# Non locally trivializable $CR$ line bundles over compact Lorentzian $CR$ manifolds

## TL;DR

This paper constructs examples of compact Lorentzian CR manifolds with line bundles that are topologically trivial but not locally CR trivializable, revealing complex geometric obstructions.

## Contribution

It introduces a deformation of the trivial CR line bundle on compact CR manifolds that is topologically trivial but not locally CR trivializable, especially in Lorentzian cases.

## Key findings

- Existence of non-locally trivializable CR line bundles
- Application to compact Lorentzian CR manifolds of hypersurface type
- Demonstration of geometric obstructions to local trivialization

## Abstract

We consider compact $CR$ manifolds of arbitrary $CR$ codimension that satisfy certain geometric conditions in terms of their Levi form. Over these compact $CR$ manifolds, we construct a deformation of the trivial $CR$ line bundle over $M$ which is topologically trivial over $M$ but fails to be even locally $CR$ trivializable over any open subset of $M$. In particular, our results apply to compact Lorentzian $CR$ manifolds of hypersurface type.