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Regularization of local CR-automorphisms of real-analytic CR-manifolds. (English) Zbl 1261.32005
The authors study the class of generic real-analytic CR-submanifolds of a complex vector space \(E\) with the following property: For every point \(a\) of the manifold, the Lie algebra of germs of infinitesimal real-analytic CR automorphisms at \(a\) is finite-dimensional and its complexification contains all constant vector fields and the Euler vector field. This class includes the Levi non-degenerate CR-quadrics and a number of Levi degenerate tube manifolds.
The main results are: (I) The germs of infinitesimal real-analytic CR-automorphisms extend to global polynomial vector fields. (II) The local real-analytic CR-automorphisms extend to birational transformations of \(E\). In particular, the authors specify conditions under which the local birational transformations of the CR-manifold form a group. (III) The group generated by those birational transformations can be realised as a group of projective transformations.
These results can be viewed as a generalization and plausible explanation of the heuristic formula for quadric automorphisms described by V. V. Ezhov and G. Schmalz in [J. Geom. Anal. 11, No. 3, 441–467 (2001; Zbl 1042.32014)].

MSC:
32V40 Real submanifolds in complex manifolds
14M17 Homogeneous spaces and generalizations
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