9 problems

Let $X$ be a smooth projective variety of dimension $n\geq 2$, let $K_X$ be its canonical line bundle, and let $L$ be an ample line bundle on $X$. Moraga–Yeong's conjecture. The co…

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 $k$ be an algebraically closed field of characteristic zero, let $X$ be a projective variety over $k$, let $\Delta\subset X$ be a closed subset, and let $p$ be a positive integ…

Let $k$ be an algebraically closed field of characteristic zero, let $X$ be a projective variety over $k$, and let $pgeq 1$ be an integer. The notions of general type, pseudo-Morde…

Lang–Vojta's pseudo-algebraic-hyperbolicity conjecture. A projective integral variety $X$ over $k$ is of general type if and only if $X$ is pseudo-algebraically hyperbolic over $k$…

Demailly–Green–Griffiths–Lang–Vojta conjecture. The following statements are equivalent: $X$ is algebraically hyperbolic over $k$; $X$ is bounded over $k$; for all $n\geq 1$ and…

Fibration property. If $Y$ is algebraically hyperbolic over $k$ and, for all $y\in Y(k)$, the projective scheme $X_y$ is algebraically hyperbolic over $k$, then $X$ is algebraicall…

Green--Griffiths--Lang conjecture. The following are equivalent:

A complex projective variety $X$ is algebraically hyperbolic if there exists an $\epsilon>0$ such that every curve $C\subset X$ of geometric genus $g(C)$ satisfies … It is Brody hy…