${\mathbb{A}}^1$-connected components and characterisation of ${\mathbb{A}}^2$

Abstract

In this article we prove that any ${\mathbb{A}}^1$-contractible smooth complex surface is isomorphic as a variety to ${\mathbb{C}}^2$. We show that the ${\mathbb{A}}^1$-connected component of a variety $X$ contains the information about ${\mathbb{A}}^1$'s in $X$.

arXiv commentProof of almost all the main theorems remain same. Corrected several typos and citations. Changed the title. Theorem 1.1 is true now in characteristic 0