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 H(X,C) (see (1.1) where V = H(X,Q)); they depend only on the complex structure. The second data, known as the Lefschetz isomorphism and the Lefschetz decomposition on cohomology (see (1.5) with AR = H (X,R)) depend only on the choice of a Kähler class, but remain satisfied by any symplectic class close to a Kähler class. Both are combined to give the so-called Lefschetz bilinear relations, which lead for example to the Hodge index theorem (cf [22], 6.3.2) which computes the signature of the intersection form on the middle cohomology of an even dimensional compact Kähler manifold as an alternate sum of its Hodge numbers. If we want to extract topological restrictions using these informations, we are faced to the following problem: neither the complex structure (or even the Hodge numbers), nor the (deformation class) of the symplectic structure (or even the symplectic class) are topological. For this reason, only a very small number of purely topological restrictions have been extracted so far from these data. (Note however that the formality theorem [7], which uses more than the data above, is a topological statement. Similarly, non-abelian Hodge theory has provided strong restrictions on π1(X) (cf [3]).) The classically known restrictions are the following: