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 Mat. Sb., 1991, Volume 182, Number 2, Pages 203–219 (Mi msb1292)

This article is cited in 36 scientific papers (total in 36 papers)

On holomorphic transformations of a quadric

V. K. Beloshapka

Abstract: Holomorphic transformations of a quadric of arbitrary codimension are studied. General formulas are derived for infinitesimal automorphisms of such a surface. The phenomenon of “rigidity” of quadrics in general position is demonstrated. Examples of quadrics with rich automorphism groups are considered. A classification is given for positive-definite quadrics of codimension two.

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English version:
Mathematics of the USSR-Sbornik, 1992, 72:1, 189–205

Bibliographic databases:

UDC: 517.5
MSC: Primary 32M05, 32H99; Secondary 32C05, 14J25
Received: 01.11.1989

Citation: V. K. Beloshapka, “On holomorphic transformations of a quadric”, Mat. Sb., 182:2 (1991), 203–219; Math. USSR-Sb., 72:1 (1992), 189–205

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. E. M. Chirka, “Introduction to the geometry of $CR$-manifolds”, Russian Math. Surveys, 46:1 (1991), 95–197
2. A. V. Abrosimov, “A description of locally biholomorphic automorphisms of standard quadrics of codimension two”, Russian Acad. Sci. Sb. Math., 80:1 (1995), 137–178
3. Vladimir V. Ežov, Gerd Schmalz, “Holomorphic automorphisms of quadrics”, Math Z, 216:1 (1994), 453
4. Vladimir V. Ežov, Gerd Schmalz, “Poincaré automorphisms for nondegenerate CR quadrics”, Math Ann, 298:1 (1994), 79
5. V. K. Beloshapka, “Geometric invariants of a $CR$-manifold”, Math. Notes, 55:5 (1994), 439–445
6. N. F. Palinchak, “On quadrics of higher codimension”, Math. Notes, 55:5 (1994), 512–516
7. N. F. Palinchak, “Real quadrics of codimension 3 in $\mathbb C^6$ and their non-linear automorphisms”, Izv. Math., 59:3 (1995), 597–617
8. V. K. Beloshapka, “Invariants of CR-manifolds associated with the tangent quadric”, Math. Notes, 59:1 (1996), 31–38
9. A. F. Arbatskii, “Structure of groups of nonlinear automorphisms for $(3,3)$ quadrics”, Math. Notes, 59:2 (1996), 116–122
10. N. F. Palinchak, “About $c$-rigid quadrics”, Math. Notes, 59:2 (1996), 158–162
11. V. K. Beloshapka, “Local invariants and prohibitions on mappings of CR-manifolds”, Math. Notes, 60:4 (1996), 438–441
12. V. K. Beloshapka, “Homogeneous real hypersurfaces in $\mathbb C^2$”, Math. Notes, 60:5 (1996), 569–572
13. Beloshapka V., “Automorphisms of Degenerate Hypersurfaces in C-2 and a Dimension Conjecture”, Russ. J. Math. Phys., 4:3 (1996), 393–396
14. V. V. Ezhov, G. Schmalz, “A simple proof of the Beloshapka theorem on the parametrization of the automorphism group of a CR manifold”, Math. Notes, 61:6 (1997), 783–786
15. E. G. Anisova, “Null-quadrics of codimension 4 in $\mathbb C^7$”, Math. Notes, 62:5 (1997), 549–556
16. Beloshapka V., “Cr-Varieties of the Type (1,2) as Varieties of “Super-High” Codimension”, Russ. J. Math. Phys., 5:3 (1997), 399–404
17. V. V. Ezov, G. Schmalz, “Automorphisms and holomorphic mappings of standard CR-manifolds and Siegel domains”, Journal of Mathematical Sciences (New York), 92:2 (1998), 3712
18. E. G. Anisova, “Quadrics of codimension 4 in $\mathbb C^7$ and their automorphisms”, Math. Notes, 64:6 (1998), 697–703
19. Vladimir V. Ežov, Gerd Schmalz, “A Matrix Poincaré formula for holomorphic automorphisms of quadrics of higher codimension. Real Associative Quadrics”, J Geom Anal, 8:1 (1998), 27
20. V. V. EZHOV, A. V. ISAEV, G. SCHMALZ, “INVARIANTS OF ELLIPTIC AND HYPERBOLIC CR-STRUCTURES OF CODIMENSION 2”, Int. J. Math, 10:01 (1999), 1
21. E. N. Shananina, “Models of CR-manifolds of type $(\ell,K)$ for $3\leqslant K\leqslant 7$ and their automorphisms”, Math. Notes, 67:3 (2000), 382–388
22. Baouendi M., Ebenfelt P., Rothschild L., “Local Geometric Properties of Real Submanifolds in Complex Space”, Bull. Amer. Math. Soc., 37:3 (2000), 309–336
23. V. K. Beloshapka, “A Cubic Model of a Real Variety”, Math. Notes, 70:4 (2001), 457–470
24. A. E. Ershova, “Automorphisms of 2-Nondegenerate Hypersurfaces in $\mathbb C^3$”, Math. Notes, 69:2 (2001), 188–195
25. V. K. Beloshapka, “Polynomial models of real manifolds”, Izv. Math., 65:4 (2001), 641–657
26. V. K. Beloshapka, “A Quasiperiodic System of Polynomial Models of CR-Manifolds”, Proc. Steklov Inst. Math., 235 (2001), 1–28
27. V. K. Beloshapka, “Real submanifolds in complex space: polynomial models, automorphisms, and classification problems”, Russian Math. Surveys, 57:1 (2002), 1–41
28. P. B. Utkin, “The Dimension Conjecture for Quadrics of Codimension Three and Higher”, Math. Notes, 72:1 (2002), 138–141
29. A. B. Sukhov, “On transformations of analytic CR-structures”, Izv. Math., 67:2 (2003), 303–332
30. V. K. Beloshapka, “Universal Models For Real Submanifolds”, Math. Notes, 75:4 (2004), 475–488
31. E. N. Shananina, “Polynomial Models of Degree 5 and Algebras of Their Automorphisms”, Math. Notes, 75:5 (2004), 702–716
32. Gaussier H., Merker J., “Nonalgebraizable Real Analytic Tubes in C-N”, Math. Z., 247:2 (2004), 337–383
33. Wilhelm Kaup, “CR-quadrics with a symmetry property”, manuscripta math, 2010
34. Wu Q., “On Holomorphic Automorphisms of a Class of Non-Homogeneous Rigid Hypersurfaces in a", (N+1)”, Chin. Ann. Math. Ser. B, 31:2 (2010), 201–210
35. Isaev A., “Spherical Tube Hypersurfaces”, Spherical Tube Hypersurfaces, Lect. Notes Math., 2020, Springer-Verlag Berlin, 2011, 1–217
36. I. B. Mamai, “Moduli spaces of model surfaces with one-dimensional complex tangent”, Izv. Math., 77:2 (2013), 354–377
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