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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the holomorphic closure dimension of real analytic sets


Authors: Janusz Adamus and Rasul Shafikov
Journal: Trans. Amer. Math. Soc. 363 (2011), 5761-5772
MSC (2010): Primary 32B20, 32V40
Published electronically: June 9, 2011
MathSciNet review: 2817408
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Abstract: Given a real analytic (or, more generally, semianalytic) set $ R$ in $ \mathbb{C}^n$ (viewed as $ \mathbb{R}^{2n}$), there is, for every $ p\in\bar{R}$, a unique smallest complex analytic germ $ X_p$ that contains the germ $ R_p$. We call $ \dim_{\mathbb{C}}X_p$ the holomorphic closure dimension of $ R$ at $ p$. We show that the holomorphic closure dimension of an irreducible $ R$ is constant on the complement of a closed proper analytic subset of $ R$, and we discuss the relationship between this dimension and the CR dimension of $ R$.


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Additional Information

Janusz Adamus
Affiliation: Department of Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B7 – and – Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
Email: jadamus@uwo.ca

Rasul Shafikov
Affiliation: Department of Mathematics, The University of Western Ontario, London, Ontario, Canada N6A 5B7
Email: shafikov@uwo.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05276-1
PII: S 0002-9947(2011)05276-1
Keywords: Real analytic sets, semianalytic sets, holomorphic closure dimension, complexification, Gabrielov regularity, CR dimension
Received by editor(s): June 3, 2008
Received by editor(s) in revised form: April 3, 2009, and September 17, 2009
Published electronically: June 9, 2011
Additional Notes: This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 2011 American Mathematical Society