12 problems

Let $F$ be a projective manifold of dimension $d$ with Kodaira dimension $\kappa(F)=0$, and let $L$ be an almost strictly nef line bundle, meaning that $L$ is the pullback of a str…

Let $X$ be a projective manifold of dimension $n$, and let $L$ be a strictly nef line bundle, meaning that $L\cdot C>0$ for every irreducible curve $C\subset X$. Serrano's conjectu…

Let $X$ be a projective variety of dimension $n$, and let $D$ be a strictly nef divisor on $X$, meaning that $D\cdot C>0$ for every irreducible curve $C$ in $X$. Serrano's conjectu…

Let $X$ be a projective log canonical variety, and suppose that $-K_X$ is a strictly nef divisor. Log version of Campana--Peternell's conjecture. The anti-canonical divisor $-K_X$…

Let $X$ be a projective log canonical variety with numerically trivial canonical divisor, so that $K_X\equiv 0$, and let $L$ be a strictly nef $\mathbb Q$-Cartier divisor on $X$. A…

Let $(X,\Delta)$ be a projective log canonical pair of dimension $n$, and let $L$ be a strictly nef Cartier divisor on $X$. Log version of Serrano's conjecture. The $\mathbb Q$-Car…

Weak birational Serrano conjecture. The divisor

Non-vanishing conjecture for strictly nef anti-canonical divisors. One should have

Rational connectedness conjecture. If $-(K_X+\Delta)$ is strictly nef, then $X$ is rationally connected.

Log version of Serrano's conjecture. The $\mathbb Q$-Cartier divisor $K_X+\Delta+tL$ is ample for $t>2n$.

Ampleness conjecture. Any strictly nef divisor on $X$ is ample.

Let $X$ be a projective manifold of dimension $n$, and let $L$ be a strictly nef divisor, meaning that $L\cdot C>0$ for every irreducible curve $C$ on $X$. Adjoint strict-nefness c…