Higher Order Tangents to Analytic Varieties along Curves. II

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

Canadian journal of mathematics, Tome 60 (2008) no. 1, pp. 33-63

Résumé

Let $V$ be an analytic variety in some open set in ${{\mathbb{C}}^{n}}$ . 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 Phragmén-Lindelöf 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.

Détail

DOI : 10.4153/CJM-2008-002-7

Mots-clés : 32C25

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