Germs of local automorphisms of real-analytic CR structures and analytic dependence on $k$-jets

Abstract

The topic of the paper is the study of germs of local holomor- phisms f between C n and C n ' such that f(M) � Mand df(T c M) = T c M ' for MC n and M ' � C n ' generic real-analytic CR submanifolds of arbitrary codimensions. It is proved that for M minimal and Mfinitely nondegener- ate, such germs depend analytically on their jets. As a corollary, an analytic structure on the set of all germs of this type is obtained. ' be connected locally closed real-analytic submanifolds, x ∈ M,x ' ∈ Mbe arbitrary points. The complex tangent subspace TxM ∩iTxM will be denoted by T c x M. M is a CR manifold, if dimT c x M is constant. In this case dimCRM := dimCT c M is called the CR dimension and codimCRM := dimIRTM − dimIRT c M the CR codimension. A CR submanifold M ⊂ C n is called generic, if TM +iTM = C n . Suppose for the moment that M,M ' ⊂ C n are generic real-analytic CR subman- ifolds of the same CR dimension and the same CR codimension. Baouendi, Ebenfelt and Rothschild found optimal nondegeneracy conditions on M and Msuch that a germ at x of a local biholomorphism f (between some neighborhoods in C n ) with f(M) ⊂ M ' ,f(x) = x ' , is uniquely determined by itsk-jetj k xf, wherek is an integer