As I understand it, the Nakai-Moishezon criterion gives conditions for the existence of an ample divisor class on an arbitrary proper scheme, and Kleiman's criterion does the same for arbitrary projective schemes (in fact, more generally for 'quasi-divisorial' schemes, as defined in Kleiman's paper).

Are there similar numerical criteria for other 'nice' classes of spaces? In particular, what about Moishezon manifolds? These are not quite schemes, but on the other hand, they nearly are, and they are smooth, so one might hope for a useful result.

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    $\begingroup$ Moishezon manifolds are the same as smooth proper algebraic spaces over $\mathbb{C}$. The Nakai-Moishezon criterion is asserted to hold for algebraic spaces in mathoverflow.net/a/4586/519 . $\endgroup$
    – naf
    Commented Sep 12, 2013 at 13:56
  • $\begingroup$ A 2021 paper of Kollár explains that the Nakai-Moishezon criterion holds, but Kleiman's criterion is not known to hold, for algebraic spaces: arxiv.org/abs/2105.06242 $\endgroup$ Commented Jul 4, 2022 at 18:45
  • $\begingroup$ On the other hand, as Kollár mentions, Kleiman's criterion has been established by Villalobos-Paz for Q-factorial log terminal algebraic spaces (including smooth ones) in characteristic zero: arxiv.org/abs/2105.14630 $\endgroup$ Commented Jul 4, 2022 at 18:50

1 Answer 1


This is adding to ulrich's comment. One reference for this result is the following article of Kollár.

MR1064874 (92e:14008) Reviewed Kollár, János(1-UT) Projectivity of complete moduli. J. Differential Geom. 32 (1990), no. 1, 235–268. 14D22 (14H10 14J10)


Edit. Although you did not ask about this, also Seshadri's criterion has been extended to algebraic spaces. Here is one reference by Cornalba.

MR1241126 (94i:14031) Reviewed Cornalba, M. D. T.(I-PAVI) On the projectivity of the moduli spaces of curves. J. Reine Angew. Math. 443 (1993), 11–20. 14H10 (14D22)


  • $\begingroup$ Thanks for the reference. I had never seen Nakai-Moishezon stated in this generality before. $\endgroup$ Commented Sep 13, 2013 at 14:45

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