29 problems

Let $X$ be a compact complex manifold, and let a Nevanlinna current be a positive closed $(1,1)$-current arising from an entire curve. By Siu's decomposition, write such a current…

Lelong-number conjecture. If $X$ admits an exact positive $(1,1)$-current not in $L^2_{-1}(X)$, then there exists on $X$ an exact positive current with a non-vanishing Lelong numbe…

Sullivan's conjecture. A positive, closed current might be written locally in this form on a dense, open set. Sullivan's conjecture concerns whether positive closed currents locall…

Pali's potential conjecture. There exists a unique function $f\in Psh(X,J)\cap L^1_{loc}(X)$ whose associated distribution coincides with the distribution $u$.

Pali's conjecture. The function $u$ is plurisubharmonic if and only if the $(1,1)$-current

Demailly's conjecture. $X$ is Moishezon if and only if there exists such a current $T$ satisfying the conditions stated in the continuation of the source.

Tosatti's uniqueness conjecture. The class $[\alpha]$ contains a unique closed positive $(1,1)$-current.

Tosatti's minimal-singularities conjecture. If $\eta\in\mathcal{T}_{[\alpha]}$ and

Tosatti's bounded-potential conjecture. There is $\varphi\in L^\infty(X)$ such that

Let $X$ be a compact Kähler manifold with $K_X$ pseudoeffective, and let $[\alpha]$ be a nef $(1,1)$-class such that … is nef for some $\lambda\in\mathbb{R}_{>0}$. Generalized boun…

Let $X$ be a Calabi–Yau manifold and let $[\alpha]$ be a nef $(1,1)$-class. Bounded-potential conjecture. Then $[\alpha]$ contains a closed positive current with bounded potentials…

Extension conjecture. The vanishing assertion $c_s(chat{E})\cap T=0$ for $s>\operatorname{rank}E$ and the positivity assertion that $c_s(chat{E})$ is a closed positive $(s,s)$-curr…

Pointwise Lelong-number refinement. If $T$ is not in $L^2_{-1}(X)$ and $x otin L^2{-1}(T|X)$, then there exists on $X$ an exact positive $(1,1)$-current with non-vanishing Lelong n…

Hyperbolicity criterion. If all exact positive $(1,1)$-currents $T$ on $X$ are in $L^2_{-1}(X)$ and satisfy $I(T)=0$, then $X$ is hyperbolic.

Rational-cycle conjecture. If every such $T$ belongs to $L^2_{-1}(X)$, but not every such $T$ satisfies $I(T)=0$, then $X$ admits a cycle of rational curves.

Let $X$ be a compact complex manifold of complex dimension $n$. Let $\alpha$ be a closed real $(1,1)$-form, and let $X(\alpha,\leq 1)$ be the set where $\alpha$ is nondegenerate an…

Let $X$ be a compact complex manifold of complex dimension $n$, and let $T$ be a closed positive $(1,1)$-current on $X$. Write $T_{ac}$ for its absolutely continuous part and…

Let $X$ be a compact Kähler manifold, and let $mu_1,mu_2$ be positive strong submeasures on $X$ with equal mass, $\lVert\mu_1\rVert=\lVert\mu_2\rVert$, and $mu_1\geq\mu_2$. A posit…

Let $X$ be a compact Kähler manifold of dimension $k$, let $T$ be a positive closed $(1,1)$ current on $X$, and let $A\reflexprsubset X$ be a subset. The notation…

Let $X$ be a compact complex manifold and let $T$ be a closed positive $(1,1)$-current on $X$. Write $T_{ac}$ for the absolutely continuous part of $T$ in its Lebesgue decompositio…

Demailly's Hodge conjecture for strongly positive currents. The current $\mathscr{T}$ is a weak limit

Let $X$ be a compact Kähler manifold and let $[\alpha]$ be a nef $(1,1)$ class on $X$. Suppose that $\lambda[\alpha]-c_1(K_X)$ is nef and big for some $\lambda>0$. Transcendental b…

Fu–Xiao's conjecture.

Log-concavity conjecture. It was conjectured that

Let $X$ be a compact K4hler manifold of complex dimension $n$. The cone $\mathcal{E}$ of pseudoeffective classes consists of classes in $H^{1,1}_{\mathbb{R}}(X)$ containing a close…