Defect and evaluations

Abstract

Let $S$ be a generic submanifold of $C^N$ of real codimension m. In this work we continue the study, carried over by various authors, of the set of analytic discs attached to S. Let $M$ be the set of analytic discs attached to $S.$ Given $q \in S$ let $M_q$ be the set of discs $ϕ$ in M such that $ϕ_(1).$ B. Trepreau and other authors gave sufficient conditions for $M$ to be a manifold in a neighborhood of a given disc. We give conditions for $M_q$ to be a manifold. When this conditions are satisfied we look at the map on $M$ given by $ϕ\rightarrow ϕ(0),$ and we describe the image of its differential, (in particular we determine its dimension). We then do the same for the map $ϕ\rightarrow ϕ(-1)$ on $M_q.$ For example we find as a corollary that if S has only minimal points, then there exists an open dense subset $Omega$ in M such that the restriction of the map $ϕ\rightarrow ϕ(0)$ to $Ω$ is an open map with value in $C^N.$