Publication details
On geometry of 2-nondegenerate CR structures of hypersurface type and flag structures on leaf spaces of Levi foliations
| Authors | |
|---|---|
| Year of publication | 2023 |
| Type | Article in Periodical |
| Magazine / Source | Advances in Mathematics |
| MU Faculty or unit | |
| Citation | |
| Web | https://doi.org/10.1016/j.aim.2022.108850 |
| Doi | http://dx.doi.org/10.1016/j.aim.2022.108850 |
| Keywords | 2-nondegenerate CR structures; Absolute parallelism; Homogeneous models; Tanaka prolongation; Curves in flag varieties |
| Description | We construct canonical absolute parallelisms over realanalytic manifolds equipped with 2-nondegenerate, hypersurface-type CR structures of arbitrary odd dimension not less than 7 whose Levi kernel has constant rank belonging to a broad subclass of CR structures that we label as recoverable. For this we develop a new approach based on a reduction to a special flag structure, called the dynamical Legendrian contact structure, on the leaf space of the CR structure's associated Levi foliation. This extends the results of [23] from the case of regular CR symbols constituting a discrete set in the set of all CR symbols to the case of the arbitrary CR symbols for which the original CR structure can be uniquely recovered from its corresponding dynamical Legendrian contact structure. We find an explicit criterion for this recoverability. In particular, if the rank of the Levi kernel is 1 and the dimension of the CR manifold is not less than 7, then for each given signature of the reduced Levi form in the space of all CR symbols (which depend on continuous parameters) there are no more than 2 symbols for which the aforementioned recoverability fails, and while the present method is applicable for all but those 2 cases, they can be treated separately by the method of [23]. Our method clarifies the relationship between the bigraded Tanaka prolongation of regular symbols developed in [23] and their usual Tanaka prolongation, providing a geometric interpretation of conditions under which they are equal. Motivated by the search for homogeneous models with given nonregular symbols, we also describe a process of reduction from the original natural frame bundle, which is inevitable for interpretation they equal. Motivated by the search for homogeneous models with given nonregular symbols, we also describe a process of reduction from the original natural frame bundle, which is inevitable for structures with nonregular CR symbols. We demonstrate this reduction procedure for examples whose underlying manifolds have dimension 7 and 9. We show that, for any fixed rank r > 1, in the set of all CR symbols associated with 2-nondegenerate, hypersurface-type CR manifolds of odd dimension greater than 4r +1 with rank r Levi kernel, the CR symbols not associated with any homogeneous model are generic, and, for r = 1, the same result holds if the CR structure is pseudoconvex. |