Algebraic Geometry versus Kähler geometry

These notes discuss Hodge theory in the algebraic and Kähler context. We introduce the notion of (polarized) Hodge structure on a cohomology algebra and show how to extract from it topological restrictions on compact Kähler manifolds, and stronger topological restrictions on projective complex manifolds. The second part of the notes is devoted to the discussion of the Hodge conjecture, showing in particular that there is no way to extend it to the Kähler context. We will also discuss algebraic de Rham cohomology which is specific to projective complex manifolds and allows to formulate a number of arithmetic questions related to the Hodge conjecture.