Algebraic and complex geometry·OpenAI Math·Result 051
Kobayashi’s canonical-ampleness conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebraic and complex geometry: 1 preprint, the latest dated 23 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 proof of Kobayashi’s canonical-ampleness conjecture (algebraic and complex geometry) in a preprint dated 23 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 It has not been peer reviewed.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 42 of 83 in Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · starts at 14:14 · watch from there on YouTube →
What OpenAI claims
Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.
Paper
Canonical ampleness of compact hyperbolic Kähler manifolds
OpenAI's abstract We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler category.
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 OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.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?
- Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.
- Where is the proof?
- In the paper, Canonical ampleness of compact hyperbolic Kähler manifolds (PDF).
- Is there a video?
- Yes, on YouTube (28:48, chapter 42).
Also in
- Chapter 2 of 10 in Kobayashi's Ampleness Conjecture, Kuznetsov Conjecture, Virasoro Conjecture and 7 More · starts at 0:21 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 1 of 4: Goldfeld's Conjecture and 82 More · Compilation · 28:48 · chapter 42 from 14:14 · description, sources and chapters →
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