  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 $n>1$.
