Algebraic and complex geometry · Result 057

Fundamental groups of special complex varieties and root orbifolds

3 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 3

The abelianity conjecture for special compact Kähler manifolds

OpenAI · September 23, 2026

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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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}
}

Two-step monodromy of special quasi-projective varieties

OpenAI · September 24, 2026

We give an independent proof that every complex linear representation of the ordinary fundamental group of a connected smooth special complex quasi-projective variety has virtually nilpotent image of class at most two. This conclusion was previously announced by Cao–Deng–Hacon–Păun. We also construct special open surfaces whose general quasi-Albanese fibres are not special.

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Cite

@misc{OAI:Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026,
  author = {{OpenAI}},
  title = {{Two-step monodromy of special quasi-projective varieties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026/paper.pdf}{OAI:Two-step-monodromy-of-special-quasi-projective-varieties-September-24-2026}},
  year = {2026}
}

A conditional abelianity theorem for special fourfold pairs with a half-weight divisor

OpenAI · October 5, 2026

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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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.