Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation

Curtis Porter; Igor Zelenko

doi:10.1515/crelle-2021-0012

Article

Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation

Curtis Porter and Igor Zelenko

Published/Copyright: April 2, 2021

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2021 Issue 777

Abstract

This article is devoted to the local geometry of everywhere 2-nondegenerate CR manifolds M of hypersurface type. An absolute parallelism for such structures was recently constructed independently by Isaev and Zaitsev, Medori and Spiro, and Pocchiola in the minimal possible dimension (dim⁡M=5), and for dim⁡M=7 in certain cases by the first author. In the present paper, we develop a bigraded (i.e., ℤ×ℤ-graded) analog of Tanaka’s prolongation procedure to construct an absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol – a complex bigraded vector space – containing all essential information about the CR structure. Under the additional regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation, endowed with an anti-linear involution, and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of the bigraded universal algebraic prolongation. Moreover, we show that for each regular symbol there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded universal algebraic prolongation of the symbol. In the case of 1-dimensional Levi kernel we classify all regular symbols and calculate their bigraded universal algebraic prolongations. In this case, the regular symbols can be subdivided into nilpotent, strongly non-nilpotent, and weakly non-nilpotent. The bigraded universal algebraic prolongation of strongly non-nilpotent regular symbols is isomorphic to the complex orthogonal algebra 𝔰⁢𝔬⁢(m,ℂ), where m=12⁢(dim⁡M+5). Any real form of this algebra – except 𝔰⁢𝔬⁢(m) and 𝔰⁢𝔬⁢(m-1,1) – corresponds to the real part of the bigraded universal algebraic prolongation of exactly one strongly non-nilpotent regular CR symbol. However, for a fixed dim⁡M≥7 the dimension of the bigraded universal algebraic prolongations of all possible regular CR symbols achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is 14⁢(dim⁡M-1)2+7.

Funding statement: Igor Zelenko was partly supported by NSF grant DMS-1406193 and Simons Foundation Collaboration Grant for Mathematicians 524213.

Acknowledgements

We are grateful to Andrea Santi for pointing out several gaps in the earlier version of the article, to Boris Doubrov for valuable discussions, especially concerning the construction of hypersurface realizations of maximally symmetric models, and to David Sykes for carefully reading the paper and offering helpful comments.

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Received: 2020-07-24

Revised: 2021-02-19

Published Online: 2021-04-02

Published in Print: 2021-08-01

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https://doi.org/10.1515/crelle-2021-0012