# Pseudoeffective line bundle

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?

• Well, yes, even if $L$ is ample one can take some K\"ahler metric $\omega$ and subtract some large multiple of $dd^c\phi$ for some $\phi$. If you choose a decent $\phi$ this will no longer be semi-positive. – Ruadhaí Dervan Jul 12 '16 at 10:33
• @Jarek: When you say "not semipositive", do you mean "nowhere semipositive"? Then the trivial bundle on the projective line is pseudoeffective, but any connection on it has semipositive curvature somewhere: no, trivially. On the other hand, if you mean "not semipositive at some point", then Ruadhai's comment applies: yes. – Ben McKay Jul 12 '16 at 12:28

## 1 Answer

There is a singular Hermitian metric on $L$ whose curvature is semipositive. See Demailly, On the cohomology of pseudoeffective line bundles, for a very clear survey.