Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Abstract

We consider (small) algebraic deformations of germs of realalgebraic Cauchy-Riemann 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.

Algebraic approximation in CR geometry

“…For nowhere minimal real-algebraic CR manifolds, partial results towards Corollary 1.4 were previously established by Baouendi, Rothschild and Zaitsev [6] and more recently by Lamel and the author in [11]. In these two works, the conclusion given by Corollary 1.4 is obtained for all points p in a certain Zariski open subset of M. The proofs of [6,11] require to exclude from M a thin set of points corresponding to the locus of the degeneracy set of a certain holomorphic foliation. Corollary 1.4 answers one of the main questions left open from [6,11] which was to decide whether one could get rid off this locus.…”

“…Our main result is the following: The study of algebraicity properties of holomorphic mappings sending real-algebraic sets into each other has been extensively studied in the past years (see e.g. [16,9,10,2,14,8,17,6,4,13,11]). Most of the mentioned results are concerned with the automatic algebraic extension of all holomorphic mappings between two given real-algebraic sets and hold under some geometric nondegeneracy conditions on these sets.…”

“…One noteworthy and typical application of Theorem 1.1 deals with the problem of deciding whether the holomorphic equivalence of two germs of real-algebraic CR manifolds of the same dimension implies their algebraic equivalence (see the works [6,4,11]). Specializing Theorem 1.1 to this situation, we have: As for Theorem 1.1, Corollary 1.4 was known only when M and M are (connected and somewhere) minimal and goes back in this case from the work of Baouendi, Rothschild and the author [4].…”

Artin’s approximation theorems and Cauchy-Riemann geometry

“…At this point, we should mention that the use of Theorem 3.11 in the equivalence and approximation problem in CR geometry appeared for the first time in [LM10]. One should also note that Theorem 3.11 is specific to the algebraic category.…”

“…Lamel and the author [LM10] improved later Theorem 2.14 by exhibiting another real-algebraic subvariety V ′ ⊂ V ⊂ M , that is in general strictly contained in the subvariety V , and for which the conclusion of Theorem 2.14 still holds with V replaced by V ′ . This subvariety V ′ is however in general strictly larger than the subvariety V 1 consisting of the set of points of p ∈ M where the dimension of the CR orbits is not constant in any neighborhood of p (as defined in Section 2.2).…”