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

TL;DR

This paper proves that any $A^1$-contractible smooth complex surface is isomorphic to $C^2$, and explores how $A^1$-connected components reveal the structure of $A^1$'s within varieties.

Contribution

It establishes that $A^1$-contractibility implies isomorphism to $C^2$ for smooth complex surfaces and characterizes $A^1$-connected components.

Findings

Any $A^1$-contractible smooth complex surface is isomorphic to $C^2

The $A^1$-connected component contains information about $A^1$'s in the variety

$A^1$-connected components help understand the structure of $A^1$-related properties

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 $X$.