Kobayashi’s canonical-ampleness conjecture

The explanation

Start with closed surfaces, counted by their holes. A sphere has none, a doughnut has one, and a pretzel shape has two. Each can be given a complex structure, which makes it a compact Riemann surface. On such a surface one can ask about entire curves. An entire curve is a non-constant smooth complex map from the whole complex plane into the surface.

The sphere and the doughnut both receive entire curves. The plane wraps onto the doughnut by rolling it up in two directions. Surfaces with two or more holes receive none. Every such map from the plane is constant. A space with no entire curves is called hyperbolic.

There is a second test. Each surface carries a canonical bundle, the line bundle built from its holomorphic one-forms. Its degree is 2g − 2, where g counts the holes. That gives −2 for the sphere, 0 for the doughnut, and 2 for two holes. Three holes give 4. The bundle is ample, a strong algebraic kind of positivity, exactly when the degree is positive. So among curves, the two tests agree. No entire curves happens exactly when the canonical bundle is ample.

Kobayashi’s canonical-ampleness conjecture asks for one direction of this match in every dimension. Take a compact Kähler manifold, a compact complex space with a compatible way to measure lengths. If it has no entire curves, its canonical bundle should be ampleSource 1. Ample means some power of the bundle has enough sections to embed the space in projective space. So the conjecture links a ban on maps from the plane to positivity and to being described by polynomial equationsSource 1.

This paper claims to prove the conjecture in every positive dimensionSource 2. If every holomorphic map from the plane is constant, the canonical bundle is ample. In particular the manifold is projective, meaning it sits inside a complex projective spaceSource 2.

Earlier results needed an extra curvature assumption. Wu and Yau showed in 2016 that negative holomorphic sectional curvature forces canonical ampleness on projective manifoldsSource 3. Tosatti and Yang removed the projectivity assumption in 2017Source 4. Diverio and Trapani allowed curvature that is only quasi-negativeSource 5. Chen and Yang proved ampleness under Gromov’s Kähler hyperbolicity, a different conditionSource 6. The paper notes that these metric assumptions give curvature inequalities that the absence of entire curves alone does not supplySource 7.

A weaker conclusion was already known. Ou, and Cao and Höring, showed the canonical bundle is nef when there are no rational curvesSource 8. Nef is a borderline form of positivity. The remaining issue for ampleness is positive canonical volumeSource 9. Volume here is a number measuring how much positivity the bundle has in total.

The obstacle is that hyperbolicity gives no curvature at all. It only bounds how fast a holomorphic disc can stretch at its centreSource 10. The paper works with extremal discs, maps of the unit disc that maximise a chosen quantity. The hard part sits at the disc’s boundary circle. There, the usual limit arguments do not justify varying an integral over the whole discSource 11. The paper first shows enough regularity at the boundary. It then builds holomorphic families of discs with a fixed centre. One second-derivative identity from these families then gives two positivity estimatesSource 12. One estimate gives a uniform lower bound on volume. A limit then gives positive top self-intersection of the canonical classSource 13. Standard theorems turn that into bigness, projectivity and ampleness.

The paper also claims two consequences. Let n be the dimension. Every power of the canonical bundle of order n + 2 or more is generated by its sectionsSource 14. In dimension 2, that means every power from 4 upward. This step uses a separate OpenAI preprint on Fujita freeness. The second describes a finite cover when the universal cover has a semialgebraic formSource 15. It uses another OpenAI preprint. Neither is used in the proof of the main theoremSource 16.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

What it doesn't show. This does not address singular spaces, non-Kähler manifolds, or the converse direction from ampleness to hyperbolicity.