9 problems

Let $X$ be a compact Kähler manifold that is special, and let $p>0$. For every rank-one coherent subsheaf $L\subset\Omega_X^p$, the numerical cotangent bound conjecture. One has ……

Let $X$ be a smooth projective variety and let $D$ be a pseudo-effective $\mathbb R$-divisor on $X$. Nakayama's numerical-dimension growth conjecture. There exist a positive intege…

Let $X$ be a compact Kähler manifold. For a nef class $L$, write $\operatorname{nd}(L)$ for its numerical dimension, and let $M$ be another nef class and $\alpha$ a movable class.…

Let $X$ be a smooth complex projective manifold. If $K_X$ is pseudoeffective, write $\nu_1(X):=\nu(X,\Omega_X^1)$. If $X$ is uniruled, let $r:X\to R_X$ be its maximal rationally co…

Work over $\mathbb{C}$. Let $(X,\Delta)$ be a projective klt log pair of dimension $n\geq 2$. Let $\kappa(X,\Delta)$ denote the Kodaira dimension of $K_X+\Delta$, and let…

Numerical-dimension conjecture. The numerical dimension satisfies

Let $X$ be a smooth projective variety, let $Y\subset X$ be a smooth curve with ample normal bundle, and let $\eta\in\operatorname{N}^{1}(X)$ be a pseudoeffective class such that…

Let $(L,\varphi)$ be a pseudo-effective line bundle on a compact Kähler manifold $X$ of dimension $n$ such that $\frac{i}{2\pi}\Theta_{\varphi}(L)\geq 0$, and suppose that…

Numerical-dimension conjecture. If