Kähler manifolds and projective manifolds. The goal of this first volume is to explain the existence of special structures on the cohomology of Kähler manifolds, namely, the Hodge decomposition and the Lefschetz decomposition, and to discuss their basic properties and consequences. The second volume will be devoted to the systematic application of these results in different directions, relating Hodge theory, topology and the study of algebraic cycles on smooth projective complex manifolds.
Indeed, smooth projective complex manifolds are special cases of compact Kähler manifolds. A Kähler manifold is a complex manifold equipped with a Hermitian metric whose imaginary part, which is a 2-form of type (1, 1) relative to the complex ...