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Modifications of the Levi core
Authors
Tanuj Gupta
Emil J. Straube
John N. Treuer
Publication date
28 August 2023
Publisher
View
on
arXiv
Abstract
We construct a family of subdistributions of the Levi core
C
(
N
)
\mathfrak{C}(\mathcal{N})
C
(
N
)
called modified Levi cores
{
M
C
A
}
A
\{\mathcal{M}\mathfrak{C}_{\mathcal{A}}\}_{\mathcal{A}}
{
M
C
A
​
}
A
​
indexed over closed distributions
A
\mathcal{A}
A
that contain the Levi null distribution
N
\mathcal{N}
N
and are contained in the complex tangent bundle
T
1
,
0
b
Ω
T^{1, 0}b\Omega
T
1
,
0
b
Ω
of a smooth bounded pseudoconvex domain
Ω
\Omega
Ω
. We show that Catlin's Property (
P
P
P
) holds on
b
Ω
b\Omega
b
Ω
if and only if Property (
P
P
P
) holds on the support of one, and hence all, of the modified Levi cores. In
C
2
\mathbb{C}^2
C
2
, all of the modified Levi cores coincide. For a smooth bounded pseudoconvex complete Hartogs domain in
C
2
\mathbb{C}^2
C
2
that satisfies Property (
P
P
P
), we show that its modified Levi core is trivial. This contrasts with
C
(
N
)
\mathfrak{C}(\mathcal{N})
C
(
N
)
, which can be nontrivial for such domains.Comment: 13 page
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oai:arXiv.org:2308.14807
Last time updated on 10/09/2023