31 problems

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…

Let $X$ be a projective contact manifold with nef canonical divisor $K_X$. Since projective contact manifolds have $\kappa(X)=-\infty$, the abundance conjecture predicts that this…

Let $(X,B)$ be a log minimal model with $\dim(X)=4$, and let $C$ be a very general curve in $X$. Four-dimensional numerical non-vanishing conjecture. If $(K+B)\cdot C>0$ for very g…

Let $(X,B)$ be a complex projective log variety with Kawamata log terminal singularities such that $K+B$ is nef. For a positive integer $m$, write $|m(K+B)|$ for the pluricanonical…

Let $X$ be a Calabi–Yau $3$-fold and let $L$ be a nef divisor on $X$. Nef abundance conjecture. The divisor $L$ is semiample. The surrounding discussion derives this assertion from…

Let $X$ be a minimal model, meaning a complex projective variety with at most terminal singularities and nef canonical divisor $K$. For a nef divisor $K$, its nef dimension…

Let $(X,\Delta)$ be a Kähler log canonical pair. A divisor is semiample if some positive multiple is base-point free. Abundance conjecture. If $K_X+\Delta$ is nef, then…

Abundance conjecture. If the restriction $[mD]|_Y$ is holomorphically trivial for some positive integer $m$, then the complete linear system $|mD|$ should yield a fibration on $X$;…

Let $X$ be a projective Calabi–Yau manifold, meaning a compact Kähler manifold with $c_1(K_X)=0$ in $H^{1,1}(X,\mathbb{R})$, and let $L\to X$ be a nef line bundle. Numerical abunda…

Let $X$ be a projective manifold with $K_X$ pseudoeffective, and let $L\to X$ be a nef line bundle such that … is nef for some $m\geqslant 1$. Lazić–Peternell's generalized abundan…

Let $X$ be a $k$-coframed projective manifold, let $q=h^0(\Omega^1_X)$, and suppose that $q=k$. Assume also that the canonical divisor $K_X$ is nef. Pseudo-Abelian variety conjectu…

Let $(X,\Delta)$ be a $\mathbb{Q}$-factorial projective klt pair such that $K_X+\Delta$ is pseudoeffective. A divisor $D$ is abundant if its Iitaka dimension equals its BDPP numeri…

Let $X$ be a Calabi–Yau variety, meaning a normal projective variety with only ${\mathbb Q}$-factorial terminal singularities, $h^1(X,{\mathscr O}_X)=0$, and numerically trivial ca…

Abundance conjecture. If $K_X+\Delta$ is nef, then it is semiample.

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

Let $f\colon (X, B_X) \to (Y,D)$ be a very basic slc-trivial fibration, where $D$ is the divisor in its structure decomposition. The abundance conjecture. If $D$ is nef, then $D$ i…

Let $X$ be a projective manifold with nef canonical class $K_X$. The canonical class is semi-ample if some positive tensor power of the associated line bundle is generated by its g…

Let $Y$ be a variety of Kodaira dimension $0$, and let $K_Y$ denote its canonical divisor. A divisor is nef if it has nonnegative intersection with every curve, and semiample if so…

Let $M$ be a hyperkähler manifold admitting a Lagrangian fibration, and let $\pi\colon M\to B$ be such a fibration. If $\omega\in H^2(B,\mathbb Z)$ is an ample class, then…

Let $X$ be a minimal projective variety, and let $K_X$ denote its canonical divisor. Abundance conjecture. The canonical divisor $K_X$ is always semi-ample. The conjecture is a cen…

Generalized abundance conjecture.

Let $X$ be a compact Kähler manifold and let $\omega(t)$ be a solution of the Kähler-Ricci flow defined on $[0,\infty)$. Volume-growth conjecture. The Kodaira dimension satisfies…

Lower-control and limit conjecture. Uniformly on $X\times[0,\infty)$, there is a constant $C$ such that

A line bundle $L$ is semiample if some positive tensor power $L^{\otimes r}$ is generated by global sections, and it is nef if it has nonnegative degree on every curve. Assume that…

Let $(X,\Delta)$ be a projective log canonical pair with $\Delta$ a $\mathbb Q$-divisor. Let $\kappa_\sigma(X,K_X+\Delta)$ denote Nakayama's numerical dimension, and let…