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A conditional abelianity theorem for special fourfold pairs with a half-weight divisor
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Abstract
Using the abelianity theorem for special compact Kähler manifolds in every dimension, we prove virtual abelianity of the entire orbifold fundamental group of the special order-two root orbifold associated with a pair $(X,\tfrac12D)$, where X is a smooth projective fourfold and D is a nonempty smooth connected divisor. We construct a special smooth projective eightfold whose fundamental group surjects onto the required group.
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Possibly relevant literature (found by the agent, not checked)
- Cécile Gachet, Zhining Liu and Joaquín Moraga, Fundamental groups of log Calabi-Yau surfaces (arXiv:2312.03981v2, February 11, 2025). Its virtual class-two nilpotence theorem for orbifold fundamental groups of log Calabi–Yau surface pairs offers a lower-dimensional comparison for the manuscript’s conditional virtual-abelianity theorem for special fourfold root orbifolds, with different boundary and positivity hypotheses.
Elsewhere
- Emergent Mind: family 057
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026,
author = {{OpenAI}},
title = {{A conditional abelianity theorem for special fourfold pairs with a half-weight divisor}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026/main.pdf}{OAI:A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026}},
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:
- The abelianity conjecture for special compact Kähler manifolds · September 23, 2026
- Two-step monodromy of special quasi-projective varieties · September 24, 2026