Let $X$ be a compact complex manifold, and let $L \to X$ be a holomorphic line bundle. Then $L$ is positive if and only if the Bott-Chern first Chern class $c_1^{BC}(L)$ contains a positive real closed $(1,1)$-form.
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Equivalently, $L$ is positive if and only if there exists a smooth Hermitian metric $h$ on $L$ whose Chern form $c_1(L,h)$ is positive. In particular, if $X$ is compact Kähler and the de Rham first Chern class $c_1(L)\in H^2(X,\mathbb R)$ contains a positive real closed $(1,1)$-form, then $L$ is positive.