Bia{\l}ynicki-Birula decomposition for reductive groups

TL;DR

This paper extends the Bia{ }ynicki-Birula decomposition from $G_m$ actions to actions of linearly reductive groups on schemes and algebraic spaces, introducing a functorial and more flexible framework.

Contribution

It generalizes the classical decomposition to reductive group actions, providing a functorial approach and a relative version, enriching the theory beyond the $G_m$ case.

Findings

Generalization to linearly reductive groups on schemes and spaces

Introduction of a functorial BB decomposition parameterized by monoids

Enrichment of the theory through flexible choices of compactification monoids

Abstract

We generalize the Bia{\l}ynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Bia{\l}ynicki-Birula decomposition functorially: for a fixed ${\bf G}$-scheme $X$ and a monoid $\overline{\bf G}$ which partially compactifies ${\bf G}$, the BB decomposition parameterizes ${\bf G}$-schemes over $X$ for which the ${\bf G}$-action extends to the $\overline{\bf G}$-action. The freedom of choice of $\overline{\bf G}$ makes the theory richer than the $G_m$-case.