arXiv subjects

Nordine Mir

Publications and source records attributed to Nordine Mir.

14 recordsLinked to original sources

arXiv2020

Unique jet determination and extension of germs of CR maps into spheres

We provide a new way of simultaneously parametrizing arbitrary local CR maps from real-analytic generic manifolds $M\subset {\mathbb C}^N$ into spheres ${\mathbb S}^{2N'-1}\subset {\mathbb C}^{N'}$ of any dimension. The parametrization is obtained as a composition of universal rational maps with a holomorphic map depending only on $M$. As applications, we obtain rigidity results of different flavours such as unique jet determination and global extension of local CR maps.

math.CV↗

arXiv2012

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 $p\in M$, for every real-algebraic subset $S'\subset \C^N\times\C^{N'}$ and every positive integer $\ell$, 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 $f$ at $p$ up to order $\ell$.

math.CV↗

arXiv2011

Holomorphic versus algebraic equivalence for deformations of real-algebraic CR manifolds

We consider (small) algebraic deformations of germs of real-algebraic CR submanifolds in complex space and study the biholomorphic equivalence problem for such deformations. We show that two algebraic deformations of minimal holomorphically nondegenerate real-algebraic CR submanifolds are holomorphically equivalent if and only if they are algebraically equivalent.

math.CV↗

arXiv2007

Finite jet determination of CR mappings

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M of C^N, N >= 2, which is essentially finite and of finite type at each of its points, for every point p on M there exists an integer l(p), depending upper-semicontinuously on p, such that for every smooth generic submanifold M' of C^N of the same dimension as M, if h_1 and h_2: (M,p)->M' are two germs of smooth finite CR mappings with the same l(p) jet at p, then necessarily their k-jets agree for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in C^N of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Omega and Omega' are two bounded domains in C^N with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary of Omega, such that if H_1 and H_2: Omega -> Omega' are two proper holomorphic mappings extending smoothly up to the boundary of Omega near some point boundary point p and agreeing up to order k at p, then necessarily H_1=H_2.

math.CV↗

arXiv2006

Lie group structures on automorphism groups of real-analytic CR manifolds

Given any real-analytic CR manifold M, we provide general conditions on M guaranteeing that the group of all its global real-analytic CR automorphisms is a Lie group (in an appropriate topology). Our conditions are in particular satisfied when M is an arbitrary compact real-analytic hypersurface embedded in some Stein manifold.

math.CV↗

arXiv2006

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.

math.CV↗

arXiv2006

Remarks on the rank properties of formal CR maps

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.

math.CV↗

arXiv2006

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.

math.CV↗

arXiv2003

Analytic regularity of CR maps into spheres

Let $M$ be a connected real-analytic hypersurface in $\C^N$ and $§$ the unit real sphere in $\C^{N'}$, $N'> N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of $\C^N$ and that there exists at least one strongly pseudoconvex point on $M$. We show that any CR map $f\colon M\to §$ of class $C^{N'-N+1}$ extends holomorphically to a neighborhood of $M$ in $\C^N$.

math.CV↗

arXiv2003

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 $M$ and $M'$ 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.

math.CV↗

arXiv2002

Approximation and convergence of formal CR-mappings

Let $M\subset C^N$ be a minimal real-analytic CR-submanifold and $M'\subset C^{N'}$ a real-algebraic subset through points $p\in M$ and $p'\in M'$. We show that that any formal (holomorphic) mapping $f\colon (C^N,p)\to (C^{N'},p')$, sending $M$ into $M'$, can be approximated up to any given order at $p$ by a convergent map sending $M$ into $M'$. If $M$ is furthermore generic, we also show that any such map $f$, that is not convergent, must send (in an appropriate sense) $M$ into the set $E'\subset M'$ of points of D'Angelo infinite type. Therefore, if $M'$ does not contain any nontrivial complex-analytic subvariety through $p'$, any formal map $f$ as above is necessarily convergent.

math.CV↗

arXiv2000

Reflection ideals and mappings between generic submanifolds in complex space

In this paper, we study formal mappings between smooth generic submanifolds in multidimensional complex space and establish results on finite determination, convergence and local biholomorphic and algebraic equivalence. Our finite determination result gives sufficient conditions to guarantee that a formal map as above is uniquely determined by its jet at a point of a preassigned order. For real-analytic generic submanifolds, we prove convergence of formal mappings under appropriate assumptions and also give natural geometric conditions to assure that if two germs of such submanifolds are formally equivalent, then they are necessarily biholomorphically equivalent. If the submanifolds are moreover real-algebraic, we address the question of deciding when biholomorphic equivalence implies algebraic equivalence. In particular, we prove that if two real-algebraic hypersurfaces in $\C^N$ are biholomorphically equivalent, then they are in fact algebraically equivalent. All the results are first proved in the more general context of ``reflection ideals" associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.

math.CV↗

arXiv2000

On the convergence of formal mappings

Let f : (M,p)\to (M',p') be a formal (holomorphic) nondegenerate map, i.e. with formal holomorphic Jacobian J_f not identically vanishing, between two germs of real analytic generic submanifolds in \C^n, p'=f(p). Assuming the target manifold to be real algebraic, and the source manifold to be minimal at p\in M in the sense of Tumanov, we prove the convergence of the so-called reflection mapping associated to f. From this, we deduce the convergence of such mappings from minimal real analytic generic submanifolds onto real algebraic holomorphically nondegenerate ones, as well as related results on partial convergence of such maps. For the proofs, we establish a principle of analyticity for formal CR power series. This principle can be used to reobtain the convergence of formal mappings between real analytic CR manifolds under a standard nondegeneracy condition.

math.CV↗

arXiv2000

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.

math.CV↗