Are these two definitions of nef-ness equivalent for Moishezon manifolds? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:41:29Z http://mathoverflow.net/feeds/question/65810 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65810/are-these-two-definitions-of-nef-ness-equivalent-for-moishezon-manifolds Are these two definitions of nef-ness equivalent for Moishezon manifolds? xiao 2011-05-24T03:38:34Z 2011-06-15T02:32:19Z <p>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 $X$.</p> <p>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).</p> <p><strong>Question:</strong> Is this equivalence also true for Moishezon manifolds?</p> <p>I don't know of any counterexamples. If it is not true, could someone give me a counterexample?</p> http://mathoverflow.net/questions/65810/are-these-two-definitions-of-nef-ness-equivalent-for-moishezon-manifolds/66428#66428 Answer by mrw for Are these two definitions of nef-ness equivalent for Moishezon manifolds? mrw 2011-05-30T08:51:15Z 2011-05-30T08:51:15Z <p>what is $\omega$ here? if it is a Kahler form, then Moishezon + Kahler implies projective, and as you said they are equivalent. </p> http://mathoverflow.net/questions/65810/are-these-two-definitions-of-nef-ness-equivalent-for-moishezon-manifolds/67821#67821 Answer by YangMills for Are these two definitions of nef-ness equivalent for Moishezon manifolds? YangMills 2011-06-15T02:32:19Z 2011-06-15T02:32:19Z <p>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</p> <p>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.</p>