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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?

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    $\begingroup$ A smooth proper complex variety is Kaehler iff it is projective. $\endgroup$
    – naf
    Commented Sep 28, 2012 at 4:12

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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.

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