15 problems

Kobayashi's conjecture. A general projective hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ with $d\geq 2n-1$ is hyperbolic.

Demailly's conjecture. For $k$ large enough, $X$ is hyperbolic if and only if $X$ admits a negative curvature $k$-jet metric.

Given a hypersurface $X_d=f_d^*(0)$ in $\mathbb{P}^n$ of degree $d$, call a (very) general small deformation of $X_d$ hyperbolic if, for any (very) general degree-$d$ hypersurface…

Hyperbolicity conjecture for general hypersurfaces. A general hypersurface

Hyperbolicity conjecture for general projective hypersurfaces. If the degree of $X$ is sufficiently high, then both $X$ and its complement $\mathbb{P}^{n}\backslash X$ are hyperbol…

Hyperbolic-index conjecture. The manifold $X$ is Kobayashi hyperbolic if and only if

Picard–analytic equivalence conjecture. The manifold $X$ is $b[31m\ellb[0m$-Picard hyperbolic if and only if it is $b[31m\ellb[0m$-analytically hyperbolic.

Equivalence conjecture. In the category of projective manifolds, the intermediate pseudo-analytic hyperbolicity notions are equivalent.

Extension property. Every non-degenerate holomorphic map

Kobayashi–Zaidenberg conjecture. Complements of generic hypersurfaces in $\mathbb P^n$ of degree at least $2n+1$ are hyperbolic.

Let $X$ be a projective variety over … to $X^{\operatorname{an}}$ whose image is analytically dense. Campana's conjecture. $X$ has a dense entire curve if and only if $X$ is specia…

Hyperbolicity implies general type conjecture. If $X$ is hyperbolic, then $X$ should be of general type; equivalently, $K_X$ should be big and $X$ should have maximal Kodaira dimen…

Campana's core-distance conjecture. The Kobayashi pseudo-distance is the pullback of the orbifold Kobayashi pseudo-distance on the core:

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…

Soit $X$ une variété complexe. Une variété est dite de type général lorsque son fibré canonique $K_X$ est big. Conjecture de Lang. La variété $X$ est hyperbolique si et seulement s…