See [Claire Voisin's amazing results on the subject,][1] or the published version: 

On the homotopy types of compact kaehler and complex projective manifolds,
*Inventiones Math*. **157** 2 (2004), 329 - 343. 

**(ArXiv) Abstract:** We show that in every dimension greater than or equal to 4, there exist compact Kaehler manifolds which do not have the homotopy type of projective complex manifolds. Thus they *a fortiori* are not deformation equivalent to a projective manifold, which solves negatively Kodaira's problem.
We give both non simply connected (of dimension at least 4) and simply connected (of dimension at least 6) such examples. 


  [1]: https://arxiv.org/abs/math/0312032