A classical way to understand the topological spaces (or manifolds) is to use its functions. For a domain in Cn, the Bergman projection is the orthogonal projection from the space of square integrable functions to the space of L2 holomorphic functions, its smooth kernel is the Bergman kernel. For a complex manifold, if it is compact, the only holomorphic functions are the constant functions. This leads naturally to consider the space of holomorphic sections of holomorphic vector bundles on complex manifolds, one of the most famous results in this direction is the Kodaira embedding theorem in 1950’s.

We will review the story of the Bergman kernel and its recent advances.