Divergence of the normalization for real Lagrangian surfaces near complex tangents

Abstract

We study real Lagrangian analytic surfaces in C^2 with a non-degenerate complex tangent. Webster proved that all such surfaces can be transformed into a quadratic surface by formal symplectic transformations of C^2. We show that there is a certain dense set of real Lagrangian surfaces which cannot be transformed into the quadratic surface by any holomorphic (convergent) transformation of C^2. The divergence is contributed by the parabolic character of a pair of involutions generated by the real Lagrangian surfaces.