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

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 $n>1$.