The Levi $q$-core and Property ($P_q$)

Abstract

We introduce the Grassmannian $q$-core of a distribution of subspaces of the tangent bundle of a smooth manifold. This is a generalization of the concept of the core previously introduced by the first two authors. In the case where the distribution is the Levi null distribution of a smooth bounded pseudoconvex domain $\hat{I}\text{c◯}\subseteq {\mathbb{C}}^n$, we prove that for $1\le q\le n$, the support of the Grassmannian $q$-core satisfies Property $(P_q)$ if and only if the boundary of $\hat{I}\text{c◯}$ satisfies Property $(P_q)$. This generalizes a previous result of the third author in the case $q=1$. The notion of the Grassmannian $q$-core offers a perspective on certain generalized stratifications appearing in a recent work of Zaitsev.