12 problems

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

Irreducibility conjecture. For every $cepsilon>0$, there exists a nonzero negatively twisted invariant logarithmic $2$-jet differential $comega$ in this space with

Let $H_1,\dots,H_c\subset\mathbb{P}^N$ be a generic choice of high-degree hypersurfaces, with $c\geqslant N/2$, and set … The cotangent bundle $T_X^*$ should be ample. Equivalently…

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…

Let $Z\subset \mathbb{P}^n$ be the complete intersection of $c$-general hypersurfaces of sufficiently high degree. Let $E_{k,m}T_Z^*$ denote the invariant jet bundle of order $k$ a…

Let $X$ be a projective manifold, let $k$ and $m$ be positive integers, and let $E_{k,m}T_X^\star$ denote the relevant bundle of jet differentials. Let $L\to X$ be a line bundle sa…

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

Finite-generation conjecture. Every fibre $\mathcal{O}((J_{k,p})_x)^{\mathbb{G}_{k,p}'}$ of $E_{k,p}^n$ is a finitely generated graded complex algebra.

Finite-generation conjecture. The group $\mathbb{G}_{k,p}'$ is a subgroup of $\mathrm{SL}(\mathrm{sym}^{\leq k}p)$ such that

Affine orbit-closure conjecture. This point has stabiliser $\mathbb{G}_{k,p}'$ in $\mathrm{SL}(n)$, and the closure of its $\mathrm{SL}(n)$-orbit is the union of its orbit and fini…

Let $X\subset \mathbb{P}^N$ be a generic complete intersection variety of dimension $n$ and codimension $c$. For integers $k$ and $m$ as in the jet-differential construction, let…

Higher-order jet-differential ampleness conjecture. The bundle map