Harmonic Forms, Hodge Theory and the Kodaira Embedding Theorem

TL;DR

This paper provides an accessible overview of harmonic forms, Hodge theory, and their applications, including proofs of key theorems like the Hodge theorem, Hodge decomposition, and Kodaira embedding, with additional insights into vector bundles.

Contribution

It offers self-contained proofs of fundamental results in differential geometry and complex manifold theory, emphasizing the interplay between harmonic forms and complex projective embeddings.

Findings

Proof of Hodge theorem for compact complex manifolds

Hodge decomposition for Kähler manifolds

Criteria for a manifold to be a projective variety

Abstract

In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a compact complex manifold, the de Rham cohomology group is isomorphic to the group of harmonic forms, (2) Hodge decomposition theorem, which states that for a K\"ahler manifold, the de Rham cohomology group decomposes into the Dolbeault cohomology groups, and (3) The Kodaira Embedding theorem, which gives a criterion of when a compact complex manifold is in fact a smooth complex projective variety. The basic theory of vector bundles is also contained for completeness.