Parametrization of local biholomorphisms of real analytic hypersurfaces

Abstract

Let $M$ be a real analytic hypersurface in $\bC^N$ which is finitely nondegenerate, a notion that can be viewed as a generalization of Levi nondegenerate, at $p_0\in M$. We show that if $M'$ is another such hypersurface and $p'_0\in M'$, then the set of germs at $p_0$ of biholomorphisms $H$ with $H(M)\subset M'$ and $H(p_0)=p'_0$, equipped with its natural topology, can be naturally embedded as a real analytic submanifold in the complex jet group of $\bC^N$ of the appropriate order. We also show that this submanifold is defined by equations that can be explicitly computed from defining equations of $M$ and $M'$. Thus, $(M,p_0)$ and $(M',p'_0)$ are biholomorphically equivalent if and only if this (infinite) set of equations in the complex jet group has a solution. Another result obtained in this paper is that any invertible formal map $H$ that transforms $(M,p_0)$ to $(M',p'_0)$ is convergent. As a consequence, $(M,p_0)$ and $(M',p'_0)$ are biholomorphically equivalent if and only if they are formally equivalent.