The Kodaira Embedding Theorem

TL;DR

The paper discusses the Kodaira Embedding Theorem, providing a sufficient condition for complex manifolds to be embedded in projective space using positive line bundles, and explores related vanishing theorems and lemmas.

Contribution

It proves the 'if' part of the Kodaira Embedding Theorem and presents related results like the Kodaira-Nakano Vanishing Theorem and key lemmas linking divisors, line bundles, and blowups.

Findings

Established a sufficient condition for embedding manifolds into projective space.

Proved the Kodaira-Nakano Vanishing Theorem for higher cohomology elimination.

Provided lemmas relating divisors, line bundles, and blowups.

Abstract

Chow's Theorem and GAGA are renowned results demonstrating the algebraic nature of projective manifolds and, more broadly, projective analytic varieties. However, determining if a particular manifold is projective is not, generally, a simple task. The Kodaira Embedding Theorem provides an intrinsic characterization of projective varieties in terms of line bundles; in particular, it states that a manifold is projective if and only if it admits a positive line bundle. We prove only the 'if' implication in this paper, giving a sufficient condition for a manifold bundle to be embedded in projective space. Along the way, we prove several other interesting results. Of particular note is the Kodaira-Nakano Vanishing Theorem, a crucial tool for eliminating higher cohomology of complex manifolds, as well as Lemmas 6.2 and 6.1, which provide important relationships between divisors, line bundles, and blowups. Although this treatment is relatively self-contained, we omit a rigorous development of Hodge theory, some basic complex analysis results, and some theorems regarding Cech cohomology (including Leray's Theorem).