Decomposable five-dimensional Lie algebras in the problem of holomorphic homogeneity in~$\mathbb{C}^3$

A. V. Atanov ; A. V. Loboda

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 4, Tome 173 (2019), pp. 86-115.

Résumé

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in the space $\mathbb{C}^3$, we study five-dimensional real Lie algebras realized as algebras of holomorphic vector fields on such manifolds. We prove that if on a holomorphically homogeneous real hypersurface $M$ of the space $\mathbb{C}^3$, there is a decomposable, solvable, five-dimensional Lie algebra of holomorphic vector fields having a full rank near some point $P\in M$, then this surface is either degenerate near $P$ in the sense of Levy or is a holomorphic image of an affine-homogeneous surface.

Détail

Keywords: homogeneous manifold, holomorphic transformation, vector field, real hypersurface in $\mathbb{C}^3$.
Mots-clés : decomposable Lie algebra

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