Are these two definitions of nef-ness equivalent for Moishezon manifolds? - MathOverflow

Are these two definitions of nef-ness equivalent for Moishezon manifolds?

2011-05-24T03:38:34Z

Recently, I have been learning about nef line bundles. I know that 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 $X$.

Demailly gave a definition of nefness that works 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 $\Theta_{h_{\varepsilon}}(L)\ge -\varepsilon\omega$. For projective manifolds, Demailly's definition coincides with the above one given by integration (this is an easy consequence of Seshadri's ampleness criterion).

Question: Is this equivalence also true for Moishezon manifolds?

I don't know of any counterexamples. If it is not true, could someone give me a counterexample?

Answer by mrw for Are these two definitions of nef-ness equivalent for Moishezon manifolds?

2011-05-30T08:51:15Z

what is $\omega$ here? if it is a Kahler form, then Moishezon + Kahler implies projective, and as you said they are equivalent.

Answer by YangMills for Are these two definitions of nef-ness equivalent for Moishezon manifolds?

2011-06-15T02:32:19Z

Yes, the equivalence is true (the second notion used to be called "metric nef" by some). This was an open problem for quite some time until it was solved in

M. Paun "Sur l'effectivité numérique des images inverses de fibrés en droites" Math. Ann. 310 (1998), no. 3, 411–421, see the Corollaire on page 412.