Generalization of Kodaira’s Embedding Theorem for Compact Kähler Manifolds with Semi-Positive Chern Class

Kodaira embedding theorem states that a compact complex manifold can only be embedded into a complex projective space PN for some N if it admits a positive line bundle. Based on Chow’s theorem, projective algebraic manifolds are algebraic. The purpose of this study is to generalize Kodaira’s known theorem. We show that if X is a compact Kähler manifold of complex dimension n and its first Chern class is semi-positive and of rank n−1 at one point of X, then X can be embedded into PN for some integer N. Furthermore, we prove that this embedding is unique up to biholomorphism. We also show that X admits a Kähler metric whose Ricci curvature is bounded from below.