The author shows the following:
a) If is a real analytic surface in with a non-degenerate complex tangent at , then there is a local biholomorphic map of , , such that is given by , .
b) If is a real analytic submanifold of of dimension 4 with a non-degenerate complex tangent at , then there is a local biholomorphic map of , , such that is given by , , .
The proofs of both theorems involve a rapid iteration argument, by solving a linearized functional equation first. The second normal form is more difficult because of the non-trivial quadratic terms. To avoid the radius of convergence of the linearized functional equation shrinking too much, the author has to find a good solution among all possible ones. This is the main novelty of the paper.
The normal form of real analytic surfaces in has been studied by J. K. Moser and S. M. Webster [Acta Math. 150, 255–296 (1983; Zbl 0519.32015)].