# Kahler manifolds and algebraic varieties

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?

## 1 Answer

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.