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?

up vote 11 down vote accepted

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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