4 problems

Let $X$ be a projective manifold, and let $L\to X$ be a big and nef line bundle such that $K_X+L$ is nef. Kawamata–Ambro–Ionescu effective non-vanishing conjecture. One should have…

Kawamata–Ambro effective nonvanishing conjecture. Then

Let $X$ be a hyperkähler variety of dimension $2n$, and let $L$ be a nef and big line bundle on $X$. Hyperkähler nonvanishing and Todd-class positivity conjecture. One has … moreov…

A perfect Fano manifold is a Fano manifold on which every nef line bundle has a nontrivial global section. Kawamata's perfect-Fano conjecture. Every Fano manifold is perfect; equiv…