TITLE: Are complex varieties Kahler? - Algebraic, non-projective complex manifolds
QUESTION [8 upvotes]: Let $X/\mathbb{C}$ a nonsingular proper variety and $X_{an}$ it's associated analytic space. Is $X_{an}$ necessarily Kahler? Certainly we know this if $X$ is projective.
A complex torus is algebraic iff it is projective. Are there Kahler manifolds which are algebraic, but not projective?

REPLY [16 votes]: Any abstract algebraic compact complex manifold is Moishezon.  By Moishezon's theorem, any Kähler Moishezon manifold is projective algebraic.  There are non-projective proper complex varieties, so $X_{an}$ is not necessarily Kähler.  This is represented in the diagram at the end of Hartshorne's Algebraic Geometry Appendix B.
In summary, all of your questions have negative answers.