Poincare Series And Very Ampleness Criterion For Pluri-canonical Bundles

TL;DR

This paper introduces a new criterion for the very ampleness of pluri-canonical bundles on compact quotients of bounded domains using Poincaré series, providing effective criteria and elementary proofs.

Contribution

It defines the notion of S very ampleness for pluri-canonical bundles and establishes an effective Seshadri constant criterion for this property.

Findings

Introduces S very ampleness for pluri-canonical bundles.

Provides an effective criterion based on Seshadri constants.

Includes an elementary proof of Poincaré map surjectivity.

Abstract

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 $mK_X$ by using the Poincar\'e series. The main result is an effective Seshadri constant criterion of $S$ very ampleness for $mK_X$. An elementary proof of surjectivity of the Poincar\'e map is also given.