location:  Publications → journals → CJM
Abstract view

# Higher Order Tangents to Analytic Varieties along Curves. II

Published:2008-02-01
Printed: Feb 2008
• Rüdiger W. Braun
• Reinhold Meise
• B. A. Taylor
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

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.
 MSC Classifications: 32C25 - Analytic subsets and submanifolds