16 problems

Campana's torus-submersion conjecture. Up to a finite étale cover of $X$, the manifold $X$ admits a torus submersion over a projective manifold $Y$ such that $K_Y$ is ample and the…

Let $X$ be a compact complex manifold, let $K_X$ denote its canonical bundle, and let holomorphic sectional curvature refer to the curvature of a Hermitian metric on $X$. A Hermiti…

Kobayashi's conjecture. The canonical bundle $K_X$ is ample.

Abundance conjecture. $K_X$ is semiample.

Let $(X,\lambda)$ be a polarized irreducible holomorphic symplectic manifold of $\mathrm{K3}^{[n]}$-type with $q_X(\lambda)=2$ and $\operatorname{div}(\lambda)=1$. Let…

Let $X$ be a smooth projective variety. A $\mathbb{Q}$-divisor is effective if it is a nonnegative $\mathbb{Q}$-linear combination of prime divisors, and two $\mathbb{Q}$-divisors…

Let $X$ be a smooth projective variety, and let $\omega_X$ denote its canonical sheaf. A line bundle is semi-ample if some positive tensor power is globally generated; in the formu…

Exterior-power ampleness conjecture. The following assertions should hold:

Strictly nef canonical bundle conjecture. If $K_X$ is strictly nef, then $K_X$ is ample.

Kawamata's conjecture. The locally free sheaf

Kobayashi–Lang conjecture. The canonical line bundle $K_X$ is ample.

Let $X$ be a compact complex manifold in class $\mathcal C$, meaning that $X$ is bimeromorphic to a compact Kähler manifold. Let $K_X$ denote its canonical bundle. A line bundle is…

The optimal rank-form extension of Yau's conjecture. One should have

Let $X$ be a projective hyperbolic manifold. Kobayashi's conjecture. The canonical bundle $K_X$ is ample. This is a necessary-condition prediction related to the Green–Griffiths–La…

Hyperbolicity implies general type conjecture. If $X$ is hyperbolic, then $X$ should be of general type; equivalently, $K_X$ should be big and $X$ should have maximal Kodaira dimen…

Let $X$ be a projective manifold, let $L$ be a numerically trivial line bundle, and let … be a non-zero section with $q<\dim X$. A finite étale cover is a finite surjective morphis…