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On the Problem of Describing Holomorphically Homogeneous Real Hypersurfaces of Four-Dimensional Complex Spaces
Proceedings of the Steklov Institute of Mathematics ( IF 0.5 ) Pub Date : 2021-02-02 , DOI: 10.1134/s0081543820060115
A. V. Loboda

Abstract

We discuss two fragments of a large problem that extends the author’s recently completed similar studies in the space \(\mathbb C^3\) to the next dimension. One of the fragments is related to the local description of nonspherical holomorphically homogeneous strictly pseudoconvex hypersurfaces in \(\mathbb C^4\) with stabilizers of submaximal dimension. Using the Moser normal form technique and the properties of subgroups of the unitary group \(\mathrm U(3)\), we show that up to holomorphic equivalence there exist only two such surfaces. Both of them are natural generalizations of known homogeneous hypersurfaces in the space \(\mathbb C^3\). In the second part of the paper, we consider a technique of holomorphic realization in \(\mathbb C^4\) of abstract seven-dimensional Lie algebras that correspond, in particular, to homogeneous hypersurfaces with trivial stabilizer. Some sufficient conditions for the Lie algebras are obtained under which the orbits of all realizations of such algebras are Levi degenerate. The schemes of studying holomorphically homogeneous hypersurfaces that were used in the two-dimensional (É. Cartan) and three-dimensional (Doubrov, Medvedev, and The; Fels and Kaup; Beloshapka and Kossovskiy; Loboda) situations and resulted in full descriptions of such hypersurfaces turn out to be quite efficient in the case of greater dimension of the ambient space as well.



中文翻译:

关于描述四维复杂空间的全纯齐次实超曲面的问题

摘要

我们讨论了一个大问题的两个片段,该片段将作者最近在空间\(\ mathbb C ^ 3 \)中完成的类似研究扩展到了下一维度。碎片之一与\(\ mathbb C ^ 4 \)中具有次最大尺寸的稳定器的非球面同态均质严格伪凸超曲面的局部描述有关。使用Moser范式技术和单一组\(\ mathrm U(3)\)的子组的性质,我们证明,直到全纯等效性,仅存在两个这样的曲面。它们都是空间\(\ mathbb C ^ 3 \)中已知齐次超曲面的自然概括。在本文的第二部分中,我们考虑了抽象的七维李代数的\(\ mathbb C ^ 4 \),特别是对应于具有琐碎稳定器的齐次超曲面。获得了李代数的一些充分条件,在这种条件下,此类代数的所有实现的轨道都是李维退化的。研究在二维(É。Cartan)和三维(Doubrov,Medvedev和The; Fels和Kaup; Beloshapka和Kossovskiy; Loboda)情况下使用的全纯同质超曲面的方案,并对其进行了完整描述在更大的周围空间尺寸的情况下,超曲面被证明是非常有效的。

更新日期:2021-02-02
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