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The abelianity conjecture for special compact Kähler manifolds
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Abstract
We prove that the fundamental group of every special compact Kähler manifold is virtually abelian, resolving Campana's abelianity conjecture in all dimensions. In particular, every smooth compact connected Kähler manifold of Kodaira dimension zero has virtually abelian ordinary fundamental group.
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Related prior literature mentioned by others
- Frédéric Campana and Benoît Claudon, Abelianity Conjecture for special threefolds (arXiv:1107.0168, 2011; Proceedings of the Edinburgh Mathematical Society, 2014). Identified in AIProofWatch’s October 18 record as prior work, this establishes virtual abelianity for special compact Kähler threefolds, the dimension-three case of the manuscript’s claimed theorem.
Possibly relevant literature (found by the agent, not checked)
- Benoît Claudon, Andreas Höring, János Kollár, Algebraic varieties with quasi-projective universal cover (arXiv:1102.2762, February 2011). Characterizes normal projective varieties with quasi-projective universal cover via finite étale covers fibred over abelian varieties with simply connected fibre; related to the paper's corollary on analytically compactifiable covers, which instead cites Claudon–Höring 2013.
- Shin-ichi Matsumura, Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature (arXiv:2502.00367, February 2025). Studies abelianness of the topological fundamental group for compact Kähler manifolds with semi-positive holomorphic sectional curvature, with reference to varieties of special type; a setting where Theorem 1.1's conclusion can be compared.
- Ya Deng and Botong Wang, $L^2$-vanishing theorem and a conjecture of Kollár (arXiv:2409.11399v2, August 6, 2025). Its $L^2$-Dolbeault vanishing theorem for universal covers of smooth projective varieties with generically large linear fundamental groups provides a comparison for the manuscript’s spectral argument, under different geometric and group-theoretic hypotheses.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 057
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026,
author = {{OpenAI}},
title = {{The abelianity conjecture for special compact K\"ahler manifolds}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026/paper.pdf}{OAI:The-abelianity-conjecture-for-special-compact-Kahler-manifolds-September-23-2026}},
year = {2026}
}Family 057: Fundamental groups of special complex varieties and root orbifolds
Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.
Other papers in this family:
- Two-step monodromy of special quasi-projective varieties · September 24, 2026
- A conditional abelianity theorem for special fourfold pairs with a half-weight divisor · October 5, 2026