Browse › Algebraic and complex geometry › 051 Kobayashi’s canonical-ampleness conjecture
Canonical ampleness of compact hyperbolic Kähler manifolds
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Abstract
We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler category.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Xiangsen Qin and Yuanhong Tian, Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples (arXiv:2609.25762, v1 22 Sep 2026, v5 19 Oct 2026). Version 5 cites this paper as [OAI26] for the extremal-disc, boundary-frame and canonical-envelope constructions and adapts them to non-Kähler Brody-hyperbolic manifolds carrying a strictly positive current $T$ with $dT$ smooth and $dd^cT\le0$ (nefness of $K_X$, and ampleness under a Lelong-level-set dimension condition); the paper includes an AI-use disclosure.
Related prior literature mentioned by others
- Simone Diverio, Kobayashi hyperbolicity, negativity of the curvature and positivity of the canonical bundle (arXiv:2011.11379, 2020; revised December 15, 2020). Recommended as background by the AI Proved What? episode on result 051 and already cited in the manuscript, this survey explains the curvature-based Wu–Yau theorem and its relationship to Kobayashi's canonical-ampleness conjecture. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Aleksei Golota, On negativity of total $k$-jet curvature and ampleness of the canonical bundle (arXiv:1708.02866, 2017). This earlier partial result on the Kobayashi–Lang conjecture proves $K_X$ ample for projective manifolds with nondegenerate negative total $k$-jet curvature; the paper's survey of partial results does not cite it.
- Ping Li, Kähler hyperbolic manifolds and Chern number inequalities (Trans. Amer. Math. Soc. 372 (2019), 6853–6868). It shows that the canonical bundle of a "Kähler exact" manifold of general type is ample, which is close to the Gromov/Chen–Yang Kähler-hyperbolic line discussed in the introduction.
- Xueyuan Wan, Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature (arXiv:2610.08111, 6 Oct 2026). It proves $K_X$ ample for compact Kähler manifolds carrying a Hermitian metric of quasi-negative Chern holomorphic sectional curvature, a criterion parallel to the Wu–Yau, Diverio–Trapani and Broder–Stanfield results surveyed in the introduction.
Digests and expositions
- AI Proved What?, Hyperbolic, so ample? Kobayashi's canonical-ampleness conjecture (episode 051, October 2026). This illustrated exposition and transcript outline the extremal-disc and volume argument and distinguish the main theorem from the two corollaries that use other release papers. [AI-generated per source] [family-level]
- Gradient Damage, Kobayashi's Ampleness Conjecture, Kuznetsov Conjecture, Virasoro Conjecture and 7 More (video compilation, October 15, 2026). The chapter beginning at 0:21 introduces result 051 through entire curves and the claimed canonical-ampleness conclusion, with a link to the manuscript. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 051
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026,
author = {{OpenAI}},
title = {{Canonical ampleness of compact hyperbolic K{\"a}hler manifolds}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026/canonical-ampleness.pdf}{OAI:Canonical-ampleness-of-compact-hyperbolic-Kahler-manifolds-September-23-2026}},
year = {2026}
}Family 051: Kobayashi’s canonical-ampleness conjecture
Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.
This is the only paper in this family.