Nullity of the Levi-form and the associated subvarieties for pseudo-convex CR structures of hypersurface type

TL;DR

This paper studies the structure of pseudo-convex CR manifolds of hypersurface type, showing how the nullity of the Levi-form defines invariant subvarieties and providing conditions for the existence of complex submanifolds.

Contribution

It characterizes the nullity sets of the Levi-form as zero sets of characteristic polynomial coefficients and offers conditions for local complex submanifold existence.

Findings

Nullity sets are zero loci of Levi-form characteristic polynomial coefficients.

Provides criteria for the local existence of complex submanifolds.

Establishes a stratification of CR manifolds based on Levi-form nullity.

Abstract

Let $M^{2n+1}$, $n\ge 1$, be a smooth manifold with a pseudo-convex integrable CR structure of hypersurface type. We consider a sequence of CR invariant subsets $ M=\mathcal S_0 \supset \mathcal S_1 \supset \cdots \supset \mathcal S_{n}, $ where $\mathcal S_q$ is the set of points where the Levi-form has nullity $\ge q$. We prove that $\mathcal S_q$'s are locally given as common zero sets of the coefficients $A_j,$ $j=0,1,\ldots, q-1,$ of the characteristic polynomial of the Levi-form. Some sufficient conditions for local existence of complex submanifolds are presented in terms of the coefficients $A_j$.