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 $(JC_n)$~:} {\it If $f_1, \cdots, f_n$ are elements in a polynomial ring $k[X_1, \cdots, X_n]$ over $k$ such that $ \det(\partial f_i/ \partial X_j) $ is a nonzero constant, then $k[f_1, \cdots, f_n] = k[X_1, \cdots, X_n]$. } For this purpose, we consider the following Result~: \noindent {\sf Open Embeddings of Affine Spaces in Affine Varieties.} {\it Let $X$ be an irreducible $k$-affine variety with $\dim(X) = n$ and let $U$ be an open $\mathbb{C}$-subvariety of $X$ such that $U$ is isomorphic to $\mathbb{A}^n_k$. Then $X = U$.} This is effective for a more general result than our original objective $(JC_n)$. \noindent {\sf The Generalized Jacobian Conjecture $(GJC)$.} {\it Let $\varphi : X \rightarrow Y$ be an unramified morphism of normal $k$-affine varieties. If both $X$ and $Y$ are simply connected, then $\varphi$ is an isomorphism.} These results derived by $\mathbb{C}$-topological-method hold for $\mathbb{C}$ instead of $k$ by ``Lefschetz-principle''.