Hodge structures on cohomology algebras and geometry

It is well-known (see eg [22]) that the topology of a compact Kähler manifold X is strongly restricted by Hodge theory. In fact, Hodge theory provides two sets of data on the cohomology of a compact Kähler manifold. The first data are the Hodge decompositions on the cohomology spaces Hk(X,C) (see (1.1) where V