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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).

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    $\begingroup$ 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. $\endgroup$
    – Henri
    Nov 8, 2021 at 13:25

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