8 problems

Let $X$ be a projective manifold such that the cotangent bundle $\Omega_X$ is pseudoeffective. Assume that $X$ admits an abelian fibration $f:X\to C$ onto a curve $C$, and let $F$…

Let $X$ be a projective manifold, let $q$ satisfy $1 e q$ and $q e dim X$ only if excluded?

Let $X$ be a Fano manifold of Picard number $1$ with dimension at least $3$. The normalized tangent bundle of $X$ is pseudoeffective if and only if $X$ is one of the following vari…

MRC-fibration pseudoeffectivity conjecture. One has

Reflexive differential nonvanishing conjecture. The sheaf $\Omega^{[q]}_X$ is pseudoeffective if and only if there exists a positive integer $m$ such that

Symmetric differential nonvanishing conjecture. The cotangent bundle is pseudoeffective if and only if there exists a positive integer $m$ such that

Let $2n$ be an even natural number. Let $(X,H)$ be a polarised hyperkähler manifold, where $X$ has dimension $2n$ and $H$ is an ample Cartier divisor. Write … for the projectivised…

Pseudoeffectivity–uniruledness conjecture. The canonical class $K_X$ is pseudoeffective if and only if $X$ is not uniruled.