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Let $X$ be a smooth complete algebraic variety over $\mathbb{C}$. Can it happen that the underlying complex manifold is not Kahler? If yes, are there explicit examples? If not - how to prove this?

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    $\begingroup$ Yes, that can and does happen. Please see "Hironaka's example" in the appendices of Hartshorne's "Algebraic geometry". $\endgroup$ – Jason Starr Oct 10 '18 at 19:17
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    $\begingroup$ Relevant: mathoverflow.net/questions/108307/… $\endgroup$ – M.G. Oct 10 '18 at 19:25
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Nonprojective compact algebraic manifolds are never Kähler. Any compact algebraic manifold is Moishezon, and Moishezon's theorem says that a Moishezon manifold is Kähler if and only if it is a projective variety.

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