Higher Order Tangents to Analytic Varieties along Curves

Braun, Rüdiger W. ; Meise, Reinhold ; Taylor, B. A.

Canadian journal of mathematics, Tome 55 (2003) no. 1, pp. 64-90

Résumé

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 ${{V}_{t}}\,:=\,{{t}^{-d}}\,\left( V\,-\,\gamma \left( t \right) \right)$ . Then the currents defined by ${{V}_{t}}$ converge to a limit current ${{T}_{\gamma ,d}}\left[ V \right]$ as $t$ tends to zero. ${{T}_{\gamma ,d}}\left[ V \right]$ 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 Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on ${{\mathbb{R}}^{n}}$ .

Détail

DOI : 10.4153/CJM-2003-003-3

Mots-clés : 32C25

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