Analytic extensions of algebraic isomorphisms

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Let Ψ: X₁ → X₂ be an isomorphism of closed affine algebraic subvarities of ⁿ such that n > (2 X₁, TX₁). We prove that Ψ can be extended to a holomorphic automorphism of ⁿ. Furthermore, when Ψ is an isomorphism of curves such an extension exists for every n≥ 3 even when TX₁=n.