On Open Embeddings of Affine Spaces in Affine Varieties and the Jacobian Conjecture

Abstract

Let $k$ be a field of \uline{characteristic $0$}. Our final goal is the following faded problem~: {\sf The Jacobian Conjecture $(J{C}_2)$~:} {\it If $f_1,f_2$ are elements in a polynomial ring $k[X_1,X_2]$ over $k$ such that $\det (\partial f_i/\partial X_j)$ is a nonzero constant, then $k[f_1,f_2]=k[X_1,X_2]$. } For this purpose, we consider the following Result~: \noindent {\sf Open Embeddings of Affine Spaces in Affine Varieties.} {\it Let $X$ be a normal $k$-affine variety of dimension $2$ and let $U$ be an open $k$-subvariety of $X$ which is isomorphic to ${\mathbb{A}}_k^2$. Then $X=U$.} This is effective for a more general result than our original objective $(J{C}_2)$. \noindent {\sf The Generalized Jacobian Conjecture $(GJ{C}_2)$.} {\it Let $\phi :X\to Y$ be an unramified morphism of normal $k$-affine varieties of dimension $2$. If both $X$ and $Y$ are simply connected, then $\phi$ is an isomorphism.} These results derived by $\mathbb{C}$-topological-method hold for $\mathbb{C}$ instead of $k$ by ``Lefschetz-principle''.

19 pages. I have revised the previous draft because AI (anthoropic) constructed the counterexample to the Jacobian conjecture $(J{C}_n)$ for $n\ge 3$ on July 26, 2026. Any comments are welcome