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.
released 2026-09-23 | 8 theorems · 18 lemmas · 30 proofs · 23,899 words |
PLAY LEVEL 1 »(pdf)
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.
released 2026-09-24 | 3 theorems · 1 lemma · 3 proofs · 8,733 words |
PLAY LEVEL 2 »(pdf)
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.
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.