# Smooth contractible threefolds with hyperbolic $\mathbb{G}_{m}$-actions via ps-divisors

## TL;DR

This paper provides an alternative proof of a theorem characterizing smooth contractible affine threefolds with hyperbolic -actions, using polyhedral divisors to generalize previous approaches.

## Contribution

It introduces a new proof method employing polyhedral divisors for classifying certain contractible threefolds with -actions.

## Key findings

- Characterization of smooth contractible affine threefolds with hyperbolic -actions
- Application of polyhedral divisors to this classification
- Alternative proof of Koras and Russell's theorem

## Abstract

The aim of this note is to give an alternative proof of a theorem of Koras and Russell, that is, a characterization of smooth contractible affine varieties endowed with a hyperbolic action of the group $\mathbb{G}_{m}\simeq\mathbb{C}^{\text{*}}$, using the language of polyhedral divisors developed by Altmann and Hausen as generalization of $\mathbb{Q}$-divisors.