The Levi q-core and Property (Pq)

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 Ω⊆Cn, we prove that for 1≤q≤n, the support of the Grassmannian q-core satisfies Property (Pq) if and only if the boundary of Ω satisfies Property (Pq). 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.