Very ampleness of the bicanonical line bundle on compact complex two ball quotients

TL;DR

This paper proves that twice the canonical bundle on most smooth compact complex two ball quotients is very ample, with specific exceptions for certain fake projective planes where the embedding may have at most two points missing.

Contribution

It establishes the very ampleness of 2K on smooth compact complex two ball quotients, except for four special fake projective planes, refining understanding of their embedding properties.

Findings

2K is very ample for most two ball quotients

Four fake projective planes are exceptions with near-embedding

Sections of 2K provide embeddings except possibly at two points

Abstract

The purpose of this note is to show that $2K$ of any smooth compact complex two ball quotient is very ample, except possibly for four pairs of fake projective planes of minimal type, where $K$ is the canonical line bundle. For the four pairs of fake projective planes, sections of $2K_M$ give an embedding of $M$ except possibly for at most two points on $M$.