8 problems

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 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 the intersection in ${\bf P}^n$ of at least $n/2$ general hypersurfaces of sufficiently high degrees. Cotangent ampleness conjecture. The cotangent bundle $\Omega_X$ is…

Let $X$ be an $n$-dimensional nonsingular projective algebraic variety over the complex numbers, and let $\Delta=\sum_{i\in I}\Delta_i$ be a reduced simple normal crossing divisor,…

Let $F$ be a totally real field in which $p$ is unramified, let $S$ denote the integral model of the Hilbert modular variety with good reduction at $p$, and let $\mathcal{L}$ be th…

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…