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

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 $A_j,$ $j=0,1,\dots ,q-1,$ of the characteristic polynomial of the Levi-form. Some sufficient conditions for local existence of complex submanifolds are presented in terms of the coefficients $A_j$.

11 pages