On the CR transversality of holomorphic maps into hyperquadrics

Abstract

Let $M_{\ell }$ be a smooth Levi-nondegenerate hypersurface of signature $\ell$ in ${\mathbf{C}}^n$ with $n\ge 3$, and write $H_{\ell }^N$ for the standard hyperquadric of the same signature in ${\mathbf{C}}^N$ with $N-n<\frac{n-1}{2}$. Let $F$ be a holomorphic map sending $M_{\ell }$ into $H_{\ell }^N$. Assume $F$ does not send a neighborhood of $M_{\ell }$ in ${\mathbf{C}}^n$ into $H_{\ell }^N$. We show that $F$ is necessarily CR transversal to $M_{\ell }$ at any point. Equivalently, we show that $F$ is a local CR embedding from $M_{\ell }$ into $H_{\ell }^N$.

To appear in Abel Symposia, dedicated to Professor Yum-Tong Siu on the occasion of his 70th birthday