Math 2.0

Several complex variables and analytic spaces Holomorphic maps and invariant metrics Conjecture

Kobayashi conjecture

Builds on Kobayashi hyperbolic manifoldProjective spaceBrody hyperbolicity

Statement

Let n≥2n \ge 2 and let X⊂ℙn+1X \subset \mathbb{P}^{n+1} be a very general hypersurface of degree dd in complex projective space. Very general means that the defining polynomial lies outside a countable union of proper algebraic subsets of the space of polynomials of degree dd.

  1. If d≥2n+1d \ge 2n + 1, then XX is Kobayashi hyperbolic.
  2. If d≥2n+3d \ge 2n + 3, then the complement ℙn+1∖X\mathbb{P}^{n+1} \setminus X is Kobayashi hyperbolic.

Remarks

Kobayashi stated these problems in his 1970 book on hyperbolic manifolds. The hypothesis n≥2n \ge 2 is needed in part 1 in this form. For n=1n = 1 the bound would be d≥3d \ge 3, but a smooth plane cubic is an elliptic curve, a complex torus, which is not hyperbolic. For n=1n = 1 the true statement is that smooth plane curves of degree at least 44 are hyperbolic, since they have genus at least 33. Part 1 is known for all dd larger than a bound that depends on nn; Yum-Tong Siu and Damian Brotbek gave proofs. Later work lowered the bound, which is still far above 2n+12n + 1 for large nn. For d≤2n−1d \le 2n - 1, every hypersurface of degree dd in ℙn+1\mathbb{P}^{n+1} contains a line, so it is not hyperbolic. Since XX is compact, part 1 asks equivalently that XX contains no entire curve, that is, that XX is Brody hyperbolic.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Brody hyperbolicity, concept, no evidence levelBrody hyperbolicityconceptKobayashi hyperbolic manifold, concept, no evidence levelKobayashi hyperbolic …conceptProjective space, concept, no evidence levelProjective spaceconceptKobayashi conjecture, conjecture, no evidence levelKobayashi conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Brody hyperbolicity concept
  2. Kobayashi hyperbolic manifold concept
  3. Projective space concept
  4. Kobayashi conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Kobayashi_metric
About this page
Tag1X3EV2TZNN
URIm2://1X3EV2TZNN
Kindconjecture
WritingW0 stub
MSC32Q45, 14J70
StatusThis copy of the site has no build data. Run m2 status 1X3EV2TZNN.
AboutKobayashi hyperbolic manifold
UsesKobayashi hyperbolic manifold, Projective space, Brody hyperbolicity

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