The equivalence theory for infinite type hypersurfaces in ${\mathbb{C}}^2$

Abstract

We develop a classification theory for real-analytic hypersurfaces in ${\mathbb{C}}^2$ in the case when the hypersurface is of {\em infinite type} at the reference point. This is the remaining, not yet understood case in ${\mathbb{C}}^2$ in the {\it Problème local}, formulated by H.\,Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results, appearing in this revised version, is a notion of {\em smooth normal forms} for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR -- DS technique in CR-geometry.