4 problems

Nef line bundle conjecture. Every smooth, complete toric variety has a nontrivial nef line bundle.

Numerical semi-ampleness conjecture. Every nef line bundle $L$ on $X$ is numerically semi-ample; that is, there exists a semi-ample line bundle $L'$ on $X$ such that

Let $X$ be a non-projective compact hyperkähler manifold and let $L$ be a non-trivial nef line bundle. Let $a(X)$ be the transcendence degree of the field of meromorphic functions…

Let $\mathcal{M}$ be an $n$-dimensional manifold and let $\mathcal{L}$ be a nef and big line bundle on $\mathcal{M}$. The sectional genus is denoted by $g(\mathcal{M},\mathcal{L})$…