Higher order Levi forms on homogeneous CR manifolds

TL;DR

This paper studies the nondegeneracy of higher order Levi forms on homogeneous CR manifolds, establishing bounds on their order and constructing examples with arbitrary nondegeneracy levels.

Contribution

It improves previous results by showing that general orbits in complex flag manifolds have Levi form order ≤ 3, and compact ones ≤ 2, and constructs CR vector bundles with any nondegeneracy order.

Findings

General orbits in complex flag manifolds have Levi form order ≤ 3.

Compact orbits have Levi form order ≤ 2.

Constructed CR vector bundles with arbitrary nondegeneracy levels.

Abstract

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous $CR$ manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal $3$ and the compact ones less or equal~$2$. Finally we construct by Lee extensions weakly nondegenerate $CR$ vector bundles with arbitrary orders of nondegeneracy.