On the holomorphic closure dimension of real analytic sets

Adamus, Janusz; Shafikov, Rasul

doi:10.1090/S0002-9947-2011-05276-1

On the holomorphic closure dimension of real analytic sets
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by Janusz Adamus and Rasul Shafikov PDF

Trans. Amer. Math. Soc. 363 (2011), 5761-5772 Request permission

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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