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).