CATALOGUE 057
Campana's abelianity conjecture and special varieties
Catalog 057 contains three papers on fundamental groups of special varieties. They concern virtual abelianity of the entire group in the compact Kähler setting, virtual nilpotence of class at most two for linear images in the open setting, and a transfer to fourfold pairs using the compact result. The articles distinguish these targets and explain the manuscripts' proof routes and the scope of editorial checking.
1. All three papers in official order
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | The abelianity conjecture for special compact Kähler manifolds →Short name: Special Kähler abelianity2026-09-23 · 50 pagesOriginal paper (GitHub) ↗ | Virtual abelianity of the entire ordinary fundamental group of a special smooth compact Kähler manifold, with consequences for Kodaira dimension zero, convexity and uniformization. Theorem 1.1; Corollaries 1.2–1.5 · pp. 2–5 ↗ | Chooses minimal quotients of Albanese fibre images; the linear and Hilbert-coefficient branches yield positivity, followed by descent retaining inertia. Supplies the third paper. Read the proof overview → |
| 02 | Two-step monodromy of special quasi-projective varieties →Short name: Two-step monodromy2026-09-24 · 19 pagesOriginal paper (GitHub) ↗ | Linear images for special smooth quasi-projective varieties are virtually nilpotent of class at most two. Under exactness, rational unipotent completion is finite-dimensional and of class at most two; a special open surface can have nonspecial general fibres. Theorems 1.1–1.3; Corollary 3.1 · pp. 1–3, 8 ↗ | Identifies the fibre image with the commutator in canonical unipotent Lie quotients and combines two Hodge-weight bounds to kill triple commutators. The surface example separates the fibre from its image. Read the proof overview → |
| 03 | A conditional abelianity theorem for special fourfold pairs with a half-weight divisor →Short name: Half-weight fourfold abelianity2026-10-05 · 14 pagesOriginal paper (GitHub) ↗ | Conditional virtual abelianity of the whole group of a special order-two root orbifold of a smooth projective fourfold with a nonempty smooth connected reduced divisor, assuming compact Kähler abelianity. Theorem 1.3 · p. 2 ↗ | Constructs a projective eightfold with four genus-one factors, transfers specialness and obtains a fundamental-group quotient before applying the first paper. Read the proof overview → |
2. Where the result for the entire group is used
The diagram keys refer to the following fixed manuscripts. External inputs to each paper are listed in its individual article.
- OpenAI26a OpenAI — The abelianity conjecture for special compact Kähler manifolds. 2026-09-23; Input for the entire compact Kähler fundamental group.
- OpenAI26b OpenAI — Two-step monodromy of special quasi-projective varieties. 2026-09-24; Comparison with linear images, completion and nonspecial fibres.
- OpenAI26c OpenAI — A conditional abelianity theorem for special fourfold pairs with a half-weight divisor. 2026-10-05; Eightfold construction and conditional transfer.
The third paper realizes the root space's fundamental group as a quotient of the fundamental group of a smooth projective eightfold $Y$. Applying the first paper requires specialness of $Y$ as well as smoothness and projectivity; Propositions 3.1 and 4.1 of the third paper supply these conditions. Virtual abelianity passes to a quotient, giving the conditional conclusion of Theorem 1.3. The third article's overview leads to the arguments establishing these conditions.
[OpenAI26c, Proposition 3.1, pp. 5–6] [OpenAI26c, Proposition 4.1, p. 6] [OpenAI26c, Proof of Theorem 1.3, p. 9]
The second paper controls linear images and rational unipotent completion through ordinary quasi-Albanese exactness and mixed Hodge structures. It cites the first paper as compact-setting background, not as a direct input to this proof. Its surface example has nonspecial general fibres whose fundamental groups have trivial image in the total group. The weight argument and surface calculation explain why these targets must be distinguished.
[OpenAI26b, Theorems 1.1–1.2, pp. 1–2] [OpenAI26b, Theorem 1.3, p. 3]
| Source → recipient | Relation | Result, hypotheses and checked use |
|---|---|---|
| Special Kähler abelianity → Half-weight fourfold abelianity | Explicit premise and direct input | Virtual abelianity in all dimensions is applied to the constructed special projective eightfold. Third paper: Assumption 1.2 and final proof on p.9. |
| Special Kähler abelianity → Two-step monodromy | Background | Comparison with the compact case in §1, p.1 of the second paper; not a proof dependency for Theorems 1.1–1.3. |
[OpenAI26a, Theorem 1.1, p. 2] [OpenAI26c, Theorem 1.3; Assumption 1.2, p. 2] [OpenAI26b, §1, p. 1]
3. Sources and checking scope
The official overview title is Campana's abelianity conjecture and special varieties, under Algebraic and complex geometry. CONTENTS instead uses Fundamental groups of special complex varieties and root orbifolds. The order and manuscripts are fixed to official repository commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. In order, the versions are September 23, 2026 (50 pages), September 24, 2026 (19 pages), and October 5, 2026 (14 pages).
OpenAI Research Catalog · Catalog 057 · p. 7
The articles report the manuscripts' claims. Editorial work checked hypotheses and all stated conclusions, selected main proof passages, and the statements and uses of direct external inputs, followed by review by a different contributor. Complete independent verification does not cover all analytic estimates and descent comparisons in the first paper, the full completed-group-algebra construction in the second, Appendix A's alternative construction in the third, or the complete external proofs. Detailed limits are recorded at the end of each article and in its source records.
The second paper's criticism of fibre specialness in earlier work is restricted to the CDHP arXiv v2 specified in its article. It is not treated as a refutation of that work's separate linear-image theorem. Unchecked generalization or containment relations are not drawn as arrows.