Regularization of Local CR-Automorphisms of Real-Analytic CR-Manifolds

Abstract

Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every a∈M the Lie algebra \({\frak{hol}}(M,a)\) of germs of all infinitesimal real-analytic CR-automorphisms of M at a is finite-dimensional and its complexification contains all constant vector fields \(\alpha\frac {\partial}{\partial z} \), α∈E, and the Euler vector field \(z\frac{\partial}{\partial z} \). Under these assumptions we show that: (I) every \({\frak{hol}}(M,a)\) consists of polynomial vector fields, hence coincides with the Lie algebra \({\frak{hol}}(M)\) of all infinitesimal real-analytic CR-automorphisms of M, (II) every local real-analytic CR-automorphism of M extends to a birational transformation of E, and (III) the group Bir (M) generated by such birational transformations is realized as a group of projective transformations upon embedding E as a Zariski open subset into a projective algebraic variety. Under additional assumptions the group Bir (M) is shown to have the structure of a Lie group with at most countably many connected components and Lie algebra \({\frak{hol}}(M)\). All of the above results apply, for instance, to Levi non-degenerate quadrics, as well as a large number of Levi degenerate tube manifolds.