This book is a written-up and expanded version of eight lectures on the Hodge theory of projective manifolds. It assumes very little background and aims at describing how the theory becomes progressively richer and more beautiful as one specializes from Riemannian, to Kahler, to complex projective manifolds. Though the proof of the Hodge Theorem is omitted, its consequences - topological, geometrical and algebraic - are discussed at some length. The special properties of complex projective manifolds constitute an important body of knowledge and readers are guided through it with the help of selected exercises. Despite starting with very few prerequisites, the concluding chapter works out, in the meaningful special case of surfaces, the proof of a special property of maps between complex projective manifolds, which was discovered only quite recently.
Calculus on Smooth Manifolds; The Hodge Theory of a Smooth, Compact, Oriented, Riemannian Manifold; Complex Manifolds; Hermitean Linear Algebra; Hermitean Manifolds; Kahler Manifolds; The Hard Lefschetz Theorem and the Hodge-Riemann Bilinear Relations; Mixed Hodge Structures, Semi-Simplicity and Approximation.