I'm reading Demailly's book 'Analytic Methods in Algebraic Geometry'.
Let $X$ be a compact complex manifold with a Hermitian metric. A line bundle $L$ is said to be nef if for every $\epsilon>0$, there is a smooth metric $h_\epsilon$ on $L$ such that $i\Theta_{L,h_\epsilon}\geq -\epsilon\omega$.
It is equivalent to nefness in projective setting. But in the proof, he says the definition is independent of choice of $\omega$ and select a special $\omega$ to do the argument.
My question:
Why nefness is independent of choice of $\omega$? It seems not obvious for me from the definition.
How does inequality of forms make sense? For example, in $i\Theta_{L,h_\epsilon}\geq -\epsilon\omega$, both sides are forms. How could we compare forms?