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

TL;DR

This paper proves that under certain conditions, local CR-automorphisms of real-analytic CR-manifolds extend to birational transformations, which can be represented as projective transformations, revealing a deep algebraic structure.

Contribution

It establishes that all local automorphisms are polynomial vector fields and extends them to birational and projective transformations, providing a new regularization framework.

Findings

Automorphisms are polynomial vector fields.

Local automorphisms extend to birational transformations.

The automorphism group can be realized as a projective group.

Abstract

Let M be a connected generic real-analytic CR-submanifold of a finite-dimensional complex vector space E. Suppose that for every point a in 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 and the Euler vector field. 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.