Math Conjectures

Green–Griffiths–Lang Conjecture

OPENLandmarkConjectureProposed 1980 · Standard version

Canonical statement

Let XX be a smooth projective complex variety of general type. There exists a proper Zariski-closed subset Y⊊XY\subsetneq X such that every nonconstant holomorphic map f:C→Xf:\mathbb C\to X satisfies f(C)⊆Yf(\mathbb C)\subseteq Y.
View source LaTeX
Let \(X\) be a smooth projective complex variety of general type. There exists a proper Zariski-closed subset \(Y\subsetneq X\) such that every nonconstant holomorphic map \(f:\mathbb C\to X\) satisfies \(f(\mathbb C)\subseteq Y\).
AG-HYP-022:[GreenGriffiths1980EntireCurves][Lang1986Hyperbolic][Demailly2020GreenGriffithsLang]
Algebraic geometryAnalysis and functional analysis

Notes

Jet-differential methods establish the predicted algebraic degeneracy for important classes, including sufficiently general high-degree hypersurfaces and varieties satisfying stronger positivity conditions. No argument produces such a proper exceptional subset for every variety of general type.
Related arithmetic and hyperbolicity formulations are often grouped under Lang’s conjectures. This record fixes the geometric entire-curve statement and does not identify it with the stronger assertion that every such XX is Kobayashi hyperbolic.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.