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

TL;DR

This paper characterizes when a real analytic submanifold in complex space, defined near a quadratic form, is holomorphically equivalent to a symmetric quadric, extending Moser's results to higher dimensions.

Contribution

It provides a pseudo-normal form for such submanifolds and establishes necessary and sufficient conditions for their formal equivalence to symmetric quadrics.

Findings

Holomorphic equivalence to the quadric is characterized by formal transformability.

A pseudo-normal form near the origin is derived for the submanifold.

Conditions for formal flattening of the submanifold are established.

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 $n=1$.