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

Abstract

Let $M\subset {\mathbb{C}}^{n+1}$ ($n\ge 2$) 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 $n=1$.

25 pages

References in corpus (2)

• Family of intersecting totally real manifolds of $({\mathbb{C}}^n,0)$ and CR-singularities
• A Bishop surface with a vanishing Bishop invariant