Poincare Series And Very Ampleness Criterion For Pluri-canonical Bundles

Let $X$ be a compact quotient of a bounded domain in ${\mathbb{C}}^n$. Let $K_X$ be the canonical line bundle of $X$. In this paper, we shall introduce the notion of $S$ very ampleness for the pluri-canonical line bundles $m{K}_X$ by using the Poincar'e series. The main result is an effective Seshadri constant criterion of $S$ very ampleness for $m{K}_X$. An elementary proof of surjectivity of the Poincar'e map is also given.