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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 isa large object that has not been described explicitly. Despite the intricaciesinvolved when $n>1$, surprisingly every $F\in \text{Aut}_{\text{hol}}(\mathbbC^n)$ is uniquely determined inside the group by only two data, ofinfinitesimal and global nature: the $1$-jet of $F$ at $0$, and the complexHessian of a certain plurisubharmonic function associated to $F$. If $n=1$ thisglobal 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 candidatefor moduli space of $ \text{Aut}_{\text{hol}} ( \mathbb C^n)$, for all $n>1$.

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