The Kodaira Embedding Theorem
Skyler Marks
TL;DR
The paper provides an intrinsic criterion for projectivity by showing that a compact complex manifold $M$ with a positive line bundle $L$ is projective, via the Kodaira Embedding Theorem. It develops the necessary analytic and geometric toolkit, including line bundles, Chern connections, curvature, and the Kodaira-Nakano vanishing theorem, and connects divisors to line bundles through the $[D]$ correspondence. Blowing up is employed to handle pointwise separation and tangential separation by global sections, while the embedding is constructed from high tensor powers $L^{\otimes k}$ and the Kodaira map $\iota_{L^{\otimes k}}$. The results bridge complex-analytic data and algebraic geometry, enabling intrinsic positivity to yield projective embeddings and thus projectivity.
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).