Szego kernels and a theorem of Tian

TL;DR

This paper provides a straightforward proof of Tian's theorem, demonstrating that Kodaira embeddings become asymptotically isometric, using Szego kernel asymptotics and the Boutet de Monvel-Sjostrand parametrix.

Contribution

It offers a simplified proof of Tian's theorem leveraging Szego kernel asymptotics and the Boutet de Monvel-Sjostrand parametrix, enhancing understanding of asymptotic isometry.

Findings

Kodaira embeddings are asymptotically isometric

Szego kernel asymptotics are key to the proof

Utilizes Boutet de Monvel-Sjostrand parametrix for Szego kernel

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.