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Abstract. We give necessary conditions for certain real analytic tube generic submanifolds in C n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C n . During the proof, we show that the local CR automorphism group of a minimal, finitely nondegenerate real algebraic generic submanifold is a real algebraic local Lie group. We may state one of the main results as follows. Let M be a real analytic hypersurface tube in C n passing through the origin, having a defining equation of the form v = ϕ(y), where (z, w) = (x + iy, u + iv) ∈ C n−1 × C. Assume that M is Levi nondegenerate at the origin and that the real Lie algebra of local infinitesimal CR automorphisms of M is of minimal possible dimension n, i.e. generated by the real parts of the holomorphic vector fields ∂z 1 , . . . , ∂z n−1 , ∂w. Then M is locally algebraizable only if every second derivative ∂ 2 y k y l ϕ is an algebraic function of the collection of first derivatives ∂y 1 ϕ, ..., ∂y m ϕ. A real analytic submanifold M in C n is called algebraic if it can be represented locally by the vanishing of a collection of Nash algebraic real analytic functions. We say that M is locally algebraizable at one of its points p if there exist some local holomorphic coordinates centered at p in which M is algebraic. For instance, every totally real, real analytic submanifold in C n of dimension k ≤ n is locally biholomorphic to a k-dimensional linear real plane, hence locally algebraizable. Also, every complex manifold is locally algebraizable. Although every real analytic submanifold M is clearly locally equivalent to its tangent plane by a real analytic (in general not holomorphic) equivalence, the question whether M is biholomorphically equivalent to a real algebraic submanifold is subtle. In this article, we study the question whether every real analytic CR submanifold is locally algebraizable. One of the interests of algebraizability lies in the reflection principle, which is better understood in the algebraic category. Indeed, in the fundamental works of Pinchuk [Pi1975] [Sha2002], the extendability of germs of CR mappings with target in a real algebraic hypersurface is achieved. On the contrary, even if some results previously shown under an algebraization hypothesis were proved recently under general assumptions (see the strong result obtained by Diederich-Pinchuk [DP2003]), most of the results cited above are still open in the case of a real analytic target hypersurface. Date: 2017Date: -8-23. 1991 Mathematics Subject Classification. Primary: 32V40. Secondary 32V25, 32H02, 32H40, 32V10. , it is known that every real analytic two-dimensional surface S ⊂ C 2 at an isolated elliptic (in the sense of Bishop) complex tangency p ∈ S is biholomorphic to one of the surfaces S γ,δ,s := {(z 1 , z 2 ) ∈ C 2 : y 2 = 0,)}, where p corresponds to the origin, where 0 < γ < 1/2 is Bishop's invariant and where δ = ±1 and s ∈ N or δ = 0. The quantities γ, δ, s form a complete system...
Abstract. We give necessary conditions for certain real analytic tube generic submanifolds in C n to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in C n . During the proof, we show that the local CR automorphism group of a minimal, finitely nondegenerate real algebraic generic submanifold is a real algebraic local Lie group. We may state one of the main results as follows. Let M be a real analytic hypersurface tube in C n passing through the origin, having a defining equation of the form v = ϕ(y), where (z, w) = (x + iy, u + iv) ∈ C n−1 × C. Assume that M is Levi nondegenerate at the origin and that the real Lie algebra of local infinitesimal CR automorphisms of M is of minimal possible dimension n, i.e. generated by the real parts of the holomorphic vector fields ∂z 1 , . . . , ∂z n−1 , ∂w. Then M is locally algebraizable only if every second derivative ∂ 2 y k y l ϕ is an algebraic function of the collection of first derivatives ∂y 1 ϕ, ..., ∂y m ϕ. A real analytic submanifold M in C n is called algebraic if it can be represented locally by the vanishing of a collection of Nash algebraic real analytic functions. We say that M is locally algebraizable at one of its points p if there exist some local holomorphic coordinates centered at p in which M is algebraic. For instance, every totally real, real analytic submanifold in C n of dimension k ≤ n is locally biholomorphic to a k-dimensional linear real plane, hence locally algebraizable. Also, every complex manifold is locally algebraizable. Although every real analytic submanifold M is clearly locally equivalent to its tangent plane by a real analytic (in general not holomorphic) equivalence, the question whether M is biholomorphically equivalent to a real algebraic submanifold is subtle. In this article, we study the question whether every real analytic CR submanifold is locally algebraizable. One of the interests of algebraizability lies in the reflection principle, which is better understood in the algebraic category. Indeed, in the fundamental works of Pinchuk [Pi1975] [Sha2002], the extendability of germs of CR mappings with target in a real algebraic hypersurface is achieved. On the contrary, even if some results previously shown under an algebraization hypothesis were proved recently under general assumptions (see the strong result obtained by Diederich-Pinchuk [DP2003]), most of the results cited above are still open in the case of a real analytic target hypersurface. Date: 2017Date: -8-23. 1991 Mathematics Subject Classification. Primary: 32V40. Secondary 32V25, 32H02, 32H40, 32V10. , it is known that every real analytic two-dimensional surface S ⊂ C 2 at an isolated elliptic (in the sense of Bishop) complex tangency p ∈ S is biholomorphic to one of the surfaces S γ,δ,s := {(z 1 , z 2 ) ∈ C 2 : y 2 = 0,)}, where p corresponds to the origin, where 0 < γ < 1/2 is Bishop's invariant and where δ = ±1 and s ∈ N or δ = 0. The quantities γ, δ, s form a complete system...
Let C 2,1 be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. We show that the CR-structures in C 2,1 are reducible to so(3, 2)-valued absolute parallelisms and give applications of this result.
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