Tameness of Complex Dimension in a Real Analytic Set

Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul

doi:10.4153/cjm-2012-019-4

Cited by 5 publications

(9 citation statements)

References 17 publications

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“…The subvariety X is 4-dimensional and coherent. It is easy to see that X [1] = X, X [2] = {z = 0, w = 0, Im ξ = 0}, and X [3] = ∅. The set X [2] is not complex.…”

Section: Examples Of Segre Variety Degeneraciesmentioning

confidence: 99%

“…But at the point (0, 0) the Segre variety is the complex line {w = 0}. In other words, X [0] = X \{(0, 0)}, X [1] = {(0, 0)}, and…”

Section: Examples Of Segre Variety Degeneraciesmentioning

confidence: 99%

“…For a good introduction in their use for submanifolds, see the book by Baouendi-Ebenfelt-Rothschild [2]. They started to be used for singular subvarieties recently, see for example Burns-Gong [4], Diederich-Mazzilli [6], the author [9], Adamus-Randriambololona-Shafikov [1], Fernández-Pérez [7], Pinchuk-Shafikov-Sukhov [11], and many others. However, the reader should be careful that sometimes in the literature on singular subvarieties a Segre variety is defined with respect to a single defining function and it is not made clear that it is then not well-defined if the point moves.…”

Section: Introductionmentioning

confidence: 99%

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Segre-Degenerate Points Form a Semianalytic Set

Lebl¹

2021

Preprint

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We prove that the set of Segre-degenerate points of a real-analytic subvariety X in C n is a closed semianalytic set. If X is coherent, it is a subvariety. More precisely, the set of points where the germ of the Segre variety is of dimension k or greater is a closed semianalytic set in general, and for a coherent X, it is a real-analytic subvariety of X. For a hypersurface X in C n , the set of Segre-degenerate points, X [n] , is a semianalytic set of dimension at most 2n − 4. If X is coherent, then X [n] is a complex subvariety of (complex) dimension n − 2. Example hypersurfaces are given showing that X [n] need not be a subvariety and that it also needs not be complex; X [n] can, for instance, be a real line.

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“…The subvariety X is 4-dimensional and coherent. It is easy to see that X [1] = X, X [2] = {z = 0, w = 0, Im ξ = 0}, and X [3] = ∅. The set X [2] is not complex.…”

Section: Examples Of Segre Variety Degeneraciesmentioning

confidence: 99%

“…But at the point (0, 0) the Segre variety is the complex line {w = 0}. In other words, X [0] = X \{(0, 0)}, X [1] = {(0, 0)}, and…”

Section: Examples Of Segre Variety Degeneraciesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Segre-Degenerate Points Form a Semianalytic Set

Lebl¹

2021

Preprint

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“…That is, a pure-dimensional semialgebraic arc-symmetric set need not have AR-closed filtration by the complex dimension. We will use the following notation from [3]: For a real analytic set R in C m and d ∈ N, let A d (R) denote the set of points p ∈ R such that R p contains a complex analytic germ of (complex) dimension d. Now, the irreducible real-algebraic hypersurface…”

Section: Holomorphic Closure Of An Arc-symmetric Setmentioning

confidence: 99%

CR-continuation of arc-analytic maps

Adamus

2015

Proc. Amer. Math. Soc.

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Given a set E in C m and a point p ∈ E, there is a unique smallest complex-analytic germ Xp containing Ep, called the holomorphic closure of Ep. We study the holomorphic closure of semialgebraic arc-symmetric sets. Our main application concerns CR-continuation of semialgebraic arc-analytic mappings: A mapping f : M → C n on a connected real-analytic CR manifold which is semialgebraic arc-analytic and CR on a non-empty open subset of M is CR on the whole M .

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“…On the other hand, our personal bias is to study the real-analytic and semianalytic sets for themselves. It seems natural to expect the sets S d (S) to remain close to the class of S. By comparison, in [1], we studied the sets A d (S) of those ξ ∈ S for which S ξ contains a complex analytic germ of dimension at least d. Theorem 1.1 of [1] asserts that the A d (S) are semianalytic (not necessarily real-analytic though). Alas, things do not look so good for the holomorphic closure dimension.…”

mentioning

confidence: 99%

Tameness of holomorphic closure dimension in a semialgebraic set

Adamus

Randriambololona

2012

Math. Ann.

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Given a semianalytic set S in a complex space and a point p in S, there is a unique smallest complex-analytic germ at p which contains the germ of S, called the holomorphic closure of S at p. We show that if S is semialgebraic then its holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic filtration by the holomorphic closure dimension. As a consequence, every semialgebraic subset of a complex vector space admits a semialgebraic stratification into CR manifolds satisfying a strong version of the condition of the frontier.Comment: Published versio

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Tameness of Complex Dimension in a Real Analytic Set

Abstract: Abstract. Given a real analytic set X in a complex manifold and a positive integer d, denote by A d the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that A d is a closed semianalytic subset of X.

Cited by 5 publications

References 17 publications

Segre-Degenerate Points Form a Semianalytic Set

Segre-Degenerate Points Form a Semianalytic Set

CR-continuation of arc-analytic maps

Tameness of holomorphic closure dimension in a semialgebraic set

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