Szego kernels and a theorem of Tian

Abstract

We give a simple proof of Tian's theorem that the Kodaira embeddings associated to a positive line bundle over a compact complex manifold are asymptotically isometric. The proof is based on the diagonal asymptotics of the Szego kernel (i.e. the orthogonal projection onto holomorphic sections). In deriving these asymptotics we use the Boutet de Monvel-Sjostrand parametrix for the Szego kernel.