CR-manifolds of infinite type in the sense of Bloom and Graham

Stepanova, M.

doi:10.1090/mosc/330

CR-manifolds of infinite type in the sense of Bloom and Graham
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by M. A. Stepanova
Translated by: Nikolai Kruzhilin

Trans. Moscow Math. Soc. 2021, 289-304

DOI: https://doi.org/10.1090/mosc/330

Published electronically: March 15, 2022

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Abstract:

An analogue of the Bloom–Graham theorem for germs of real analytic CR-manifolds of infinite type is devised, and a certain standard form to which they can be transformed (a reduced form) is described. The concept of Bloom–Graham type is refined (as a stratified type). The refined type is also holomorphically invariant. The concept of a quasimodel surface is introduced and it is shown that for biholomorphically equivalent manifolds such surfaces are quasilinearly equivalent. A criterion for the Lie algebra of infinitesimal holomorphic automorphisms to be finite-dimensional is obtained in the case when the type is uniformly infinite (that is, infinite at all points). In combination with the criterion of a finite-dimensional automorphism algebra for manifolds of finite type almost everywhere, this yields a complete criterion for this algebra to be finite-dimensional. The sets of fixed Blooom–Graham type are shown to be semi-analytic and the type of a generic point (lying outside a proper analytic subset) is minimal in a certain sense.

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