18 problems

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…

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…

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…

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 $X^n$ be a Calabi–Yau manifold and let $\alpha$ be a closed real $(1,1)$-form such that $[\alpha]\in\partial\mathcal{C}_X$ is nef. Bounded-potential conjecture. There is a boun…

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 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…

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…

Let $X$ be a compact Kähler manifold and let $[\alpha]$ be a nef $(1,1)$ class such that $[\alpha]+\lambda c_1(X)$ is a Kähler class for some $\lambda>0$. A closed positive current…

Weakly pseudo-effective pushforward conjecture. Any weakly pseudo-effective class $\alpha\in H^{2l}(X,\mathbf{R})$ such that $\varphi_*\alpha=0$ belongs to the real vector space sp…