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# NeflinebundleonArethesetwodefinitionsofnef-nessequivalentfor Moishezon manifoldmanifolds?

Hi,everyone! Nowadays

Recently, I has have been learning something about nef line bundle,I bundles. I know that when $X$ is projective or Moishezon,a Moishezon, a line bundle $L$ over $X$ is said to be nef iff $L.C=\int_{C}C_{1}(L)\ge 0$ $L.C=\int_{C}c_{1}(L)\ge 0$$for every curve$C$in$X$.Moverover,Demailly had given X$.

Demailly gave a definition of nefness that works on an arbitrary compact complex manifold,i.e.,a manifold, i.e., a line bundle $L$ over $X$ is said to be nef if for every $\varepsilon >0$ there exists a smooth hermitian metric $h_{\varepsilon}$ on $L$ such that its curvature $\Theta_{h_{\varepsilon}}(L)\ge -\varepsilon\omega$.And for the \varepsilon\omega\$. For projective manifolds,Demailly's manifolds, Demailly's definition coincides with the above one given by integration (this is an easy consequence of Seshadri's ampleness criterion).I wonder whether it is criterion).

Question: Is this equivalence also true for the Moishezon manifolds.By now,I have no manifolds?

I don't know of any counter examples.If counterexamples. If it is not true,can true, could someone gives give me an counter-examplea counterexample?

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# Nef line bundle on Moishezon manifold

Hi,everyone! Nowadays I has been learning something about nef line bundle,I know when $X$ is projective or Moishezon,a line bundle $L$ over $X$ is said to be nef iff $L.C=\int_{C}C_{1}(L)\ge 0$ for every curve $C$ in $X$.Moverover,Demailly had given a definition of nefness on an arbitrary compact complex manifold,i.e.,a line bundle $L$ over $X$ is said to be nef if for every $\varepsilon >0$ there exists a smooth hermitian metric $h_{\varepsilon}$ on $L$ such that its curvature $\Theta_{h_{\varepsilon}}(L)\ge -\varepsilon\omega$.And for the projective manifolds,Demailly's definition coincides with the above one given by integration (this is an easy consequence of Seshadri's ampleness criterion).I wonder whether it is also true for the Moishezon manifolds.By now,I have no any counter examples.If it is not true,can someone gives me an counter-example?
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