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

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 ${\text{Aut}}_{\text{hol}}(\mathbb{C})=\text{Aff}(\mathbb{C})\cong \mathbb{C}\times {\mathbb{C}}^{\ast }$. 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 $n>1$.

8 pages