OpenAI · 057-03 · 2026-10-05 · Algebraic and complex geometry
A conditional abelianity theorem for special fourfold pairs with a half-weight divisor
preprints/A-conditional-abelianity-theorem-for-special-fourfold-pairs-with-a-half-weight-divisor-October-5-2026
Family 057. Campana's abelianity conjecture and special varieties.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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 , 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.
Preprint source and Lean scope
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