Green–Griffiths–Lang conjecture

An analytic curve follows a point as a complex parameter varies. Here its formulas must work on the whole complex plane, not just near one input. In a smooth projective algebraic shape of general type, must every nonconstant curve of this kind stay inside one fixed smaller algebraic part? The same part must contain all the curve images. A whole-plane holomorphic map of this kind is called an entire curve. General type describes maximal growth of the shape’s canonical differential forms.[1]

Local analytic curves can be drawn in smooth coordinate charts. Green–Griffiths–Lang asks whether curves whose parameter extends over the whole complex plane are forced into one smaller algebraic part of a general-type projective variety. The quantifier is uniform: choosing a different exceptional subset for each curve would not suffice. A whole-plane holomorphic map of this kind is called an entire curve.

If $X/\mathbb C$ is smooth projective with $K_X$ big, does there exist a proper closed algebraic $Z\subsetneq X$ such that $f(\mathbb C)\subseteq Z$ for every nonconstant holomorphic $f:\mathbb C\to X$? General type means maximal growth of pluricanonical sections for the dimension. The conclusion permits curves inside $Z$; it does not require their complete absence.[1] [2]

The predicted confinement is proved for selected classes, including generic hypersurfaces of sufficiently high degree. Computations for particular plane-curve complements address a related problem with points removed. Neither establishes one common exceptional subset for every general-type projective shape.[1] [2]

The logarithmic complement results have extra geometric and jet-section hypotheses. General projective degeneracy remains unresolved; it need not mean the absence of all entire curves.[1] [2]

Local coordinates allow small analytic pieces through every smooth point. A whole-plane curve has to extend through all complex parameter values, which is a much stronger requirement. A common proper exceptional set would identify a region outside which no such global curve can pass, even though local analytic pieces still exist there.[1]

The conjecture connects global analytic parametrizations to the algebraic geometry of canonical forms. Jet-differential methods illustrate the link: equations involving derivatives constrain lifts of entire curves, and enough constraints can force their images into an algebraic exceptional locus. The generic-hypersurface and logarithmic-complement results show this mechanism in restricted settings. A uniform projective result would organize the surviving curves without confusing algebraic degeneracy with hyperbolicity.[1] [2]