  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.
