---
tag: AD9GD9EM9R
kind: conjecture
name: Green–Griffiths–Lang conjecture
msc: [32Q45, "14E99"]
uses: [A4H4CZ2V35, ZW00Q0SAGR, Y7W36GJDJD, HRPAX8V7HA]
about: [Y7W36GJDJD]
sources:
  - url: https://en.wikipedia.org/wiki/Kobayashi_metric
history:
  posed_by: "Mark Green and Phillip Griffiths"
  first_stated: "1980"
---

## Statement

Let $X$ be a complex [[projective variety|A4H4CZ2V35]] of general type, that is, one whose [[Kodaira dimension|ZW00Q0SAGR]] equals its dimension.
Then there is a proper algebraic subvariety $Y \subsetneq X$ such that every nonconstant holomorphic map $f \colon \C \to X$ has image contained in $Y$.

## 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 $2$, so its universal covering is the unit disc.
Hence every holomorphic map from $\C$ to it is constant.
Michael McQuillan proved it for surfaces of general type with $c_1^2 > c_2$.
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|HRPAX8V7HA]] if and only if all its subvarieties are of general type.

## Explanations

### L1 human

A variety of general type is, in a precise sense, a variety with a lot of negative curvature.
The conjecture predicts that holomorphic copies of the complex plane in such a variety cannot spread everywhere: they all lie inside one smaller algebraic subvariety.
In other words, such a variety is [[Brody hyperbolic|Y7W36GJDJD]] outside a small exceptional set.