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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:

  1. Why nefness is independent of choice of $\omega$? It seems not obvious for me from the definition.

  2. 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?

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    $\begingroup$ There are several notions of positivity for $(p,p)$ forms, they all agree for $p=1$ or $p=n-1$, see Demailly's book, Chapter III. Therefore, if $\alpha$ and $\beta$ are $(1,1)$ forms, $\alpha\geq -\beta$ simply means that $\alpha+\beta$ is positive. And the definition of a nef class doesn't depend on the fixed metric $g$ because if you use another metric $h$, then there exist two constants $C_1$ and $C_2$ such that $g\leq C_1h$ and $h\leq C_2g$. $\endgroup$
    – user48958
    Jan 19, 2023 at 21:14
  • $\begingroup$ @user48958 How to show that two metrics can control each other? Or is there any reference for it? $\endgroup$
    – Hydrogen
    Jan 20, 2023 at 0:41

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