Hodge theory on compact Kähler manifolds

Abstract

The purpose of this master thesis is to study the Hodge Decomposition Theorem for compact Kähler manifolds. Hodge theory, named after W.V.D. Hodge, is a branch of mathematics belonging to both algebraic topology and differential geometry that enables us to find topological information about a smooth or complex manifold from the study of differential forms and differential operators on these manifolds. Namely, we can find the singular cohomology groups or deduce properties of them with new tools derived from the Hodge theory. It was first developed in the 1930s as an extension of the de Rham cohomology. Recall that the de Rham cohomology gives an isomorphism between the singular cohomology of a smooth manifold and the de Rham cohomology, which is given by the study of the differential forms on that manifold. Hodge theory includes this particular case and extends the results to more general types of manifolds. The Theorem we are going to study, the Hodge Decomposition Theorem on compact Kähler manifolds, gives a decomposition of the singular kth cohomology group of a manifold of a specific type called Kähler. Even though the condition of being K ̈ahler may seem very restrictive, there exists lots of examples of manifolds satisfying this condition. For example, we can think of Kähler manifolds as submanifolds of the complex projective space.