On the defect of an analytic disc

Abstract

Although the concept of defect of an analytic disc attached to a generic manifold of $\C^{n}$ seems to play a merely technical role, it turns out to be a rather deep and fruitful notion for the extendability of CR functions defined on the manifold. In this paper we give a new geometric description of defect, drawing attention to the behaviour of the interior points of the disc by infinitesimal perturbations. For hypersurfaces a stronger result holds because these perturbations describe a complex vector space of $\C^{n}$. For a big analytic disc the defect does not need to be smaller than the codimension of the manifold. Indeed we show by an example that it can be arbitrarily large independently of the codimension of the manifold. Nevertheless we also prove that the defect is always finite. In the case of a hypersurface we give a geometric upper bound for the defect.