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Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case. (English) Zbl 1032.32025
The authors prove the finite jet determination of local analytic CR automorphisms of a real analytic hypersurface in $$\mathbb{C}^2$$. More precisely, let $$M$$ be a real analytic hypersurface in $$\mathbb{C}^2$$, which is not Levi flat near a point $$p$$. Then there exists an integer $$k$$ such that if $$f$$ and $$g$$ are germs of real analytic CR automorphisms of $$M$$ at $$p$$ with the same $$k$$-jet at $$p$$, then $$f=g$$.
The proof relies on the parametrization of local biholomorphisms along the zero Segre variety and a finite determination result for solutions of a system of singular differential equations. In the case of finite type, the authors prove that local biholomorphisms (or real analytic CR automorphisms) are not only uniquely determined, but also analytically parametrized by their 2-jets.

##### MSC:
 32V35 Finite-type conditions on CR manifolds 32V40 Real submanifolds in complex manifolds
##### Keywords:
analytic CR automorphism; finite type; jet
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