7 problems

Let $X$ be a compact Kähler manifold of dimension $n$, and let $k\in\mathbb{N}$ satisfy $1\leq k\leq n-1$. Let $\alpha\in H^{1,1}(X,\mathbb{R})$ be a big class. Say that $\alpha$ i…

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable $n$-pointed genus-zero curves. A divisor $E$ on $\overline{\mathcal{M}}_{0,n}$ is $F$-nef if $C(a,b,c,d).E\geq 0$ f…

F-conjecture. The cone $\overline{\text{NE}}_1(\overline{\rm{M}}_{g,n})$ is generated by the cycle classes of F-curves. Equivalently, a line bundle on $\overline{\rm{M}}_{g,n}$ is…

Stein universal-cover conjecture. If the universal cover of $Y$ is a Stein manifold, then the cotangent bundle of $Y$ is nef.

Demailly–Peternell–Schneider conjecture. The Albanese morphism of $X$ is surjective.

Let $X_n$ be the blow-up of the projective plane at $n$ general points, and let $\mathcal{L}$ be a nef linear system, meaning that $\mathcal{L}\cdot\mathcal{L}'\geq0$ for every eff…

Let $X$ be a non-uniruled projective manifold. A vector bundle $E$ on $X$ is sufficiently nef if, for every $x \in X$, there is a family of curves through $x$ covering $X$ such tha…