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?



Are complex varieties Kahler? - Algebraic, non-projective complex manifolds
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.


