13 problems

Strong Green–Griffiths conjecture. There exists a closed subvariety $E\subsetneq X$ such that $E$ contains the image $f(\mathbf{C})$ for every non-constant holomorphic map

Pointed Brody conjecture. If the infinitesimal Kobayashi–Royden pseudometric is degenerate on $T_xX$, then there exists a non-constant holomorphic map

Let $(X, \Delta)$ be a smooth C-pair over $\mathbb{C}$, where a dense entire curve means a holomorphic C-pair map $\mathbb{C} \to (X, \Delta)^{\an}$ whose image is Zariski-dense in…

Let $X$ be a compact complex manifold, and let an Ahlfors current mean a current obtained as an asymptotic limit from the concentric holomorphic discs associated to an entire curve…

Green-Griffiths-Lang conjecture. Every entire curve $f: \mathbb{C} \to X\setminus D$ is algebraically degenerate; that is, the image of $f$ is contained in a proper algebraic subva…

Lang's entire-curve conjecture. (i) A projective manifold of general type does not contain a Zariski dense entire curve. Equivalently, (ii) a projective manifold containing a Zaris…

Universal entire-curve conjecture. There should be some entire curve $g:\mathbb{C}\rightarrow X$ such that the concentric discs

Brunella's mixed Nevanlinna-current conjecture. Some entire curve shall produce a Nevanlinna current $\mathsf{T}=\mathsf{T}_{\text{alg}}+\mathsf{T}_{\text{diff}}$ with both nontriv…

Let $X$ be a smooth projective variety and $D$ a simple normal crossings divisor such that $K_X+D$ is big. Let $L$ be an ample line bundle on $X$, and let $g(C)$ denote the geometr…

Let $X$ be a projective variety defined over ${\mathbb C}$. An entire curve on $X$ is a holomorphic map from ${\mathbb C}$ to $X$. Let $Y_{\mathrm{exc}}$ be a proper closed subset,…

Let $X$ be a projective manifold, let $X_1:=\mathbb P(T_X)$ be the projectivized tangent bundle, and for every entire curve $f:\mathbb C\to X$ let $f_{[1]}:\mathbb C\to X_1$ be its…

Orbifold Green–Griffiths–Lang conjecture. If $\mathcal{X}$ is of general type, then there exists a proper sub-orbifold $\mathcal{Z}\subset\mathcal{X}$ containing the image of every…

Lang's conjecture. There exists a non-constant holomorphic map