Abstract
In the paper, the groups of holomorphic automorphisms of model surfaces of types (1,8), (1,9), ..., (1,12) are studied. This completes the investigation of automorphisms of the model surfaces with one-dimensional complex tangent whose Levi-Tanaka algebra is of length not exceeding five. As a corollary, the following assertion is proved: the graded Lie algebra of infinitesimal holomorphic symmetries of a model surface of type (1,K) has no positive component for any K, 2 ⩽ K ⩽ 12. This is another confirmation, after Kossovskii’s theorem, of the Beloshapka conjecture on the rigidity of model surfaces.
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References
H. Poincaré, “Les fonction analytiques de deux variables et la représentation conforme,” Rend. Circ. Math. Palermo 23, 185–220 (1907).
S. S. Chern and J. K. Moser, “Real Hypersurfaces in Complex Manifold,” Acta Math. 133(3–4), 219–271 (1974).
V. K. Beloshapka, “Real Submanifolds of a Complex Space: Their Polynomial Models, Automorphisms, and Classification Problems,” Uspekhi Mat. Nauk 57(1), 3–44 (2002) [Russian Math. Surveys 57 (1), 1–41 (2002)].
V. K. Beloshapka, “A Universal Model for a Real Submanifold,” Mat. Zametki 75(4) (2004), 507–522 [Math. Notes 75 (3–4), 475–488 (2004)].
V. K. Beloshapka, “CR-Varieties of the Type (1, 2) as Varieties of ’super-High’ Codimension,” Russ. J. Math. Phys. 5(3), 399–404 (1997).
E. N. Shananina, “Models of CR-Manifolds of Type (1,K) for 3 ≤ K ≤ 7 and Their Automorphisms,” Mat. Zametki 67(3), 452–459 (2000) [Math. Notes 67 (3–4), 382–388 (2000)].
R. V. Gammel’ and I.G. Kossovskii, “The Envelope of Holomorphy of a Model Surface of the Third Degree and the ‘Rigidity’ Phenomenon,” Tr. Mat. Inst. Steklova 253, 30–45 (2006) [Proc. Steklov Inst. Math. 2 (253), 22–36 (2006)].
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Financially supported by RFBR under grants nos. 05-01-0981 and NSh-2040.2003.1.
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Mamai, I.B. Model CR-manifolds with one-dimensional complex tangent. Russ. J. Math. Phys. 16, 97–102 (2009). https://doi.org/10.1134/S1061920809010075
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DOI: https://doi.org/10.1134/S1061920809010075