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Search: MSC category 32C25 ( Analytic subsets and submanifolds )

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1. CJM 2012 (vol 65 pp. 721)

Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul
Tameness of Complex Dimension in a Real Analytic Set
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
Categories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15

2. CJM 2008 (vol 60 pp. 33)

Braun, RĂ¼diger W.; Meise, Reinhold; Taylor, B. A.
Higher Order Tangents to Analytic Varieties along Curves. II
Let~$V$ be an analytic variety in some open set in~$\C^n$. For a real analytic curve~$\gamma$ with $ \gamma(0) = 0 $ and $ d \ge 1 $ define $ V_t = t^{-d}(V - \gamma(t)) $. It was shown in a previous paper that the currents of integration over~$V_t$ converge to a limit current whose support $ T_{\gamma,d} V $ is an algebraic variety as~$t$ tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the~$V_t$. As a corollary, it is shown that $ T_{\gamma,d} V $ is either inhomogeneous or coincides with $ T_{\gamma,\delta} V $ for all~$\delta$ in some neighborhood of~$d$. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragm\'en--Lindel\"of conditions. Corresponding results for limit varieties $ T_{\sigma,\delta} W $ of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.


3. CJM 2003 (vol 55 pp. 64)

Braun, RĂ¼diger W.; Meise, Reinhold; Taylor, B. A.
Higher Order Tangents to Analytic Varieties along Curves
Let $V$ be an analytic variety in some open set in $\mathbb{C}^n$ which contains the origin and which is purely $k$-dimensional. For a curve $\gamma$ in $\mathbb{C}^n$, defined by a convergent Puiseux series and satisfying $\gamma(0) = 0$, and $d \ge 1$, define $V_t := t^{-d} \bigl( V-\gamma(t) \bigr)$. Then the currents defined by $V_t$ converge to a limit current $T_{\gamma,d} [V]$ as $t$ tends to zero. $T_{\gamma,d} [V]$ is either zero or its support is an algebraic variety of pure dimension $k$ in $\mathbb{C}^n$. Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragm\'en-Lindel\"of condition that was used by H\"ormander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on $\mathbb{R}^n$.


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