524 research outputs found
Finite jet determination of local CR automorphisms through resolution of degeneracies
Let M be a connected real-analytic hypersurface in N-dimensional complex
euclidean space whose Levi form is nondegenerate at some point. We prove that
for every point p in M, there exists an integer k=k(M,p) such that germs at p
of local real-analytic CR automorphisms of M are uniquely determined by their
k-jets (at p). To prove this result we develop a new technique that can be seen
as a resolution of the degeneracies of M. This procedure consists of blowing up
M near an arbitrary point p in M regardless of its minimality or nonminimality;
then, thanks to the blow-up, the original problem can be reduced to an
analogous one for a very special class of nonminimal hypersurfaces for which
one may use known techniques to prove the finite jet determination property of
its CR automorphisms.Comment: 16 page
Convergence of the Chern-Moser-Beloshapka normal forms
In this article, we first describe a normal form of real-analytic,
Levi-nondegenerate submanifolds of of codimension d 1 under the
action of formal biholomorphisms, that is, of perturbations of
Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal
normal form that ensures that the normalizing transformation to this normal
form is holomorphic. We show that our techniques can be adapted in the case d =
1 in order to obtain a new and direct proof of Chern-Moser normal form theorem
Parametrization of local CR automorphisms by finite jets and applications
For any real-analytic hypersurface M in complex euclidean space of dimension
>= 2 which does not contain any complex-analytic subvariety of positive
dimension, we show that for every point p in M the local real-analytic CR
automorphisms of M fixing p can be parametrized real-analytically by their
l(p)-jets at p. As a direct application, we derive a Lie group structure for
the topological group Aut(M,p). Furthermore, we also show that the order l(p)
of the jet space in which the group Aut(M,p) embeds can be chosen to depend
upper-semicontinuously on p. As a first consequence, it follows that that given
any compact real-analytic hypersurface M in complex euclidean space, there
exists an integer k depending only on M such that for every point p in M germs
at p of CR diffeomorphisms mapping M into another real-analytic hypersurface in
a complex space of the same dimension are uniquely determined by their k-jet at
that point. Another consequence is a boundary version of H. Cartan's uniqueness
theorem.
Our parametrization theorem also holds for the stability group of any
essentially finite minimal real-analytic CR manifold of arbitrary codimension.
One of the new main tools developed in the paper, which may be of independent
interest, is a parametrization theorem for invertible solutions of a certain
kind of singular analytic equations, which roughly speaking consists of
inverting certain families of parametrized maps with singularities.Comment: to appear in J. Amer. Math. So
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