45 research outputs found
Convergence of formal embeddings between real-analytic hypersurfaces in codimension one
We show that every formal embedding sending a real-analytic strongly
pseudoconvex hypersurface in M\subset \C^N into another such hypersurface in
M'\subset \C^{N+1} is convergent. More generally, if and are merely
Levi-nondegenerate, the same conclusion holds for any formal embedding provided
either that the embedding is CR transversal or the target hypersurface does not
contain any complex curves.Comment: 8 page
Formal biholomorphic maps of real analytic hypersurfaces
Let f : (M,p)\to (M',p') be a formal biholomorphic mapping between two germs
of real analytic hypersurfaces in \C^n, p'=f(p). Assuming the source manifold
to be minimal at p, we prove the convergence of the so-called reflection
function associated to f. As a consequence, we derive the convergence of formal
biholomorphisms between real analytic minimal holomorphically nondegenerate
hypersurfaces. Related results on partial convergence of formal biholomorphisms
are also obtained.Comment: 15 pages, Late
Analytic regularity of CR maps into spheres
Let be a connected real-analytic hypersurface in \C^N and the unit
real sphere in \C^{N'}, . Assume that does not contain any
complex-analytic hypersurface of \C^N and that there exists at least one
strongly pseudoconvex point on . We show that any CR map
of class extends holomorphically to a neighborhood of in
\C^N.Comment: 11 page
Algebraic approximation in CR geometry
We prove the following CR version of Artin's approximation theorem for
holomorphic mappings between real-algebraic sets in complex space. Let
M\subset \C^N be a real-algebraic CR submanifold whose CR orbits are all of
the same dimension. Then for every point , for every real-algebraic
subset S'\subset \C^N\times\C^{N'} and every positive integer , if
f\colon (\C^N,p)\to \C^{N'} is a germ of a holomorphic map such that {\rm
Graph}\, f \cap (M\times \C^{N'})\subset S', then there exists a germ of a
complex-algebraic map f^\ell \colon (\C^N,p)\to \C^{N'} such that {\rm
Graph}\, f^\ell \cap (M\times \C^{N'})\subset S' and that agrees with at
up to order .Comment: To appear in J. Math. Pures App
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
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
Approximation and convergence of formal CR-mappings
Let be a minimal real-analytic CR-submanifold and a real-algebraic subset through points and . We show
that that any formal (holomorphic) mapping ,
sending into , can be approximated up to any given order at by a
convergent map sending into . If is furthermore generic, we also
show that any such map , that is not convergent, must send (in an
appropriate sense) into the set of points of D'Angelo
infinite type. Therefore, if does not contain any nontrivial
complex-analytic subvariety through , any formal map as above is
necessarily convergent