alexander Grothendieck (standard conjectures); the abelian-variety reduction is due to Milne (2002)
- Year posed
- —
- Years open
- —
- 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
- 50 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims the Hodge standard conjecture for l-adic etale cohomology for all abelian varieties in arbitrary characteristic, deduced from the paper's CM Hodge theorem via Milne 2002. It does not address the standard conjectures for other varieties, the Lefschetz standard conjecture beyond what the abelian case already gives, or crystalline cohomology.
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. This entry is a corollary in that same manuscript, obtained through Milne's 2002 theorem.
Verification
No independent mathematician has checked this yet. Checked here: Corollary (Hodge standard conjecture for abelian varieties) in the TeX source, read against the posed problem; it states positivity of the primitive intersection form for every abelian variety over every algebraically closed field and every ample class, and coincidence of numerical and l-adic homological equivalence. The deduction applies Milne's Theorem 3.3 (2002) with his addendum to the paper's CM Hodge theorem, so it depends on that unrefereed theorem. No Lean formalization is listed.