Asymptotic of Bergman Kernel (Master Thesis)

TL;DR

This thesis presents a new semi-classical symbolic calculus approach to prove pointwise asymptotic expansions of Bergman kernels on positive and semi-positive line bundles, advancing understanding in complex geometry.

Contribution

Introduces a novel semi-classical symbolic calculus inspired by recent work, providing a new proof for Bergman kernel asymptotics on certain line bundles.

Findings

Established pointwise asymptotic expansion on positive line bundles.

Developed a semi-classical symbol space and calculus.

Extended results to semi-positive line bundles.

Abstract

In this master thesis, we give a new proof on the pointwise asymptotic expansion for Bergman kernel of a hermitian holomorphic line bundle on the points where the curvature of the line bundle is positive and satisfy local spectral gap condition. The main point is to introduce a suitable semi-classical symbol space and related symbolic calculus inspired from recent work of Hsiao and Savale. Particularly, we establish the existence of pointwise asymptotic expansion on the positive part for certain semi-positive line bundles.