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Higher Order Tangents to Analytic Varieties along Curves

  Published:2003-02-01
 Printed: Feb 2003
  • Rüdiger W. Braun
  • Reinhold Meise
  • B. A. Taylor
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Abstract

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 $\mathbb{R}^n$.
MSC Classifications: 32C25 show english descriptions Analytic subsets and submanifolds 32C25 - Analytic subsets and submanifolds
 

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