Can someone provide examples of Kähler manifolds which are not algebraic?
This question came to my mind seeing the post of Andrea Ferretti.
Can someone provide examples of Kähler manifolds which are not algebraic? This question came to my mind seeing the post of Andrea Ferretti. 


generic complex tori in complex dimension 2 or higher. MR 


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. 


You might want to take a look at this previous MO question. There, I mentioned Voisin's results disproving Kodaira's conjecture (every Kahler manifold is deformation equivalent to a projective manifold). 

