On the ampleness of positive CR line bundles over Levi-flat manifolds

TL;DR

This paper presents a counterexample of a Levi-flat CR 3-manifold with a positive CR line bundle that is not ample in a certain smoothness context, impacting the understanding of CR embedding theorems.

Contribution

It provides the first example showing limitations of the regularity in Kodaira type embedding theorems for Levi-flat CR manifolds.

Findings

Counterexample of a Levi-flat CR 3-manifold with a positive CR line bundle that is not ample in smooth CR sections.

Demonstrates the impossibility of improving regularity in existing CR embedding theorems.

Highlights limitations in the current understanding of positivity and ampleness in CR geometry.

Abstract

We give an example of a compact Levi-flat CR 3-manifold with a positive-along-leaves CR line bundle which is not ample with respect to transversely infinitely differentiable CR sections. This example shows that we cannot improve the regularity of Kodaira type embedding theorem for compact Levi-flat CR manifolds obtained by Ohsawa and Sibony.