Equivalences of Real Submanifolds in Complex Space

Abstract

We show that for any real-analytic submanifold M in C N there is a proper real-analytic subvariety V contained in M such that for any p ∈ M \ V , any real-analytic submanifold M ′ in C N , and any p ′ ∈ M ′ , the germs of the submanifolds M and M ′ at p and p ′ respectively are formally equivalent if and only if they are biholomorphically equivalent. More general results for k-equivalences are also stated and proved.

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“…Indeed, in view of an example of Moser-Webster [MW83], there exist realalgebraic surfaces M, M ′ ⊂ C 2 that are formally but not biholomorphically equivalent. However, our first main result shows that this phenomenon cannot happen if M is a minimal CR-submanifold (not necessarily algebraic) in C N (see §2.1 for the notation and definitions): Approximation results in the spirit of Theorem 1.1 have been recently obtained in [BRZ00,BMR00] in the important case when N = N ′ and f is invertible. Note that under the assumptions of Theorem 1.1, there may exist nonconvergent maps f sending M into M ′ .…”

Mappings between real submanifolds in complex space

“…In the first case, one may use the implicit function theorem to show that there is a dense open set U ⊂ M such for any p ∈ U, after a change of local holomorphic coordinates in C N near p, one may assume p = 0 and M = M 1 × C s , where M 1 is a holomorphically nondegenerate generic submanifold of C N −s , with s a positive integer (see [BRZ01]). Thus if M is connected and holomorphically degenerate, near all points p ∈ U, after a change of holomorphic coordinates near p, one may assume that p = 0 and M is given by a defining function which is independent of one of the coordinates, say Z N .…”

On some rigidity properties of mappings between CR-submanifolds in complex space

“…More precisely, one has the following problem: Given two real-analytic submanifolds M, M ⊂ C N of the same dimension, characterize those points p ∈ M and p ∈ M for which formal equivalence of the germs (M, p) and (M , p ) implies their biholomorphic equivalence. A recent result along these lines in [BRZ01a] states that notions of formal and biholomorphic equivalence coincide at all points of nonempty Zariski open subsets of the submanifolds (by Zariski open subsets we shall always mean complements of real-analytic subvarieties). More precisely, the following holds: The subvariety V in Theorem 1.1 is explicitly described as a union of three distinguished proper real-analytic subvarieties.…”

“…We refer the reader to [BRZ01a] and [BMR02] for more details on this matter. In view of Theorem 1.7 and Theorem 1.8, one may formulate a more general approximation problem for formal holomorphic maps in complex spaces of (possibly) different dimensions: …”

“…[BER99a] for the definition). It can be shown that the set of points of constant degeneracy on a real-analytic CR-submanifold is indeed an (analytic) Zariski open subset of M (see [BRZ01a]). …”