Compact Complex Manifolds

Abstract

In the first half of the twentieth century, new tools were developed for the study of the algebraic topology of manifolds, and in particular, complex manifolds: de Rham’s representation of algebraic topology with differential forms, Kahler’s introduction of specific types of Riemannian metrics for complex manifolds, and Hodge’s theory of harmonic differential forms. These tools, along with the theory of sheaf cohomology developed by Leray somewhat later, allowed Kodaira to formulate and prove in 1954 an embedding theorem for compact complex manifolds which completely characterized the abstract manifolds that could be embedded as a submanifold of a complex projective space.