New examples of $2$-nondegenerate real hypersurfaces in $\mathbb{C}^N$ with arbitrary nilpotent symbols

TL;DR

This paper constructs an explicit infinite-dimensional family of 2-nondegenerate CR hypersurfaces in complex spaces with arbitrary dimension, classifies their symmetry algebras, and solves the equivalence problem for certain structures.

Contribution

It introduces the first explicit infinite-dimensional family of 2-nondegenerate hypersurfaces with nilpotent symbols in complex dimensions greater than three.

Findings

Constructed explicit examples of hypersurfaces with nilpotent CR symbols.

Classified the symmetry algebras of these structures.

Solved the equivalence problem for structures with single Jordan block symbols.

Abstract

We introduce a class of uniformly $2$-nondegenerate CR hypersurfaces in $\mathbb{C}^N$, for $N>3$, having a rank $1$ Levi kernel. The class is first of all remarkable by the fact that for every $N>3$ it forms an {\em explicit} infinite-dimensional family of everywhere $2$-nondegenerate hypersurfaces. To the best of our knowledge, this is the first such construction. Besides, the class an infinite-dimensional family of non-equivalent structures having a given constant nilpotent CR symbol for every such symbol. Using methods that are able to handle all cases with $N>5$ simultaneously, we solve the equivalence problem for the considered structures whose symbol is represented by a single Jordan block, classify their algebras of infinitesimal symmetries, and classify the locally homogeneous structures among them. We show that the remaining considered structures, which have symbols represented by a direct sum of Jordan blocks, can be constructed from the single block structures through simple linking and extension processes.