Green–Griffiths conjecture on algebraic degeneracy of holomorphic curves

Let $X$ be a smooth complex projective variety of general type, meaning that its canonical line bundle $K_X$ is pseudo ample. A holomorphic curve in $X$ is algebraically non-degenerate if its image is not contained in any proper algebraic subvariety of $X$.

Green–Griffiths conjecture. $X$ contains no algebraically non-degenerate holomorphic curves.

This is a foundational conjecture on the algebraic degeneracy of entire curves in varieties of general type. The cited passage also describes the stronger Green–Griffiths–Lang conjecture, which predicts a single proper algebraic subvariety containing every nonconstant holomorphic curve, but the present row records the stated Green–Griffiths conjecture.