Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(1): 169-178

Online first article July 3, 2018      Printed January 31, 2019

https://doi.org/10.4134/BKMS.b180170

Copyright © The Korean Mathematical Society.

Nullity of the Levi-form and the associated subvarieties for pseudo-convex CR structures of hypersurface type

Kuerak Chung, Chong-Kyu Han

Korea Institute for Advanced Study; Seoul National University

Abstract

Let $M^{2n+1}$, $n\ge 1$, be a smooth manifold with a pseudo-convex integrable CR structure of hypersurface type. We consider a sequence of CR invariant subsets $ M=\mathcal S_0 \supset \mathcal S_1 \supset \cdots \supset \mathcal S_{n}, $ where $\mathcal S_q$ is the set of points where the Levi-form has nullity $\ge q$. We prove that $\mathcal S_q$'s are locally given as common zero sets of the coefficients $A_j,$ $j=0,1,\ldots, q-1,$ of the characteristic polynomial of the Levi-form. Some sufficient conditions for local existence of complex submanifolds are presented in terms of the coefficients $A_j$.

Keywords: CR structure, invariant subvarieties, nullity of Levi-form, complex submanifolds

MSC numbers: Primary 32V05, 53A55; Secondary 32V25, 35N10

Supported by: The authors were partially supported by National Research Foundation of Korea with grant NRF-2017R1A2A2B4007119.

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