Math 2.0

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

Green–Griffiths–Lang conjecture

Builds on Projective varietyKodaira dimensionBrody hyperbolicityKobayashi hyperbolic manifold

Statement

Let XX be a complex projective variety of general type, that is, one whose Kodaira dimension equals its dimension. Then there is a proper algebraic subvariety Y⊊XY \subsetneq X such that every nonconstant holomorphic map f ⁣:ℂ→Xf \colon \C \to X has image contained in YY.

Remarks

Green and Griffiths posed the problem for entire curves in varieties of general type. Serge Lang stated it as part of a group of conjectures that link hyperbolicity with the arithmetic of varieties. The conjecture is open in general. For curves it holds: a smooth projective curve of general type has genus at least 22, so its universal covering is the unit disc. Hence every holomorphic map from ℂ\C to it is constant. Michael McQuillan proved it for surfaces of general type with c12>c2c_1^2 > c_2. It is also known for very general hypersurfaces of high degree in projective space. A related conjecture of Lang says that a projective variety is Kobayashi hyperbolic if and only if all its subvarieties are of general type.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Brody hyperbolicity, concept, no evidence levelBrody hyperbolicityconceptKobayashi hyperbolic manifold, concept, no evidence levelKobayashi hyperbolic …conceptKodaira dimension, concept, no evidence levelKodaira dimensionconceptProjective variety, concept, no evidence levelProjective varietyconceptGreen–Griffiths–Lang conjecture, conjecture, no evidence levelGreen–Griffiths–Lang …conjecture
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. Kodaira dimension concept
  4. Projective variety concept
  5. Green–Griffiths–Lang conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Kobayashi_metric
About this page
TagAD9GD9EM9R
URIm2://AD9GD9EM9R
Kindconjecture
WritingW0 stub
MSC32Q45, 14E99
StatusThis copy of the site has no build data. Run m2 status AD9GD9EM9R.
AboutBrody hyperbolicity
UsesProjective variety, Kodaira dimension, Brody hyperbolicity, Kobayashi hyperbolic manifold

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