CATALOGUE 051
Kobayashi's canonical-ampleness conjecture
This catalogue concerns canonical ampleness for compact hyperbolic Kähler manifolds. The official list contains one manuscript. The article presents its claims and proof route, with the scope of source checking stated at the end.
The single paper in official catalogue 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 | Canonical ampleness of compact hyperbolic Kähler manifolds →Short name: Canonical ampleness2026-09-23 · 31 pagesOriginal paper (GitHub) ↗ | The canonical bundle of a compact connected Brody-hyperbolic Kähler manifold is ample. Consequences give generation for every m≥n+2 and a finite étale product cover under a semialgebraic universal-cover hypothesis. Theorem 1.1 · p. 1; Corollaries 10.1–10.2 · pp. 26–27 ↗ | Boundary-regular disc variations yield nefness and a volume bound; Monge–Ampère, Morse, projectivity and cone inputs complete the proof. The corollaries use separate inputs from 038 and 058. Read the proof overview → |
Entering the proof and its two consequences
Start with the main proof overview, then follow disc variations and the Hessian, nefness, the uniform volume bound, and positive intersection. The overview also explains the subsequent steps through bigness and projectivity to ampleness.
After the main theorem, the all-exponents generation result uses Fujita freeness from 038, while the finite étale product cover uses universal-cover classification from 058. These inputs do not enter the proof of the main theorem.
Sources and checking scope
The official subject is Algebraic and complex geometry. The reference catalogue is dated October 6, 2026 and pinned to commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.
OpenAI Research Catalog · Catalogue 051 · p. 7
The manuscript is by OpenAI, dated September 23, 2026, and has 31 pages including references.
Theorem 1.1 · p. 1; Corollaries 10.1–10.2 · pp. 26–27
The principal reductions in the proof and the statements and application hypotheses of twelve external references were compared. This does not independently verify every boundary-regularity, functional-analytic or smoothing estimate, re-prove the external theorems, or exhaust all dependencies. See the article’s closing section for the checking scope and unresolved items.