Kobayashi–Lang conjecture on ampleness of the canonical bundle
Let be a compact complex manifold that is Kobayashi hyperbolic, meaning that its intrinsic Kobayashi pseudometric is a metric. Let denote its canonical line bundle.
Kobayashi–Lang conjecture. The canonical line bundle is ample.
This conjecture predicts a strong birational-geometric consequence of Kobayashi hyperbolicity. The paper proves ampleness under the stronger assumption of nondegenerate negative total -jet curvature, but the conjecture itself is not resolved here.
References
Primary source
Aleksei Golota, “On negativity of total k-jet curvature and ampleness of the canonical bundle”, arXiv:1708.02866 (2017).
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 1
RemarkAI-assistedClaimed by OpenAI. The manuscript claims ample canonical bundle for every positive-dimensional compact connected Kahler manifold without a nonconstant entire curve. By the compact Brody criterion this addresses the Kahler subcase of Kobayashi–Lang; the source also claims projectivity. It does not treat arbitrary non-Kahler compact complex manifolds.See full solution
Claimed by OpenAI. The manuscript claims ample canonical bundle for every positive-dimensional compact connected Kahler manifold without a nonconstant entire curve. By the compact Brody criterion this addresses the Kahler subcase of Kobayashi–Lang; the source also claims projectivity. It does not treat arbitrary non-Kahler compact complex manifolds.
GitHub repository: https://github.com/openai/math
- OpenAI-051-01-Canonical-ampleness-of-compact-hyperbolic-K-hler-manifolds.pdfOpen