21 problems

Let $X$ be a complex algebraic variety of log general type. Green-Griffiths-Lang conjecture. There exists a proper closed subset $Z\subsetneq X$ such that, for every non-constant h…

Green–Griffiths conjecture. Every entire curve $f\colon\mathbb C\to X$ in a projective manifold $X$ of general type is algebraically degenerate.

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…

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…

In the complex projective plane $bbP^2$, let $cmathcal{C} csubset cmathbb{P}^2$ be a generic algebraic curve of degree $d cgeqslant 5$. Kobayashi's logarithmic hyperbolicity conjec…

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…

Griffiths' conjecture. For any $\epsilon>0$, the inequality

Campana's exceptional-locus conjecture. The subset $W$ can be chosen to be the union of the Zariski closures of all entire curves on $X$, and hence is also the union of all special…

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

Brunella's zero–one conjecture. For every entire curve $f:\mathbb{C}\rightarrow X$, one can choose some increasing radii $\{r_i\}_{i\geqslant 1}\nearrow\infty$ such that the obtain…

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 smooth projective variety, let $D$ be a simple normal crossing divisor on $X$, and let $A$ be an ample line bundle on $X$. Let $f:\mathbb{C}\to X$ be a Zariski dense e…

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,…

Dense-entire-curve conjecture. Every holomorphically symplectic manifold contains a Zariski-dense entire curve.

Nevanlinna weak Hilbert property conjecture. There exists a Zariski-dense entire curve $h:\mathbb C\to X$ which does not lift to an entire curve in $Y$.

Special-manifold characterization conjecture. The manifold $X$ is special if and only if both of the following properties, conjecturally equivalent to each other, hold: $d_X$ vanis…

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…

Green–Griffiths conjecture. There exists a sufficiently large integer $k=k(X)$ such that this growth is maximal, namely asymptotic to $m^{(k+1)n-1}$.

Soit $X$ une variété de type général. Une courbe entière est une application holomorphe $f:\mathbb{C}\rightarrow X$. Conjecture de Green–Griffiths. Toute courbe entière…