Analytic extensions of algebraic isomorphisms

Let $Ψ: X_1 \to X_2$ 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 $Ψ$ can be extended to a holomorphic automorphism of $\C^n$. Furthermore, when $Ψ$ is an isomorphism of curves such an extension exists for every $n\geq 3$ even when $\dim TX_1=n$.

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