Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the same)?

Does there exist a smooth proper complex variety whose analytification is not homotopy equivalent to the analytification of a smooth projective complex variety?

  • $\begingroup$ By the way, not every Moishezon manifold comes from a proper complex variety; it might be an algebraic space instead of a scheme. There is an example due to Kleiman which is a quotient of Hironaka's example. $\endgroup$ – Ben Dec 9 '18 at 12:47
  • $\begingroup$ @Ben that is true of course but we are interested in whether that distinction is visible at the topological level $\endgroup$ – user132250 Dec 9 '18 at 13:03
  • $\begingroup$ Sorry, I'm a bit off today. I misunderstood something you wrote.. (I thought you were suggesting hat every Moishezon manifold is the analytification of a proper variety.) $\endgroup$ – Ben Dec 9 '18 at 13:12

Edit: I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.

The only example I am aware of is due to Oguiso, Two remarks on Calabi-Yau Moishezon threefolds, J. reine angew. Math. 452 (1994). There, you find a construction of a Moishezon threefold $X$ with $H^2(X) = \mathbb{Z}c$ where $c^3 = 0$. In particular, $X$ is not even homotopy equivalent to a Kähler manifold.

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