A note on the group extension problem to semi-universal deformation

TL;DR

This paper investigates the limitations of extending automorphism group actions to semi-universal deformations, revealing that such extensions are generally only valid up to first order and highlighting the local nature of these actions for reductive groups.

Contribution

It clarifies the extent to which automorphism groups can act on semi-universal deformations, providing detailed explanations and establishing the optimal locality of these actions.

Findings

Automorphism group actions extend only up to first order in semi-universal deformations.

The locality of extended actions for reductive groups is optimal in general.

Provides detailed clarification of a previously noted remark in deformation theory.

Abstract

The aim of this note is twofold. Firstly, we explain in detail Remark 4.1 in \cite{doan-a} by showing that the action of the automorphism group of the second Hirzebruch surface $\mathbb{F}_2$ on itself extends to its formal semi-universal deformation only up to the first order. Secondly, we show that for reductive group actions, the locality of the extended actions on the Kuranishi space constructed in \cite{doan-equivariant} is the best one could expect in general.