CR Singularities and Generalizations of Moser's Theorem II

Abstract

Let a real-analytic manifold $M$ formally (holomorphically) equivalent to the following model \begin{equation*}w=z_{1}\overline{z}_{1}+\dots+z_{N}\overline{z}_{N}+\lambda_{1}\left(z_{1}^{2}+\overline{z}_{1}^{2}\right)+\dots+\lambda_{N}\left(z_{N}^{2}+\overline{z}_{N}^{2}\right),{equation*} assuming that $$ \lambda_{1},\dots, \lambda_{N}\in \left[0,\frac{1}{2}\right).$$ It is proven that $M$ is holomorphically equivalent to this model by developing a partial normal form for such real-analytic submanifold using generalized Fischer Decompositions. In particular, there are defined certain Spaces of Normalizations used also in proving other analogues of The Theorem of Moser in certain non-equidimensional situations. There presented also other applications for the methods considered.