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Higher Order Tangents to Analytic Varieties along Curves. II

  Published:2008-02-01
 Printed: Feb 2008
  • Rüdiger W. Braun
  • Reinhold Meise
  • B. A. Taylor
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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 show english descriptions Analytic subsets and submanifolds 32C25 - Analytic subsets and submanifolds
 

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