A codimension two CR singular submanifold that is formally equivalent to a symmetric quadric

Abstract

Let $M\subset \mathbb{C}^{n+1}$ ($n\geq 2$) be a real analytic submanifold defined by an equation of the form: $w=|z|^2+O(|z|^3)$, where we use $(z,w)\in \mathbb{C}^{n}\times \mathbb{C}$ for the coordinates of $\mathbb{C}^{n+1}$. We first derive a pseudo-normal form for $M$ near 0. We then use it to prove that $(M,0)$ is holomorphically equivalent to the quadric $(M_\infty: w=|z|^2,0)$ if and only if it can be formally transformed to $(M_\infty,0)$. We also use it to give a necessary and sufficient condition when $(M,0)$ can be formally flattened. The result is due to Moser for the case of $n=1$.