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

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?

• I've cleaned up the layout of your question a bit. We usually put spaces after periods and commas, because this makes sentences easier to read. May 24, 2011 at 4:46
• The proof that Demailly's definition implies the classical one carries over word-for-word for Moishezon manifolds (or any manifold that admits curves). The other direction is less clear, as we only have big bundles on a Moishezon manifold, and I don't know of a Nakai-Moishezon type criterion for bigness. I think one can work something out, probably by replacing the smooth hermitian metric $h_\epsilon$ by a singular hermitian metric and passing to curvature currents, but I'm not sure that will be equivalent to Demailly's definition with smooth metrics. May 30, 2011 at 11:47

what is $\omega$ here? if it is a Kahler form, then Moishezon + Kahler implies projective, and as you said they are equivalent.
• $\omega$ is the Kahler form of a hermitian metric - i.e. it is the associated real $(1,1)$-form $\omega = -i \Im h$, but it need not be closed. May 30, 2011 at 11:43