Equivalence of Cauchy-Riemann manifolds and multisummability theory

We prove that if two real-analytic hypersurfaces in ${\mathbb{C}}^2$ are equivalent formally, then they are also $C^{\mathrm{\infty }}$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in ${\mathbb{C}}^2$ are algebraic (in particular are convergent). The result is obtained by using the recent {\em CR

• DS technique}, connecting degenerate CR-manifolds and Dynamical Systems, and employing subsequently the {\em multisummability theory} of divergent power series used in the Dynamical Systems theory.