Abelianity of special compact Kähler fundamental groups
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
Must the fundamental group of every connected smooth special compact Kähler manifold have an abelian subgroup of finite index? Special has Campana's standard meaning, excluding fibrations with general-type orbifold base.Short English statement
Must the fundamental group of every connected smooth special compact Kähler manifold have an abelian subgroup of finite index? Special has Campana's standard meaning, excluding fibrations with general-type orbifold base.Sources
- S1 Original question source or mathematical context. http://aimpl.org/amenablediscrete/5/.
- S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2014/09.
- S3 Original question source or mathematical context. https://link.springer.com/article/10.1007/s10231-024-01501-8.
- S4 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2537
Manuscript 1 (family 057, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.
Abelianity of special compact Kähler fundamental groups
OpenAI's claim
For a special compact Kähler manifold, its entire ordinary fundamental group is claimed to be virtually abelian, including when the manifold has Kodaira dimension zero. The cited manuscript argues for that result; its relation to this problem is specified below.Mathematical statement of the claimed result
Mathematical claim 1.
Call a compact complex manifold $X$ special when no line bundle $L$ admits a nonzero map $L\to\Omega_X^p$ with $\kappa(X,L)=p$ for any $1\le p\le\dim X$. For every connected smooth compact Kähler special $X$, OpenAI claims \[ \exists A\le\pi_1(X):\quad A\text{ abelian},\quad[\pi_1(X):A]<\infty. \] Equivalently it claims a finite holomorphic covering, connected and unramified, whose entire discrete fundamental group is abelian. All complex dimensions are included.Outline of the argument
Manuscript 1 (family 057, 2026-09-23): After maximizing the Albanese first Betti number on finite covers, the proof seeks to make the fiber-group image finite. A minimal infinite quotient is treated either by a semisimple linear representation or by Hilbert-coefficient tensors from low spectrum; both produce positivity that descends, with the actual meridian orders, to a differential sheaf forbidden by specialness.Formal verification
Family 057 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.
Status caveat
OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.Sources
- OpenAI, The abelianity conjecture for special compact Kähler manifolds, 2026-09-23; citation key
OAI:The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:abelianity. - OpenAI catalogue entry.