Higher Order Tangents to Analytic Varieties along Curves. II

Abstract

AbstractLet V be an analytic variety in some open set in ℂn. For a real analytic curve γ with γ(0) = 0 and d ≥ 1, define Vt = t−d(V − γ(t)). It was shown in a previous paper that the currents of integration over Vt converge to a limit current whose support Tγ,δV 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 Vt. As a corollary, it is shown that Tγ,δV is either inhomogeneous or coincides with Tγ,δV for all δ 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 Tσ,δW of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.