Abstract
Given a semianalytic set S in \({\mathbb{C}^n}\) and a point \({p \in \bar{S}}\), there is a unique smallest complex-analytic germ X p which contains S p , called the holomorphic closure of S p . We show that if S is semialgebraic then X p 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.
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References
Adamus, J., Randriambololona, S., Shafikov, R.: Tameness of complex dimension in a real analytic set. Electronic preprint. arXiv:1006.4190v2
Adamus J., Shafikov R.: On the holomorphic closure dimension of real analytic sets. Trans. Am. Math. Soc. 363(11), 5761–5772 (2011)
Baouendi M.S, Ebenfelt P., Rothschild L.P.: Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series, no. 47. Princeton University Press, Princeton (1999)
Bochnak J., Coste M., Roy M.-F.: Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), no. 36. Springer, Berlin (1998)
Boggess A.: CR Manifolds and the Tangential Cauchy-Riemann complex. Studies in Advanced Mathematics. CRC Press, Boca Raton (1991)
Cartan H.: Variétés analytiques réelles et variétes analytiques complexes. Bull. Soc. Math. France 85, 77–99 (1957)
Coste, M.: An Introduction to Semialgebraic Geometry. Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000)
D’Angelo J.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton (1993)
Fortuna E., Łojasiewicz S., Raimondo M.: Algébricité de germes analytiques. J. Reine Angew. Math. 374, 208–213 (1987)
Grauert H., Remmert R.: Analytische Stellenalgebren. Die Grundlehren der Mathematischen Wissenschaften, vol. 176. Springer, Berlin (1971)
Grothendieck, A., Dieudonné, J.: Eléments de géométrie algébrique IV. Etudes locale des schémas et des morphismes de schémas. Publ. Math. I.H.E.S. 20 (1964); 24 (1965); 28 (1966)
Łojasiewicz S.: Introduction to Complex Analytic Geometry. Birkhäuser, Basel (1991)
Marker D.: Semialgebraic expansions of \({\mathbb{C}}\). Trans. Am. Math. Soc. 320(2), 581–592 (1990)
Peterzil Y., Starchenko S.: Complex analytic geometry and analytic-geometric categories. J. Reine Angew. Math. 626, 39–74 (2009)
Shafikov R.: Real analytic sets in complex spaces and CR maps. Math. Z. 256, 757–767 (2007)
Tworzewski P.: Intersections of analytic sets with linear subspaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17(4), 227–271 (1990)
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J. Adamus’s research was partially supported by Natural Sciences and Engineering Research Council of Canada.
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Adamus, J., Randriambololona, S. Tameness of holomorphic closure dimension in a semialgebraic set. Math. Ann. 355, 985–1005 (2013). https://doi.org/10.1007/s00208-012-0808-y
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DOI: https://doi.org/10.1007/s00208-012-0808-y