8 problems

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

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…

Demailly's conjecture. The variety $X$ is Kobayashi hyperbolic if and only if there exists $k\in\mathbb{N}$ such that $(X,T_X)$ has negative $k$-jet curvature.

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…

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.

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