Result 051, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

The manuscript reports that ruling out nonconstant holomorphic maps from the complex plane forces strong positivity on a compact Kähler manifold. This links a restriction on complex-analytic maps to an algebraic description of the space.

What changes?

The claim covers every compact, connected, smooth Kähler manifold of positive complex dimension. Here Kähler means that a compatible metric has a closed associated two-form. An entire curve is a holomorphic map from the complex plane; the assumption excludes every nonconstant such map. The conclusion is that the canonical bundle, the line bundle of top-degree holomorphic differential forms, is ample, a strong positivity property. Consequently, the manifold is projective: it embeds into a finite-dimensional complex projective space.

What does that help mathematicians do?

Ampleness says more than the existence of some projective embedding. It means that global sections of a sufficiently high tensor power of the canonical bundle itself provide such an embedding. Thus the claimed result identifies an intrinsic source of algebraic coordinates for these manifolds. It also rules out nonprojective examples under the stated assumptions, sharpening the connection between restrictions on holomorphic maps and algebraic structure.

Are there practical applications?

Its immediate value is foundational: it places this analytically defined class of spaces inside projective algebraic geometry, where geometric questions can be expressed using homogeneous polynomial equations. The stated result supplies a structural constraint on possible manifolds, not a computational procedure or a demonstrated technological application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Canonical ampleness of compact hyperbolic Kähler manifolds

September 23, 2026 31 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.