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1. CJM 2012 (vol 65 pp. 721)
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)
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.
Category:32C25 |
3. CJM 2003 (vol 55 pp. 64)
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 $\mathbb{R}^n$.
Category:32C25 |