Analytic extensions of algebraic isomorphisms

TL;DR

The paper proves that algebraic isomorphisms between certain affine subvarieties of complex space can be extended to holomorphic automorphisms of the ambient space, under specific dimension conditions.

Contribution

It establishes conditions under which algebraic isomorphisms extend to holomorphic automorphisms, including for curves in higher dimensions.

Findings

Extension of isomorphisms to automorphisms under dimension constraints

Existence of extensions for curve isomorphisms in all dimensions ≥ 3

Extension results hold even when tangent space dimension equals ambient space dimension

Abstract

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 $\dim TX_1=n$.