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

TL;DR

This paper introduces the Grassmannian q-core of tangent bundle distributions, linking its properties to boundary conditions of pseudoconvex domains, thereby generalizing previous core concepts and connecting to recent stratification studies.

Contribution

It defines the Grassmannian q-core and establishes its equivalence with boundary Property (P_q) for pseudoconvex domains, extending prior results for q=1.

Findings

Support of the Grassmannian q-core satisfies Property (P_q) if and only if the boundary does.

Generalizes previous results from q=1 to 1 ≤ q ≤ n.

Connects the q-core concept to recent stratification work of Zaitsev.

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 $\Omega\subseteq \mathbb{C}^n$, we prove that for $1 \leq q \leq n$, the support of the Grassmannian $q$-core satisfies Property $(P_q)$ if and only if the boundary of $\Omega$ 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.