3 problems

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…

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…

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.