# 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?

• Yes, that can and does happen. Please see "Hironaka's example" in the appendices of Hartshorne's "Algebraic geometry". – Jason Starr Oct 10 '18 at 19:17
• Relevant: mathoverflow.net/questions/108307/… – M.G. Oct 10 '18 at 19:25

## 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.