Small bandwidth ${\mathbb C}^*$-actions and birational geometry

TL;DR

This paper explores the relationship between small bandwidth ${\mathbb C}^*$-actions on smooth projective varieties and their birational geometric properties, introducing new tools to reconstruct these actions from birational transformations.

Contribution

It defines birational geometric counterparts of ${\mathbb C}^*$-actions and shows how low complexity actions relate to known birational transformations like Atiyah flips and Cremona transformations.

Findings

Low bandwidth actions are determined by Atiyah flips.

Bordism rank helps classify ${\mathbb C}^*$-actions.

Reconstruction of actions from birational transformations is possible under certain conditions.

Abstract

In this paper we study smooth projective varieties and polarized pairs with an action of a one dimensional complex torus. As a main tool, we define birational geometric counterparts of these actions, that, under certain assumptions, encode the information necessary to reconstruct them. In particular, we consider some cases of actions of low complexity -- measured in terms of two invariants of the action, called bandwidth and bordism rank -- and discuss how they are determined by well known birational transformations, namely Atiyah flips and Cremona transformations.