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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2222_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subseteq \mathbb {C}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, we prove that for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2222_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le q \le n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, the support of the Grassmannian q-core satisfies Property <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2222_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$(P_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if the boundary of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2222_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> satisfies Property <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2222_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$(P_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This generalizes a previous result of the third author in the case <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2222_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$q=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The notion of the Grassmannian q-core offers a perspective on certain generalized stratifications appearing in a recent work of Zaitsev.

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

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2025年10月15日

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Gian Maria Dall’Ara,
Samuele Mongodi,
John N. Treuer

摘要

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 $\Omega \subseteq \mathbb {C}^n$ $Ω \subseteq C^{n}$ , we prove that for $1 \le q \le n$ $1 \leq q \leq n$ , the support of the Grassmannian q-core satisfies Property $(P_q)$ $(P_{q})$ if and only if the boundary of $\Omega $ $Ω$ satisfies Property $(P_q)$ $(P_{q})$ . This generalizes a previous result of the third author in the case $q=1$ $q = 1$ . The notion of the Grassmannian q-core offers a perspective on certain generalized stratifications appearing in a recent work of Zaitsev.

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

摘要