Compact homogeneous Leviflat CR-manifolds

We consider compact Leviflat homogeneous Cauchy–Riemann (CR) manifolds. In this setting, the Levi-foliation exists and we show that all its leaves are homogeneous and biholomorphic. We analyze separately the structure of orbits in complex projective spaces and parallelizable homogeneous CR-manifolds in our context and then combine the projective and parallelizable cases. In codimensions one and two, we also give a classification.

Higher order Levi forms on homogeneous CR manifolds

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 than 3 and the compact ones less or equal 2. Finally we construct by Lie extensions weakly nondegenerate CR vector bundles with arbitrary orders of nondegeneracy.

Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN

We consider the significant class of homogeneous CR manifolds represented by some weighted homogeneous polynomials and we derive some plain and useful features which enable us to set up a fast effective algorithm to compute homogeneous components of their Lie algebras of infinitesimal CR automorphisms. This algorithm mainly relies upon a natural gradation of the sought Lie algebras, and it also consists in treating separately the related graded components. While classical methods are based on constructing and solving some associated PDE systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra. Furthermore, it benefits from a divide-and-conquer strategy to break down the computations into some simpler subcomputations. Also, we consider the new and effective concept of comprehensive Gr?bner systems which provides us some powerful tools to treat the computations in the much complicated parametric case. The designed algorithm is also implemented in the Maple software, what required also implementing a recently designed algorithm of Kapur et al.