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We show that in every dimension greater than or equal to 4, there existcompact Kaehler manifolds which do not have the homotopy type of projectivecomplex manifolds. Thus they a fortiori are not deformation equivalent to aprojective manifold, which solves negatively Kodaira's problem.  We give both non simply connected (of dimension at least 4) and simplyconnected (of dimension at least 6) such examples.Comment: To appear in Inventiones Mat

On the homotopy types of compact Kähler and complex projective manifolds