Regularization of local CR-automorphisms of real-analytic CR-manifolds

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 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 α∂/∂z , α ∈ E, and the Euler vector field z∂/∂z. Under these assumptions we show that: (I) every hol(M,a) consists of polynomial vector fields, hence coincides with the Lie algebra 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 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.