27 problems

Let $X$ be a compact Kähler manifold, let $K_X$ denote its canonical bundle, and suppose that $X$ is Kobayashi hyperbolic. Kobayashi–Lang conjecture. The canonical bundle $K_X$ is…

Uniform lower-bound conjecture. There is a number $C(m,g)>0$, depending only on $m$ and $g$, such that

Let $Y$ be a projective manifold such that for every finite surjective morphism $f:X\to Y$ from a connected projective manifold, the associated trace-zero bundle ${\cal E}$ is ampl…

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…

Abundance conjecture. If $X$ is minimal, then $K_X$ is abundant; equivalently, $|mK_X|$ is base point free for some $m \gg 0$.

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…

Abundance conjecture. If $K_X$ is nef, then $K_X$ is semi-ample; equivalently, a sufficiently large power of $K_X$ is globally generated or base point free.

Kobayashi--Lang conjecture. The manifold $X$ is canonically polarised.

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…

Abundance conjecture. On any minimal projective variety $M$, some positive tensor power of the canonical bundle is spanned by its global sections:

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…

Let $X$ be a compact Kähler manifold admitting a Kähler metric whose holomorphic sectional curvature is negative at every point and in every complex tangent direction. Yau's amplen…

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.

Let $B$ be a Ramond boundary component of the moduli space $\overline{{\mathcal S}}$ of stable supercurves, with gluing map $iota:B\tooverline{{\mathcal S}}$, and let…

Kawamata's conjecture. The locally free sheaf

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

Yau's conjecture. The canonical line bundle $K_M$ is ample.

Yau's conjecture. The manifold $X$ also admits a Kähler metric with negative Ricci curvature.

Let $M$ be a projective minimal model, meaning that its canonical divisor $K_M$ is nef. A line bundle is semi-ample if some sufficiently large multiple is globally generated. Abund…

Abundance conjecture. If $K_X$ is nef, then $K_X$ is semiample; namely, there exists an integer $\ell\geqslant 1$ such that $K_X^\ell$ is base-point-free.

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…