# Semipositive curvature on holomorphic line bundle

Let $$(X,\omega)$$ be a (possibly non-Kähler) compact hermitian manifold and let $$L\rightarrow X$$ be a holomorphic line bundle. Is there an algebraic characterization of (Griffiths) semi-positivity of $$L$$? i.e. such that there is a non-singular smooth hermitian metric $$h$$ on $$L$$ such that $$iF_{\nabla^C}\geq0$$?

This condition should be stronger than nef (which implies that for any $$\epsilon>0$$ there is $$h_\epsilon$$ hermitian on $$L$$ s.t. $$iF_{\nabla^C(h_\epsilon)}\geq -\epsilon \omega$$) and a regular version of pseudoeffective (for which one has a class of singular hermitian metrics with 'minimal singularities', see "Analytic methods in algebraic geometry" $$\S$$6, J. P. Demailly).

• I don't think that there is a characterization, no. Of course, you always have the implication $L$ semi-ample (i.e. $L^{\otimes m}$ is basepoint free for some $m>0$) implies $L$ admits an hermitian metric with semipositive curvature. Nov 8, 2021 at 13:25