A canonical embedding of $\textbf{Aut}_{\textbf{hol}}({\bf \mathbb C^n})$

Abstract

The group $\text{Aut}_{\text{hol}}(\mathbb C^n)$ of self-biholomorphisms of $\mathbb C^n$ consists of affine maps if $n=1$, but in higher dimensions it is a large object that has not been described explicitly. Despite the intricacies involved when $n>1$, surprisingly every $F\in \text{Aut}_{\text{hol}}(\mathbb C^n)$ is uniquely determined inside the group by only two data, of infinitesimal and global nature: the $1$-jet of $F$ at $0$, and the complex Hessian of a certain plurisubharmonic function associated to $F$. If $n=1$ this global datum is zero for all $F$, which is then determined solely by its $1$-jet at $0$, and one recovers $\text{Aut}_{\text{hol}}(\mathbb C)= \text{Aff}(\mathbb C)\cong \mathbb C \times \mathbb C^{*}$. Our main result, formulated as the existence of a canonical embedding of $ \text{Aut}_{\text{hol}} ( \mathbb C^n)$, also singles out a natural candidate for moduli space of $ \text{Aut}_{\text{hol}} ( \mathbb C^n)$, for all $n>1$.