We show a number of applications to geometry of the study of cohomology algebras of various kinds of manifolds. The main tool is Hodge theory, and we use it to show that projective complex manifolds are more restricted topologically than compact Kähler manifolds. We also make explicit numerous constraints satisfied by cohomology algebras of compact Kähler manifolds, making them very non generic amongst cohomology algebras of symplectic manifolds satisfying the hard Lefschetz property. Proof.

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... eed, the wedge product of a closed form of type (p, q) and a closed form of type (p , q ) is a closed form of type (p + p , q + q ).