On the homotopy types of compact Kähler and complex projective manifolds

The celebrated Kodaira theorem [6] says that a compact complex manifold is projective if and only if it admits a Kähler form whose cohomology class is integral. This suggests that Kähler geometry is an extension of projective geometry, obtained by relaxing the integrality condition on a Kähler class. This point of view, together with the many restrictive conditions on the topology of Kähler manifolds provided by Hodge theory (the strongest one being the formality theorem [4]), would indicate that compact Kähler manifolds and complex projective ones cannot be distinguished by topological invariants. This is supported by the results known for Kähler surfaces, for which a much stronger statement is known, as a consequence of Kodaira’s classification : recall first that two compact complex manifolds X and X ′ are said to be deformation equivalent if there exist a family π : X → B, where B is a connected analytic space and π is smooth and proper, and two points b, b ∈ B such that