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# Tameness of Complex Dimension in a Real Analytic Set

Published:2012-07-16
Printed: Aug 2013
Department of Mathematics, The University of Western Ontario, London, ON N6A 5B7
• Serge Randriambololona,
Department of Mathematics, The University of Western Ontario, London, ON N6A 5B7
• Rasul Shafikov,
Department of Mathematics, The University of Western Ontario, London, ON N6A 5B7
 Format: LaTeX MathJax PDF

## Abstract

Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $\mathcal A^d$ 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 $\mathcal A^d$ is a closed semianalytic subset of $X$.
 Keywords: complex dimension, finite type, semianalytic set, tameness
 MSC Classifications: 32B10 - Germs of analytic sets, local parametrization 32B20 - Semi-analytic sets and subanalytic sets [See also 14P15] 32C07 - Real-analytic sets, complex Nash functions [See also 14P15, 14P20] 32C25 - Analytic subsets and submanifolds 32V15 - CR manifolds as boundaries of domains 32V40 - Real submanifolds in complex manifolds 14P15 - Real analytic and semianalytic sets [See also 32B20, 32C05]

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