Let $M$ be a compact complex surface which admits a holomorphic line bundle $L$ with $c_1^2(L)>0$. Can we prove that $M$ is projective?
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2$\begingroup$ If $c_1^2(L)$ is positive, then by Riemann-Roch and Serre duality, either $L$ or $L^\vee$ is big. So $M$ is at least Moishezon. Then use "Goodman's theorem" to prove that $M$ is in fact projective. $\endgroup$– Jason StarrCommented Oct 27, 2022 at 10:58
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Yes, this is precisely Theorem (6.2), p. 160 of
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.
answered Oct 27, 2022 at 11:41
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$\begingroup$ Just to comment on the difference between that proof and the one I sketched: in the proof using Goodman's theorem, we "forget" $L$ after we use it to prove that $M$ is Moishezon. Instead, Goodman shows that the closed complement of any Zariski open that is affine supports an ample divisor. So instead of modifying $L$ to satisfy Nakai-Moishezon or Grauert, we just use Goodman's ample divisor. $\endgroup$ Commented Oct 28, 2022 at 0:25