Campana's abelianity conjecture for special compact Kahler manifolds
Campana's classification program splits compact Kahler manifolds into special ones and fibrations over orbifold bases of general type. A manifold is special if it has no Bogomolov sheaf: no line bundle with . Rationally connected manifolds, compact tori and manifolds of Kodaira dimension zero are special. Campana (2004, Conjecture 7.1) conjectured that the fundamental group of a special manifold is virtually abelian; Campana and Claudon proved it for threefolds. It would imply, for instance, that every compact Kahler manifold with has virtually abelian . Is the fundamental group of every special compact Kahler manifold virtually abelian?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Complex algebraic geometry; fundamental groups of Kahler manifolds
- Posed by
- Frederic Campana (Conjecture 7.1, Ann. Inst. Fourier 54, 2004)
- Year posed
- 2004
- Years open
- 22y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: every special compact Kahler manifold has virtually abelian fundamental group, in every dimension. Corollaries: compact Kahler manifolds with have virtually abelian ; Campana's Conjecture S and the bimeromorphic Iitaka uniformization (finite etale covers bimeromorphic to tori when or the universal cover is ); holomorphic convexity of universal covers of special manifolds. It does not prove the orbifold or quasi-projective versions in general: the companions give a two-step nilpotent bound for linear images in the open case (credited first to Cao-Deng-Hacon-Paun) and one conditional fourfold orbifold case.
What the AI did
The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two companions in the family build on it: 'Two-step monodromy of special quasi-projective varieties' (September 24, 2026), an independent proof of a result the manuscript credits to Cao-Deng-Hacon-Paun, and 'A conditional abelianity theorem for special fourfold pairs with a half-weight divisor' (October 5, 2026), which uses this theorem.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Campana's Conjecture 7.1 (in the Claudon-Horing appendix form the manuscript cites); it states that every connected smooth compact Kahler manifold that is special has a finite-index abelian subgroup in its full topological , with no linearity, projectivity or residual finiteness assumption. The proof was not refereed. No Lean formalization of this family is in the release (no lean/docs/057.md at the pinned commit). The manuscript cites another release manuscript (Orbifold and logarithmic Iitaka subadditivity) only to record a known specialness implication, not as a proof input.