Algebraic and complex geometry·OpenAI Math·Result 050·Lean formalisation listed
A counterexample to Griffiths’ positivity conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebraic and complex geometry: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a counterexample to Griffiths’ positivity conjecture (algebraic and complex geometry) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More, is on YouTube.
Chapter 41 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 13:53 · watch from there on YouTube →
What OpenAI claims
Constructs ample rank-two bundles on ℙ1×ℙ1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface. (Lean)
Paper
Ample rank-two bundles on the quadric surface without Griffiths-positive metrics
OpenAI's abstract We give counterexamples to Griffiths' conjecture in rank two on the quadric surface ℙ1×ℙ1. We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with 𝒪(1,1), are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Constructs ample rank-two bundles on ℙ1×ℙ1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface. (Lean)
- Where is the proof?
- In the paper, Ample rank-two bundles on the quadric surface without Griffiths-positive metrics (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 41).
Also in
The video
Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · Compilation · 28:48 · chapter 41 from 13:53 · description, sources and chapters →
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