  Let $\Psi : X_1 \to X_2$ be an isomorphism of closed affine algebraic
subvarities of $\C^n$ such that $n > \max (2\dim X_1, \dim TX_1)$. We prove
that $\Psi$ can be extended to a holomorphic automorphism of $\C^n$.
Furthermore, when $\Psi$ is an isomorphism of curves such an extension exists
for every $n\geq 3$ even when $\dim TX_1=n$.
