19 problems

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 $X$ 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. The class $[\alpha]$ is semiample.

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 $X$ 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…

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…

Generalized abundance conjecture.

Uniruledness conjecture. $X$ is uniruled, meaning covered by rational curves, if and only if

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

Let $(Y/Z,B_Y)$ be a log canonical (lc) pair, and suppose that $K_Y+B_Y$ is nef over $Z$. A divisor is semi-ample over $Z$ if it defines a contraction over $Z$ after a suitable pos…

Let $X$ be a Calabi–Yau manifold and let $L\in H^2(X,\mathbb{Q})$ be a $(1,1)$-class such that … for every algebraic curve $C\subset X$. Strong abundance for Calabi–Yau manifolds.…

Let $(X\vert\Delta)$ be a log-canonical pair. Define the numerical dimension $\nu(X,D)$ for a rational divisor $D$ by asymptotic growth of sections after adding an ample divisor. A…

Abundance conjecture. The canonical divisor $K_X$ is abundant, namely