Let $L$ be a pseudoeffective line bundle on a complex manifold $X$ then is there a singular hermitian metric of $L$ which its curvature is not semi-positive?



Pseudoeffective line bundle
There is a singular Hermitian metric on $L$ whose curvature *is* semipositive. See Demailly, *[On the cohomology of pseudoeffective line bundles](http://arxiv.org/abs/1401.5432v1)*, for a very clear survey.


