5/11/2005

Nefton Pali
Princeton University

Complex analytic zeroes of smooth functions

We will explain a differential criteria which allows  to find complex analytic zeros of smooth functions. An application of our criteria concerns a method which allows to find complex analytic sets which are obtained by smooth deformations of other ones. We will give a idea of the proof in some particular case. The principal difficulty of the proof is the solution of a quasi-linear differential equation with standard $\bar{\partial}$ as its principal term. We are able to find a solution of this differential equation, using a  Nash-Moser fast iteration method.