Can someone provide examples of Kähler manifolds which are not algebraic?
This question came to my mind seeing the post of Andrea Ferretti.
In (complex) dimension greater or equal with 2 there are complex tori which are not algebraic. A criterion for projectivity is that of Riemann (see http://en.wikipedia.org/wiki/Abelian_variety#Riemann_conditions). Also there are K3 surfaces which are not algebraic.
It doesn't in itself give specific examples, but the theoretical answer to the question "When is a compact Kahler manifold algebraic?" is given by the Kodaira embedding theorem. A nice exposition is given in Richard Wells' Differential Analysis on Complex Manifolds. Indeed, the KET is the crescendo to which the entire book builds.